Daily Papers

D-peptide binders targeting L-proteins have promising therapeutic potential. Despite rapid advances in machine learning-based target-conditioned peptide design, generating D-peptide binders remains largely unexplored. In this work, we show that by injecting axial features to E(3)-equivariant (polar) vector features, it is feasible to achieve cross-chirality generalization from homo-chiral (L--L) training data to hetero-chiral (D--L) design tasks. By implementing this method within a latent diffusion model, we achieved D-peptide binder design that not only outperforms existing tools in in silico benchmarks, but also demonstrates efficacy in wet-lab validation. To our knowledge, our approach represents the first wet-lab validated generative AI for the de novo design of D-peptide binders, offering new perspectives on handling chirality in protein design. Codes are available at https://github.com/YZY010418/PepMirror

Obtaining the desired effect of drugs is highly dependent on their molecular geometries. Thus, the current prevailing paradigm focuses on 3D point-cloud atom representations, utilizing graph neural network (GNN) parametrizations, with rotational symmetries baked in via E(3) invariant layers. We prove that such models must necessarily disregard chirality, a geometric property of the molecules that cannot be superimposed on their mirror image by rotation and translation. Chirality plays a key role in determining drug safety and potency. To address this glaring issue, we introduce a novel field-based representation, proposing reference rotations that replace rotational symmetry constraints. The proposed model captures all molecular geometries including chirality, while still achieving highly competitive performance with E(3)-based methods across standard benchmarking metrics.

Recent advances in deep learning for physics have focused on discovering shared representations of target systems by incorporating physics priors or inductive biases into neural networks. While effective, these methods are limited to the system domain, where the type of system remains consistent and thus cannot ensure the adaptation to new, or unseen physical systems governed by different laws. For instance, a neural network trained on a mass-spring system cannot guarantee accurate predictions for the behavior of a two-body system or any other system with different physical laws. In this work, we take a significant leap forward by targeting cross domain generalization within the field of Hamiltonian dynamics. We model our system with a graph neural network and employ a meta learning algorithm to enable the model to gain experience over a distribution of tasks and make it adapt to new physics. Our approach aims to learn a unified Hamiltonian representation that is generalizable across multiple system domains, thereby overcoming the limitations of system-specific models. Our results demonstrate that the meta-trained model not only adapts effectively to new systems but also captures a generalized Hamiltonian representation that is consistent across different physical domains. Overall, through the use of meta learning, we offer a framework that achieves cross domain generalization, providing a step towards a unified model for understanding a wide array of dynamical systems via deep learning.

Group equivariant neural networks are used as building blocks of group invariant neural networks, which have been shown to improve generalisation performance and data efficiency through principled parameter sharing. Such works have mostly focused on group equivariant convolutions, building on the result that group equivariant linear maps are necessarily convolutions. In this work, we extend the scope of the literature to self-attention, that is emerging as a prominent building block of deep learning models. We propose the LieTransformer, an architecture composed of LieSelfAttention layers that are equivariant to arbitrary Lie groups and their discrete subgroups. We demonstrate the generality of our approach by showing experimental results that are competitive to baseline methods on a wide range of tasks: shape counting on point clouds, molecular property regression and modelling particle trajectories under Hamiltonian dynamics.

Neural networks adapt very well to distributed and continuous representations, but struggle to generalize from small amounts of data. Symbolic systems commonly achieve data efficient generalization by exploiting modularity to benefit from local and discrete features of a representation. These features allow symbolic programs to be improved one module at a time and to experience combinatorial growth in the values they can successfully process. However, it is difficult to design a component that can be used to form symbolic abstractions and which is adequately overparametrized to learn arbitrary high-dimensional transformations. I present Graph-based Symbolically Synthesized Neural Networks (G-SSNNs), a class of neural modules that operate on representations modified with synthesized symbolic programs to include a fixed set of local and discrete features. I demonstrate that the choice of injected features within a G-SSNN module modulates the data efficiency and generalization of baseline neural models, creating predictable patterns of both heightened and curtailed generalization. By training G-SSNNs, we also derive information about desirable semantics of symbolic programs without manual engineering. This information is compact and amenable to abstraction, but can also be flexibly recontextualized for other high-dimensional settings. In future work, I will investigate data efficient generalization and the transferability of learned symbolic representations in more complex G-SSNN designs based on more complex classes of symbolic programs. Experimental code and data are available at https://github.com/shlomenu/symbolically_synthesized_networks .

Developing equivariant neural networks for the E(3) group plays an important role in modeling 3D data across real-world applications. Enforcing this equivariance primarily involves the tensor products of irreducible representations (irreps). However, the computational complexity of such operations increases significantly as higher-order tensors are used. In this work, we propose a systematic approach to substantially accelerate the computation of the tensor products of irreps. We mathematically connect the commonly used Clebsch-Gordan coefficients to the Gaunt coefficients, which are integrals of products of three spherical harmonics. Through Gaunt coefficients, the tensor product of irreps becomes equivalent to the multiplication between spherical functions represented by spherical harmonics. This perspective further allows us to change the basis for the equivariant operations from spherical harmonics to a 2D Fourier basis. Consequently, the multiplication between spherical functions represented by a 2D Fourier basis can be efficiently computed via the convolution theorem and Fast Fourier Transforms. This transformation reduces the complexity of full tensor products of irreps from O(L^6) to O(L^3), where L is the max degree of irreps. Leveraging this approach, we introduce the Gaunt Tensor Product, which serves as a new method to construct efficient equivariant operations across different model architectures. Our experiments on the Open Catalyst Project and 3BPA datasets demonstrate both the increased efficiency and improved performance of our approach.

Accurate and scalable machine-learned inter-atomic potentials (MLIPs) are essential for molecular simulations ranging from drug discovery to new material design. Current state-of-the-art models enforce roto-translational symmetries through equivariant neural network architectures, a hard-wired inductive bias that can often lead to reduced flexibility, computational efficiency, and scalability. In this work, we introduce TransIP: Transformer-based Inter-Atomic Potentials, a novel training paradigm for interatomic potentials achieving symmetry compliance without explicit architectural constraints. Our approach guides a generic non-equivariant Transformer-based model to learn SO(3)-equivariance by optimizing its representations in the embedding space. Trained on the recent Open Molecules (OMol25) collection, a large and diverse molecular dataset built specifically for MLIPs and covering different types of molecules (including small organics, biomolecular fragments, and electrolyte-like species), TransIP attains comparable performance in machine-learning force fields versus state-of-the-art equivariant baselines. Further, compared to a data augmentation baseline, TransIP achieves 40% to 60% improvement in performance across varying OMol25 dataset sizes. More broadly, our work shows that learned equivariance can be a powerful and efficient alternative to equivariant or augmentation-based MLIP models.

Conformal symmetries, i.e.\ coordinate transformations that preserve angles, play a key role in many fields, including physics, mathematics, computer vision and (geometric) machine learning. Here we build a neural network that is equivariant under general conformal transformations. To achieve this, we lift data from flat Euclidean space to Anti de Sitter (AdS) space. This allows us to exploit a known correspondence between conformal transformations of flat space and isometric transformations on the AdS space. We then build upon the fact that such isometric transformations have been extensively studied on general geometries in the geometric deep learning literature. We employ message-passing layers conditioned on the proper distance, yielding a computationally efficient framework. We validate our model on tasks from computer vision and statistical physics, demonstrating strong performance, improved generalization capacities, and the ability to extract conformal data such as scaling dimensions from the trained network.

Mixup is a highly successful technique to improve generalization of neural networks by augmenting the training data with combinations of random pairs. Selective mixup is a family of methods that apply mixup to specific pairs, e.g. only combining examples across classes or domains. These methods have claimed remarkable improvements on benchmarks with distribution shifts, but their mechanisms and limitations remain poorly understood. We examine an overlooked aspect of selective mixup that explains its success in a completely new light. We find that the non-random selection of pairs affects the training distribution and improve generalization by means completely unrelated to the mixing. For example in binary classification, mixup across classes implicitly resamples the data for a uniform class distribution - a classical solution to label shift. We show empirically that this implicit resampling explains much of the improvements in prior work. Theoretically, these results rely on a regression toward the mean, an accidental property that we identify in several datasets. We have found a new equivalence between two successful methods: selective mixup and resampling. We identify limits of the former, confirm the effectiveness of the latter, and find better combinations of their respective benefits.

We present a general framework for symmetrizing an arbitrary neural-network architecture and making it equivariant with respect to a given group. We build upon the proposals of Kim et al. (2023); Kaba et al. (2023) for symmetrization, and improve them by replacing their conversion of neural features into group representations, with an optimization whose loss intuitively measures the distance between group orbits. This change makes our approach applicable to a broader range of matrix groups, such as the Lorentz group O(1, 3), than these two proposals. We experimentally show our method's competitiveness on the SO(2) image classification task, and also its increased generality on the task with O(1, 3). Our implementation will be made accessible at https://github.com/tiendatnguyen-vision/Orbit-symmetrize.

In many neural networks, different values of the parameters may result in the same loss value. Parameter space symmetries are loss-invariant transformations that change the model parameters. Teleportation applies such transformations to accelerate optimization. However, the exact mechanism behind this algorithm's success is not well understood. In this paper, we show that teleportation not only speeds up optimization in the short-term, but gives overall faster time to convergence. Additionally, teleporting to minima with different curvatures improves generalization, which suggests a connection between the curvature of the minimum and generalization ability. Finally, we show that integrating teleportation into a wide range of optimization algorithms and optimization-based meta-learning improves convergence. Our results showcase the versatility of teleportation and demonstrate the potential of incorporating symmetry in optimization.

In chemistry and biochemistry, chirality represents the structural asymmetry characterized by non-superimposable mirror images for a material like DNA. In physics, however, chirality commonly refers to the spin-momentum locking of a particle or quasiparticle in the momentum space. While seemingly disconnected, structural chirality in molecules and crystals can drive electronic chirality through orbital-momentum locking, i.e. chirality can be transferred from the atomic geometry to electronic orbitals. Electronic chirality provides an insightful understanding of the chirality-induced spin selectivity (CISS), in which electrons exhibit salient spin polarization after going through a chiral material, and electric magnetochiral anisotropy (EMCA), which is characterized by the diode-like transport. It further gives rise to new phenomena, such as anomalous circularly polarized light emission (ACPLE), in which the light handedness relies on the emission direction. These chirality-driven effects will generate broad impacts in fundamental science and technology applications in spintronics, optoelectronics, and biochemistry.

Despite the success of equivariant neural networks in scientific applications, they require knowing the symmetry group a priori. However, it may be difficult to know which symmetry to use as an inductive bias in practice. Enforcing the wrong symmetry could even hurt the performance. In this paper, we propose a framework, LieGAN, to automatically discover equivariances from a dataset using a paradigm akin to generative adversarial training. Specifically, a generator learns a group of transformations applied to the data, which preserve the original distribution and fool the discriminator. LieGAN represents symmetry as interpretable Lie algebra basis and can discover various symmetries such as the rotation group SO(n), restricted Lorentz group SO(1,3)^+ in trajectory prediction and top-quark tagging tasks. The learned symmetry can also be readily used in several existing equivariant neural networks to improve accuracy and generalization in prediction.

Computational materials discovery is limited by the high cost of first-principles calculations. Machine learning (ML) potentials that predict energies from crystal structures are promising, but existing methods face computational bottlenecks. Steerable graph neural networks (GNNs) encode geometry with spherical harmonics, respecting atomic symmetries -- permutation, rotation, and translation -- for physically realistic predictions. Yet maintaining equivariance is difficult: activation functions must be modified, and each layer must handle multiple data types for different harmonic orders. We present Facet, a GNN architecture for efficient ML potentials, developed through systematic analysis of steerable GNNs. Our innovations include replacing expensive multi-layer perceptrons (MLPs) for interatomic distances with splines, which match performance while cutting computational and memory demands. We also introduce a general-purpose equivariant layer that mixes node information via spherical grid projection followed by standard MLPs -- faster than tensor products and more expressive than linear or gate layers. On the MPTrj dataset, Facet matches leading models with far fewer parameters and under 10% of their training compute. On a crystal relaxation task, it runs twice as fast as MACE models. We further show SevenNet-0's parameters can be reduced by over 25% with no accuracy loss. These techniques enable more than 10x faster training of large-scale foundation models for ML potentials, potentially reshaping computational materials discovery.

We study the problem of learning equivariant neural networks via gradient descent. The incorporation of known symmetries ("equivariance") into neural nets has empirically improved the performance of learning pipelines, in domains ranging from biology to computer vision. However, a rich yet separate line of learning theoretic research has demonstrated that actually learning shallow, fully-connected (i.e. non-symmetric) networks has exponential complexity in the correlational statistical query (CSQ) model, a framework encompassing gradient descent. In this work, we ask: are known problem symmetries sufficient to alleviate the fundamental hardness of learning neural nets with gradient descent? We answer this question in the negative. In particular, we give lower bounds for shallow graph neural networks, convolutional networks, invariant polynomials, and frame-averaged networks for permutation subgroups, which all scale either superpolynomially or exponentially in the relevant input dimension. Therefore, in spite of the significant inductive bias imparted via symmetry, actually learning the complete classes of functions represented by equivariant neural networks via gradient descent remains hard.

Machine-learning generative methods for material design are constructed by representing a given chemical structure, either a solid or a molecule, over appropriate atomic features, generally called structural descriptors. These must be fully descriptive of the system, must facilitate the training process and must be invertible, so that one can extract the atomic configurations corresponding to the output of the model. In general, this last requirement is not automatically satisfied by the most efficient structural descriptors, namely the representation is not directly invertible. Such drawback severely limits our freedom of choice in selecting the most appropriate descriptors for the problem, and thus our flexibility to construct generative models. In this work, we present a general optimization method capable of inverting any local many-body descriptor of the chemical environment, back to a cartesian representation. The algorithm is then implemented together with the bispectrum representation of the local structure and demonstrated for a number of molecules. The scheme presented here, thus, represents a general approach to the inversion of structural descriptors, enabling the construction of efficient structural generative models.

Molecular structure recognition is the task of translating a molecular image into its graph structure. Significant variation in drawing styles and conventions exhibited in chemical literature poses a significant challenge for automating this task. In this paper, we propose MolScribe, a novel image-to-graph generation model that explicitly predicts atoms and bonds, along with their geometric layouts, to construct the molecular structure. Our model flexibly incorporates symbolic chemistry constraints to recognize chirality and expand abbreviated structures. We further develop data augmentation strategies to enhance the model robustness against domain shifts. In experiments on both synthetic and realistic molecular images, MolScribe significantly outperforms previous models, achieving 76-93% accuracy on public benchmarks. Chemists can also easily verify MolScribe's prediction, informed by its confidence estimation and atom-level alignment with the input image. MolScribe is publicly available through Python and web interfaces: https://github.com/thomas0809/MolScribe.

Large-scale datasets have enabled highly accurate machine learning interatomic potentials (MLIPs) for general-purpose heterogeneous catalysis modeling. There are, however, some limitations in what can be treated with these potentials because of gaps in the underlying training data. To extend these capabilities, we introduce AQCat25, a complementary dataset of 13.5 million density functional theory (DFT) single point calculations designed to improve the treatment of systems where spin polarization and/or higher fidelity are critical. We also investigate methodologies for integrating new datasets, such as AQCat25, with the broader Open Catalyst 2020 (OC20) dataset to create spin-aware models without sacrificing generalizability. We find that directly tuning a general model on AQCat25 leads to catastrophic forgetting of the original dataset's knowledge. Conversely, joint training strategies prove effective for improving accuracy on the new data without sacrificing general performance. This joint approach introduces a challenge, as the model must learn from a dataset containing both mixed-fidelity calculations and mixed-physics (spin-polarized vs. unpolarized). We show that explicitly conditioning the model on this system-specific metadata, for example by using Feature-wise Linear Modulation (FiLM), successfully addresses this challenge and further enhances model accuracy. Ultimately, our work establishes an effective protocol for bridging DFT fidelity domains to advance the predictive power of foundational models in catalysis.

Many theoretical results in deep learning can be traced to symmetry or equivariance of neural networks under parameter transformations. However, existing analyses are typically problem-specific and focus on first-order consequences such as conservation laws, while the implications for second-order structure remain less understood. We develop a general equivariance toolbox that yields coupled first- and second-order constraints on learning dynamics. The framework extends classical Noether-type analyses in three directions: from gradient constraints to Hessian constraints, from symmetry to general equivariance, and from continuous to discrete transformations. At the first order, our framework unifies conservation laws and implicit-bias relations as special cases of a single identity. At the second order, it provides structural predictions about curvature: which directions are flat or sharp, how the gradient aligns with Hessian eigenspaces, and how the loss landscape geometry reflects the underlying transformation structure. We illustrate the framework through several applications, recovering known results while also deriving new characterizations that connect transformation structure to modern empirical observations about optimization geometry.

We define the bicategory of Graph Convolutional Neural Networks GCNN_n for an arbitrary graph with n nodes. We show it can be factored through the already existing categorical constructions for deep learning called Para and Lens with the base category set to the CoKleisli category of the product comonad. We prove that there exists an injective-on-objects, faithful 2-functor GCNN_n to Para(CoKl(R^{n times n} times -)). We show that this construction allows us to treat the adjacency matrix of a GCNN as a global parameter instead of a a local, layer-wise one. This gives us a high-level categorical characterisation of a particular kind of inductive bias GCNNs possess. Lastly, we hypothesize about possible generalisations of GCNNs to general message-passing graph neural networks, connections to equivariant learning, and the (lack of) functoriality of activation functions.

We propose Equivariance by Contrast (EbC) to learn equivariant embeddings from observation pairs (y, g cdot y), where g is drawn from a finite group acting on the data. Our method jointly learns a latent space and a group representation in which group actions correspond to invertible linear maps -- without relying on group-specific inductive biases. We validate our approach on the infinite dSprites dataset with structured transformations defined by the finite group G:= (R_m times Z_n times Z_n), combining discrete rotations and periodic translations. The resulting embeddings exhibit high-fidelity equivariance, with group operations faithfully reproduced in latent space. On synthetic data, we further validate the approach on the non-abelian orthogonal group O(n) and the general linear group GL(n). We also provide a theoretical proof for identifiability. While broad evaluation across diverse group types on real-world data remains future work, our results constitute the first successful demonstration of general-purpose encoder-only equivariant learning from group action observations alone, including non-trivial non-abelian groups and a product group motivated by modeling affine equivariances in computer vision.

We propose, implement, and compare with competitors a new architecture of equivariant neural networks based on geometric (Clifford) algebras: Generalized Lipschitz Group Equivariant Neural Networks (GLGENN). These networks are equivariant to all pseudo-orthogonal transformations, including rotations and reflections, of a vector space with any non-degenerate or degenerate symmetric bilinear form. We propose a weight-sharing parametrization technique that takes into account the fundamental structures and operations of geometric algebras. Due to this technique, GLGENN architecture is parameter-light and has less tendency to overfitting than baseline equivariant models. GLGENN outperforms or matches competitors on several benchmarking equivariant tasks, including estimation of an equivariant function and a convex hull experiment, while using significantly fewer optimizable parameters.

Despite recent advancements, most computational methods for molecule optimization are constrained to single- or double-property optimization tasks and suffer from poor scalability and generalizability to novel optimization tasks. Meanwhile, Large Language Models (LLMs) demonstrate remarkable out-of-domain generalizability to novel tasks. To demonstrate LLMs' potential for molecule optimization, we introduce MoMUInstruct, the first high-quality instruction-tuning dataset specifically focused on complex multi-property molecule optimization tasks. Leveraging MoMUInstruct, we develop GeLLM^3Os, a series of instruction-tuned LLMs for molecule optimization. Extensive evaluations across 5 in-domain and 5 out-of-domain tasks demonstrate that GeLLM^3Os consistently outperform state-of-the-art baselines. GeLLM^3Os also exhibit outstanding zero-shot generalization to unseen tasks, significantly outperforming powerful closed-source LLMs. Such strong generalizability demonstrates the tremendous potential of GeLLM^3Os as foundational models for molecule optimization, thereby tackling novel optimization tasks without resource-intensive retraining. MoMUInstruct, models, and code are accessible through https://github.com/ninglab/GeLLMO.

A striking geometric disparity has long persisted in the practice of deep learning. While modern neural network architectures naturally exhibit rich symmetry and equivariance properties, popular optimizers such as Adam and its variants operate inherently coordinate-wise, rendering them unable to respect the equivariance structures of the parameter space. We address this disparity by introducing a symmetry-compatible principle for optimizer design: the gradient update rule should be equivariant under the symmetry group acting on the corresponding weight block. Following this principle, we first provide a unified perspective on bi-orthogonally equivariant updates for general matrix layers, as employed by stochastic spectral descent, Muon, Scion, and polar gradient methods. More importantly, by moving from orthogonal groups to permutation and shared-shift symmetries, we derive symmetry-compatible optimizers for parameter blocks whose symmetries differ from those of general matrix layers: embedding and LM head matrices, SwiGLU MLP projections, and MoE router matrices. These constructions include one-sided spectral, row-norm, hybrid row-norm/spectral, row-aware, column-aware, centered row-norm, and left-spectral updates. They yield an end-to-end layerwise optimizer stack in which each major matrix-valued parameter class is assigned an update whose equivariance matches its symmetry group. We corroborate this principle through pre-training experiments on dense and sparse MoE language models, including Qwen3-0.6B-style, Gemma 3 1B-style, OLMoE-1B-7B-style, and downsized gpt-oss architectures. Across these experiments, symmetry-compatible updates consistently improve final validation loss, and in several cases training stability, over corresponding AdamW updates.

When manipulating three-dimensional data, it is possible to ensure that rotational and translational symmetries are respected by applying so-called SE(3)-equivariant models. Protein structure prediction is a prominent example of a task which displays these symmetries. Recent work in this area has successfully made use of an SE(3)-equivariant model, applying an iterative SE(3)-equivariant attention mechanism. Motivated by this application, we implement an iterative version of the SE(3)-Transformer, an SE(3)-equivariant attention-based model for graph data. We address the additional complications which arise when applying the SE(3)-Transformer in an iterative fashion, compare the iterative and single-pass versions on a toy problem, and consider why an iterative model may be beneficial in some problem settings. We make the code for our implementation available to the community.

We provide a full characterisation of all of the possible alternating group (A_n) equivariant neural networks whose layers are some tensor power of R^{n}. In particular, we find a basis of matrices for the learnable, linear, A_n-equivariant layer functions between such tensor power spaces in the standard basis of R^{n}. We also describe how our approach generalises to the construction of neural networks that are equivariant to local symmetries.

SO(3)-equivariant networks are the dominant models for machine learning interatomic potentials (MLIPs). The key operation of such networks is the Clebsch-Gordan (CG) tensor product, which is computationally expensive. To accelerate the computation, we develop tensor decomposition networks (TDNs) as a class of approximately equivariant networks in which CG tensor products are replaced by low-rank tensor decompositions, such as the CANDECOMP/PARAFAC (CP) decomposition. With the CP decomposition, we prove (i) a uniform bound on the induced error of SO(3)-equivariance, and (ii) the universality of approximating any equivariant bilinear map. To further reduce the number of parameters, we propose path-weight sharing that ties all multiplicity-space weights across the O(L^3) CG paths into a single shared parameter set without compromising equivariance, where L is the maximum angular degree. The resulting layer acts as a plug-and-play replacement for tensor products in existing networks, and the computational complexity of tensor products is reduced from O(L^6) to O(L^4). We evaluate TDNs on PubChemQCR, a newly curated molecular relaxation dataset containing 105 million DFT-calculated snapshots. We also use existing datasets, including OC20, and OC22. Results show that TDNs achieve competitive performance with dramatic speedup in computations. Our code is publicly available as part of the AIRS library (https://github.com/divelab/AIRS/tree/main/OpenMol/TDN{https://github.com/divelab/AIRS/}).

Graph neural networks (GNNs) are the most widely adopted model in graph-structured data oriented learning and representation. Despite their extraordinary success in real-world applications, understanding their working mechanism by theory is still on primary stage. In this paper, we move towards this goal from the perspective of generalization. To be specific, we first establish high probability bounds of generalization gap and gradients in transductive learning with consideration of stochastic optimization. After that, we provide high probability bounds of generalization gap for popular GNNs. The theoretical results reveal the architecture specific factors affecting the generalization gap. Experimental results on benchmark datasets show the consistency between theoretical results and empirical evidence. Our results provide new insights in understanding the generalization of GNNs.

Many geometric learning problems require invariants on heterogeneous product spaces, i.e., products of distinct spaces carrying different group actions, where standard techniques do not directly apply. We show that, when a group G acts transitively on a space M, any G-invariant function on a product space X times M can be reduced to an invariant of the isotropy subgroup H of M acting on X alone. Our approach establishes an explicit orbit equivalence (X times M)/G cong X/H, yielding a principled reduction that preserves expressivity. We apply this characterization to Equivariant Neural Fields, extending them to arbitrary group actions and homogeneous conditioning spaces, and thereby removing the major structural constraints imposed by existing methods.

Datasets often have their intrinsic symmetries, and particular deep-learning models called equivariant or invariant models have been developed to exploit these symmetries. However, if some or all of these symmetries are only approximate, which frequently happens in practice, these models may be suboptimal due to the architectural restrictions imposed on them. We tackle this issue of approximate symmetries in a setup where symmetries are mixed, i.e., they are symmetries of not single but multiple different types and the degree of approximation varies across these types. Instead of proposing a new architectural restriction as in most of the previous approaches, we present a regularizer-based method for building a model for a dataset with mixed approximate symmetries. The key component of our method is what we call equivariance regularizer for a given type of symmetries, which measures how much a model is equivariant with respect to the symmetries of the type. Our method is trained with these regularizers, one per each symmetry type, and the strength of the regularizers is automatically tuned during training, leading to the discovery of the approximation levels of some candidate symmetry types without explicit supervision. Using synthetic function approximation and motion forecasting tasks, we demonstrate that our method achieves better accuracy than prior approaches while discovering the approximate symmetry levels correctly.

Graph Neural Networks (GNNs) show promising results for graph tasks. However, existing GNNs' generalization ability will degrade when there exist distribution shifts between testing and training graph data. The cardinal impetus underlying the severe degeneration is that the GNNs are architected predicated upon the I.I.D assumptions. In such a setting, GNNs are inclined to leverage imperceptible statistical correlations subsisting in the training set to predict, albeit it is a spurious correlation. In this paper, we study the problem of the generalization ability of GNNs in Out-Of-Distribution (OOD) settings. To solve this problem, we propose the Learning to Reweight for Generalizable Graph Neural Network (L2R-GNN) to enhance the generalization ability for achieving satisfactory performance on unseen testing graphs that have different distributions with training graphs. We propose a novel nonlinear graph decorrelation method, which can substantially improve the out-of-distribution generalization ability and compares favorably to previous methods in restraining the over-reduced sample size. The variables of the graph representation are clustered based on the stability of the correlation, and the graph decorrelation method learns weights to remove correlations between the variables of different clusters rather than any two variables. Besides, we interpose an efficacious stochastic algorithm upon bi-level optimization for the L2R-GNN framework, which facilitates simultaneously learning the optimal weights and GNN parameters, and avoids the overfitting problem. Experimental results show that L2R-GNN greatly outperforms baselines on various graph prediction benchmarks under distribution shifts.

Machine learning methods can be a valuable aid in the scientific process, but they need to face challenging settings where data come from inhomogeneous experimental conditions. Recent meta-learning methods have made significant progress in multi-task learning, but they rely on black-box neural networks, resulting in high computational costs and limited interpretability. Leveraging the structure of the learning problem, we argue that multi-environment generalization can be achieved using a simpler learning model, with an affine structure with respect to the learning task. Crucially, we prove that this architecture can identify the physical parameters of the system, enabling interpreable learning. We demonstrate the competitive generalization performance and the low computational cost of our method by comparing it to state-of-the-art algorithms on physical systems, ranging from toy models to complex, non-analytical systems. The interpretability of our method is illustrated with original applications to physical-parameter-induced adaptation and to adaptive control.

The rapid development of universal machine learning interatomic potentials (uMLIPs) has demonstrated the possibility for generalizable learning of the universal potential energy surface. In principle, the accuracy of uMLIPs can be further improved by bridging the model from lower-fidelity datasets to high-fidelity ones. In this work, we analyze the challenge of this transfer learning problem within the CHGNet framework. We show that significant energy scale shifts and poor correlations between GGA and r^2SCAN pose challenges to cross-functional data transferability in uMLIPs. By benchmarking different transfer learning approaches on the MP-r^2SCAN dataset of 0.24 million structures, we demonstrate the importance of elemental energy referencing in the transfer learning of uMLIPs. By comparing the scaling law with and without the pre-training on a low-fidelity dataset, we show that significant data efficiency can still be achieved through transfer learning, even with a target dataset of sub-million structures. We highlight the importance of proper transfer learning and multi-fidelity learning in creating next-generation uMLIPs on high-fidelity data.

Equivariant neural networks have shown improved performance, expressiveness and sample complexity on symmetrical domains. But for some specific symmetries, representations, and choice of coordinates, the most common point-wise activations, such as ReLU, are not equivariant, hence they cannot be employed in the design of equivariant neural networks. The theorem we present in this paper describes all possible combinations of finite-dimensional representations, choice of coordinates and point-wise activations to obtain an exactly equivariant layer, generalizing and strengthening existing characterizations. Notable cases of practical relevance are discussed as corollaries. Indeed, we prove that rotation-equivariant networks can only be invariant, as it happens for any network which is equivariant with respect to connected compact groups. Then, we discuss implications of our findings when applied to important instances of exactly equivariant networks. First, we completely characterize permutation equivariant networks such as Invariant Graph Networks with point-wise nonlinearities and their geometric counterparts, highlighting a plethora of models whose expressive power and performance are still unknown. Second, we show that feature spaces of disentangled steerable convolutional neural networks are trivial representations.

Neural networks efficiently encode learned information within their parameters. Consequently, many tasks can be unified by treating neural networks themselves as input data. When doing so, recent studies demonstrated the importance of accounting for the symmetries and geometry of parameter spaces. However, those works developed architectures tailored to specific networks such as MLPs and CNNs without normalization layers, and generalizing such architectures to other types of networks can be challenging. In this work, we overcome these challenges by building new metanetworks - neural networks that take weights from other neural networks as input. Put simply, we carefully build graphs representing the input neural networks and process the graphs using graph neural networks. Our approach, Graph Metanetworks (GMNs), generalizes to neural architectures where competing methods struggle, such as multi-head attention layers, normalization layers, convolutional layers, ResNet blocks, and group-equivariant linear layers. We prove that GMNs are expressive and equivariant to parameter permutation symmetries that leave the input neural network functions unchanged. We validate the effectiveness of our method on several metanetwork tasks over diverse neural network architectures.

Downsampling layers are crucial building blocks in CNN architectures, which help to increase the receptive field for learning high-level features and reduce the amount of memory/computation in the model. In this work, we study the generalization of the uniform downsampling layer for group equivariant architectures, e.g., G-CNNs. That is, we aim to downsample signals (feature maps) on general finite groups with anti-aliasing. This involves the following: (a) Given a finite group and a downsampling rate, we present an algorithm to form a suitable choice of subgroup. (b) Given a group and a subgroup, we study the notion of bandlimited-ness and propose how to perform anti-aliasing. Notably, our method generalizes the notion of downsampling based on classical sampling theory. When the signal is on a cyclic group, i.e., periodic, our method recovers the standard downsampling of an ideal low-pass filter followed by a subsampling operation. Finally, we conducted experiments on image classification tasks demonstrating that the proposed downsampling operation improves accuracy, better preserves equivariance, and reduces model size when incorporated into G-equivariant networks

The expressive power of Graph Neural Networks (GNNs) has been studied extensively through the Weisfeiler-Leman (WL) graph isomorphism test. However, standard GNNs and the WL framework are inapplicable for geometric graphs embedded in Euclidean space, such as biomolecules, materials, and other physical systems. In this work, we propose a geometric version of the WL test (GWL) for discriminating geometric graphs while respecting the underlying physical symmetries: permutations, rotation, reflection, and translation. We use GWL to characterise the expressive power of geometric GNNs that are invariant or equivariant to physical symmetries in terms of distinguishing geometric graphs. GWL unpacks how key design choices influence geometric GNN expressivity: (1) Invariant layers have limited expressivity as they cannot distinguish one-hop identical geometric graphs; (2) Equivariant layers distinguish a larger class of graphs by propagating geometric information beyond local neighbourhoods; (3) Higher order tensors and scalarisation enable maximally powerful geometric GNNs; and (4) GWL's discrimination-based perspective is equivalent to universal approximation. Synthetic experiments supplementing our results are available at https://github.com/chaitjo/geometric-gnn-dojo

Spherical equivariant graph neural networks (EGNNs) provide a principled framework for learning on three-dimensional molecular and biomolecular systems, where predictions must respect the rotational symmetries inherent in physics. These models extend traditional message-passing GNNs and Transformers by representing node and edge features as spherical tensors that transform under irreducible representations of the rotation group SO(3), ensuring that predictions change in physically meaningful ways under rotations of the input. This guide develops a complete, intuitive foundation for spherical equivariant modeling - from group representations and spherical harmonics, to tensor products, Clebsch-Gordan decomposition, and the construction of SO(3)-equivariant kernels. Building on this foundation, we construct the Tensor Field Network and SE(3)-Transformer architectures and explain how they perform equivariant message-passing and attention on geometric graphs. Through clear mathematical derivations and annotated code excerpts, this guide serves as a self-contained introduction for researchers and learners seeking to understand or implement spherical EGNNs for applications in chemistry, molecular property prediction, protein structure modeling, and generative modeling.

Generative machine learning models can potentially provide direct access to novel and relevant portions of the full chemical space, overcoming the cost of systematic sampling. However, the training of these models generally requires a large amount of data, often precluding the use of expensive high-level ab initio simulations for this task. The generation of coordination compounds of Dy with large magnetic anisotropy represents a topical example, where multireference simulations of large molecules are necessary to perform reliable predictions. Here, we show that a semi-supervised chemically-inspired training-by-proxy of generative variational autoencoders can reduce the cost associated with building a training set from multireference simulations by two orders of magnitude. We illustrate the power of this approach by generating 100s of new organic ligands for Dy(III) pentagonal bipyramidal complexes exhibiting record values of magnetic anisotropy, while starting from datasets as small as 1k multireference calculations. This work thus paves the way to the computational generation of molecules as complex coordination compounds with target electronic and magnetic properties.

Motivated by applications in chemistry and other sciences, we study the expressive power of message-passing neural networks for geometric graphs, whose node features correspond to 3-dimensional positions. Recent work has shown that such models can separate generic pairs of non-isomorphic geometric graphs, though they may fail to separate some rare and complicated instances. However, these results assume a fully connected graph, where each node possesses complete knowledge of all other nodes. In contrast, often, in application, every node only possesses knowledge of a small number of nearest neighbors. This paper shows that generic pairs of non-isomorphic geometric graphs can be separated by message-passing networks with rotation equivariant features as long as the underlying graph is connected. When only invariant intermediate features are allowed, generic separation is guaranteed for generically globally rigid graphs. We introduce a simple architecture, EGENNET, which achieves our theoretical guarantees and compares favorably with alternative architecture on synthetic and chemical benchmarks. Our code is available at https://github.com/yonatansverdlov/E-GenNet.

As SE(3)-equivariant graph neural networks mature as a core tool for 3D atomistic modeling, improving their efficiency, expressivity, and physical consistency has become a central challenge for large-scale applications. In this work, we introduce EquiformerV3, the third generation of the SE(3)-equivariant graph attention Transformer, designed to advance all three dimensions: efficiency, expressivity, and generality. Building on EquiformerV2, we have the following three key advances. First, we optimize the software implementation, achieving 1.75times speedup. Second, we introduce simple and effective modifications to EquiformerV2, including equivariant merged layer normalization, improved feedforward network hyper-parameters, and attention with smooth radius cutoff. Third, we propose SwiGLU-S^2 activations to incorporate many-body interactions for better theoretical expressivity and to preserve strict equivariance while reducing the complexity of sampling S^2 grids. Together, SwiGLU-S^2 activations and smooth-cutoff attention enable accurate modeling of smoothly varying potential energy surfaces (PES), generalizing EquiformerV3 to tasks requiring energy-conserving simulations and higher-order derivatives of PES. With these improvements, EquiformerV3 trained with the auxiliary task of denoising non-equilibrium structures (DeNS) achieves state-of-the-art results on OC20, OMat24, and Matbench Discovery.

Data arrives at our senses as a continuous stream, smoothly transforming from one instant to the next. These smooth transformations can be viewed as continuous symmetries of the environment that we inhabit, defining equivalence relations between stimuli over time. In machine learning, neural network architectures that respect symmetries of their data are called equivariant and have provable benefits in terms of generalization ability and sample efficiency. To date, however, equivariance has been considered only for static transformations and feed-forward networks, limiting its applicability to sequence models, such as recurrent neural networks (RNNs), and corresponding time-parameterized sequence transformations. In this work, we extend equivariant network theory to this regime of `flows' -- one-parameter Lie subgroups capturing natural transformations over time, such as visual motion. We begin by showing that standard RNNs are generally not flow equivariant: their hidden states fail to transform in a geometrically structured manner for moving stimuli. We then show how flow equivariance can be introduced, and demonstrate that these models significantly outperform their non-equivariant counterparts in terms of training speed, length generalization, and velocity generalization, on both next step prediction and sequence classification. We present this work as a first step towards building sequence models that respect the time-parameterized symmetries which govern the world around us.

Modeling the energy and forces of atomic systems is a fundamental problem in computational chemistry with the potential to help address many of the world's most pressing problems, including those related to energy scarcity and climate change. These calculations are traditionally performed using Density Functional Theory, which is computationally very expensive. Machine learning has the potential to dramatically improve the efficiency of these calculations from days or hours to seconds. We propose the Spherical Channel Network (SCN) to model atomic energies and forces. The SCN is a graph neural network where nodes represent atoms and edges their neighboring atoms. The atom embeddings are a set of spherical functions, called spherical channels, represented using spherical harmonics. We demonstrate, that by rotating the embeddings based on the 3D edge orientation, more information may be utilized while maintaining the rotational equivariance of the messages. While equivariance is a desirable property, we find that by relaxing this constraint in both message passing and aggregation, improved accuracy may be achieved. We demonstrate state-of-the-art results on the large-scale Open Catalyst dataset in both energy and force prediction for numerous tasks and metrics.

Effectively leveraging data from multiple quantum-chemical methods is essential for building machine-learned force fields (MLFFs) that are applicable to a wide range of chemical systems. This study systematically investigates two multi-fidelity training strategies, pre-training/fine-tuning and multi-headed training, to elucidate the mechanisms underpinning their success. We identify key factors driving the efficacy of pre-training followed by fine-tuning, but find that internal representations learned during pre-training are inherently method-specific, requiring adaptation of the model backbone during fine-tuning. Multi-headed models offer an extensible alternative, enabling simultaneous training on multiple fidelities. We demonstrate that a multi-headed model learns method-agnostic representations that allow for accurate predictions across multiple label sources. While this approach introduces a slight accuracy compromise compared to sequential fine-tuning, it unlocks new cost-efficient data generation strategies and paves the way towards developing universal MLFFs.

Graph Neural Networks (GNNs) with equivariant properties have emerged as powerful tools for modeling complex dynamics of multi-object physical systems. However, their generalization ability is limited by the inadequate consideration of physical inductive biases: (1) Existing studies overlook the continuity of transitions among system states, opting to employ several discrete transformation layers to learn the direct mapping between two adjacent states; (2) Most models only account for first-order velocity information, despite the fact that many physical systems are governed by second-order motion laws. To incorporate these inductive biases, we propose the Second-order Equivariant Graph Neural Ordinary Differential Equation (SEGNO). Specifically, we show how the second-order continuity can be incorporated into GNNs while maintaining the equivariant property. Furthermore, we offer theoretical insights into SEGNO, highlighting that it can learn a unique trajectory between adjacent states, which is crucial for model generalization. Additionally, we prove that the discrepancy between this learned trajectory of SEGNO and the true trajectory is bounded. Extensive experiments on complex dynamical systems including molecular dynamics and motion capture demonstrate that our model yields a significant improvement over the state-of-the-art baselines.

Building generalist models has recently demonstrated remarkable capabilities in diverse scientific domains. Within the realm of molecular learning, several studies have explored unifying diverse tasks across diverse domains. However, negative conflicts and interference between molecules and knowledge from different domain may have a worse impact in threefold. First, conflicting molecular representations can lead to optimization difficulties for the models. Second, mixing and scaling up training data across diverse tasks is inherently challenging. Third, the computational cost of refined pretraining is prohibitively high. To address these limitations, this paper presents Omni-Mol, a scalable and unified LLM-based framework for direct instruction tuning. Omni-Mol builds on three key components to tackles conflicts: (1) a unified encoding mechanism for any task input; (2) an active-learning-driven data selection strategy that significantly reduces dataset size; (3) a novel design of the adaptive gradient stabilization module and anchor-and-reconcile MoE framework that ensures stable convergence. Experimentally, Omni-Mol achieves state-of-the-art performance across 15 molecular tasks, demonstrates the presence of scaling laws in the molecular domain, and is supported by extensive ablation studies and analyses validating the effectiveness of its design. The code and weights of the powerful AI-driven chemistry generalist are open-sourced at: https://anonymous.4open.science/r/Omni-Mol-8EDB.

Approximate second-order optimization methods often exhibit poorer generalization compared to first-order approaches. In this work, we look into this issue through the lens of the loss landscape and find that existing second-order methods tend to converge to sharper minima compared to SGD. In response, we propose Sassha, a novel second-order method designed to enhance generalization by explicitly reducing sharpness of the solution, while stabilizing the computation of approximate Hessians along the optimization trajectory. In fact, this sharpness minimization scheme is crafted also to accommodate lazy Hessian updates, so as to secure efficiency besides flatness. To validate its effectiveness, we conduct a wide range of standard deep learning experiments where Sassha demonstrates its outstanding generalization performance that is comparable to, and mostly better than, other methods. We provide a comprehensive set of analyses including convergence, robustness, stability, efficiency, and cost.

The generalization mystery in deep learning is the following: Why do over-parameterized neural networks trained with gradient descent (GD) generalize well on real datasets even though they are capable of fitting random datasets of comparable size? Furthermore, from among all solutions that fit the training data, how does GD find one that generalizes well (when such a well-generalizing solution exists)? We argue that the answer to both questions lies in the interaction of the gradients of different examples during training. Intuitively, if the per-example gradients are well-aligned, that is, if they are coherent, then one may expect GD to be (algorithmically) stable, and hence generalize well. We formalize this argument with an easy to compute and interpretable metric for coherence, and show that the metric takes on very different values on real and random datasets for several common vision networks. The theory also explains a number of other phenomena in deep learning, such as why some examples are reliably learned earlier than others, why early stopping works, and why it is possible to learn from noisy labels. Moreover, since the theory provides a causal explanation of how GD finds a well-generalizing solution when one exists, it motivates a class of simple modifications to GD that attenuate memorization and improve generalization. Generalization in deep learning is an extremely broad phenomenon, and therefore, it requires an equally general explanation. We conclude with a survey of alternative lines of attack on this problem, and argue that the proposed approach is the most viable one on this basis.

Learning representations that generalize under distribution shifts is critical for building robust machine learning models. However, despite significant efforts in recent years, algorithmic advances in this direction have been limited. In this work, we seek to understand the fundamental difficulty of out-of-distribution generalization with deep neural networks. We first empirically show that perhaps surprisingly, even allowing a neural network to explicitly fit the representations obtained from a teacher network that can generalize out-of-distribution is insufficient for the generalization of the student network. Then, by a theoretical study of two-layer ReLU networks optimized by stochastic gradient descent (SGD) under a structured feature model, we identify a fundamental yet unexplored feature learning proclivity of neural networks, feature contamination: neural networks can learn uncorrelated features together with predictive features, resulting in generalization failure under distribution shifts. Notably, this mechanism essentially differs from the prevailing narrative in the literature that attributes the generalization failure to spurious correlations. Overall, our results offer new insights into the non-linear feature learning dynamics of neural networks and highlight the necessity of considering inductive biases in out-of-distribution generalization.

We show how the Schur-Weyl duality that exists between the partition algebra and the symmetric group results in a stronger theoretical foundation for characterising all of the possible permutation equivariant neural networks whose layers are some tensor power of the permutation representation M_n of the symmetric group S_n. In doing so, we unify two separate bodies of literature, and we correct some of the major results that are now widely quoted by the machine learning community. In particular, we find a basis of matrices for the learnable, linear, permutation equivariant layer functions between such tensor power spaces in the standard basis of M_n by using an elegant graphical representation of a basis of set partitions for the partition algebra and its related vector spaces. Also, we show how we can calculate the number of weights that must appear in these layer functions by looking at certain paths through the McKay quiver for M_n. Finally, we describe how our approach generalises to the construction of neural networks that are equivariant to local symmetries.

Despite extensive studies, the underlying reason as to why overparameterized neural networks can generalize remains elusive. Existing theory shows that common stochastic optimizers prefer flatter minimizers of the training loss, and thus a natural potential explanation is that flatness implies generalization. This work critically examines this explanation. Through theoretical and empirical investigation, we identify the following three scenarios for two-layer ReLU networks: (1) flatness provably implies generalization; (2) there exist non-generalizing flattest models and sharpness minimization algorithms fail to generalize, and (3) perhaps most surprisingly, there exist non-generalizing flattest models, but sharpness minimization algorithms still generalize. Our results suggest that the relationship between sharpness and generalization subtly depends on the data distributions and the model architectures and sharpness minimization algorithms do not only minimize sharpness to achieve better generalization. This calls for the search for other explanations for the generalization of over-parameterized neural networks.

The development of efficient machine learning models for molecular systems representation is becoming crucial in scientific research. We introduce TensorNet, an innovative O(3)-equivariant message-passing neural network architecture that leverages Cartesian tensor representations. By using Cartesian tensor atomic embeddings, feature mixing is simplified through matrix product operations. Furthermore, the cost-effective decomposition of these tensors into rotation group irreducible representations allows for the separate processing of scalars, vectors, and tensors when necessary. Compared to higher-rank spherical tensor models, TensorNet demonstrates state-of-the-art performance with significantly fewer parameters. For small molecule potential energies, this can be achieved even with a single interaction layer. As a result of all these properties, the model's computational cost is substantially decreased. Moreover, the accurate prediction of vector and tensor molecular quantities on top of potential energies and forces is possible. In summary, TensorNet's framework opens up a new space for the design of state-of-the-art equivariant models.

We expect the generalization error to improve with more samples from a similar task, and to deteriorate with more samples from an out-of-distribution (OOD) task. In this work, we show a counter-intuitive phenomenon: the generalization error of a task can be a non-monotonic function of the number of OOD samples. As the number of OOD samples increases, the generalization error on the target task improves before deteriorating beyond a threshold. In other words, there is value in training on small amounts of OOD data. We use Fisher's Linear Discriminant on synthetic datasets and deep networks on computer vision benchmarks such as MNIST, CIFAR-10, CINIC-10, PACS and DomainNet to demonstrate and analyze this phenomenon. In the idealistic setting where we know which samples are OOD, we show that these non-monotonic trends can be exploited using an appropriately weighted objective of the target and OOD empirical risk. While its practical utility is limited, this does suggest that if we can detect OOD samples, then there may be ways to benefit from them. When we do not know which samples are OOD, we show how a number of go-to strategies such as data-augmentation, hyper-parameter optimization, and pre-training are not enough to ensure that the target generalization error does not deteriorate with the number of OOD samples in the dataset.

In recent years, several results in the supervised learning setting suggested that classical statistical learning-theoretic measures, such as VC dimension, do not adequately explain the performance of deep learning models which prompted a slew of work in the infinite-width and iteration regimes. However, there is little theoretical explanation for the success of neural networks beyond the supervised setting. In this paper we argue that, under some distributional assumptions, classical learning-theoretic measures can sufficiently explain generalization for graph neural networks in the transductive setting. In particular, we provide a rigorous analysis of the performance of neural networks in the context of transductive inference, specifically by analysing the generalisation properties of graph convolutional networks for the problem of node classification. While VC Dimension does result in trivial generalisation error bounds in this setting as well, we show that transductive Rademacher complexity can explain the generalisation properties of graph convolutional networks for stochastic block models. We further use the generalisation error bounds based on transductive Rademacher complexity to demonstrate the role of graph convolutions and network architectures in achieving smaller generalisation error and provide insights into when the graph structure can help in learning. The findings of this paper could re-new the interest in studying generalisation in neural networks in terms of learning-theoretic measures, albeit in specific problems.

Developing accurate models for chemical reactors is often challenging due to the complexity of reaction kinetics and process dynamics. Traditional approaches require retraining models for each new system, limiting generalizability and efficiency. In this work, we take a step toward foundation models for chemical reactor modeling by introducing a neural network framework that generalizes across diverse reactor types and rapidly adapts to new chemical processes. Our approach leverages meta-learning to pretrain the model on a broad set of reactor dynamics, enabling efficient adaptation to unseen reactions with minimal data. To further enhance generalizability, we incorporate physics-informed fine-tuning, ensuring physically consistent adaptation to new reactor conditions. Our framework is evaluated across three integer-order fundamental reactor types - continuous stirred tank reactors, batch reactors, and plug flow reactors - demonstrating superior few-shot adaptation compared to conventional data-driven, physics-informed, and transfer learning approaches. By combining meta-learning with physics-informed adaptation, this work lays the foundation for a generalizable modeling framework, advancing the development of foundation models for chemical engineering applications. Source code is available at https://github.com/killingbear999/chemical-reactor-foundation-model.

Neural operators have shown promise in learning solution maps of partial differential equations (PDEs), but they often struggle to generalize when test inputs lie outside the training distribution, such as novel initial conditions, unseen PDE coefficients or unseen physics. Prior works address this limitation with large-scale multiple physics pretraining followed by fine-tuning, but this still requires examples from the new dynamics, falling short of true zero-shot generalization. In this work, we propose a method to enhance generalization at test time, i.e., without modifying pretrained weights. Building on DISCO, which provides a dictionary of neural operators trained across different dynamics, we introduce a neural operator splitting strategy that, at test time, searches over compositions of training operators to approximate unseen dynamics. On challenging out-of-distribution tasks including parameter extrapolation and novel combinations of physics phenomena, our approach achieves state-of-the-art zero-shot generalization results, while being able to recover the underlying PDE parameters. These results underscore test-time computation as a key avenue for building flexible, compositional, and generalizable neural operators.

Based on the theory of homogeneous spaces we derive geometrically optimal edge attributes to be used within the flexible message-passing framework. We formalize the notion of weight sharing in convolutional networks as the sharing of message functions over point-pairs that should be treated equally. We define equivalence classes of point-pairs that are identical up to a transformation in the group and derive attributes that uniquely identify these classes. Weight sharing is then obtained by conditioning message functions on these attributes. As an application of the theory, we develop an efficient equivariant group convolutional network for processing 3D point clouds. The theory of homogeneous spaces tells us how to do group convolutions with feature maps over the homogeneous space of positions R^3, position and orientations R^3 {times} S^2, and the group SE(3) itself. Among these, R^3 {times} S^2 is an optimal choice due to the ability to represent directional information, which R^3 methods cannot, and it significantly enhances computational efficiency compared to indexing features on the full SE(3) group. We support this claim with state-of-the-art results -- in accuracy and speed -- on five different benchmarks in 2D and 3D, including interatomic potential energy prediction, trajectory forecasting in N-body systems, and generating molecules via equivariant diffusion models.

Graphs are essential data structures for modeling complex interactions in domains such as social networks, molecular structures, and biological systems. Graph-level tasks, which predict properties or classes for the entire graph, are critical for applications, such as molecular property prediction and subgraph counting. Graph Neural Networks (GNNs) have shown promise in these tasks, but their evaluations are often limited to narrow datasets, tasks, and inconsistent experimental setups, restricting their generalizability. To address these limitations, we propose a unified evaluation framework for graph-level GNNs. This framework provides a standardized setting to evaluate GNNs across diverse datasets, various graph tasks (e.g., graph classification and regression), and challenging scenarios, including noisy, imbalanced, and few-shot graphs. Additionally, we propose a novel GNN model with enhanced expressivity and generalization capabilities. Specifically, we enhance the expressivity of GNNs through a k-path rooted subgraph approach, enabling the model to effectively count subgraphs (e.g., paths and cycles). Moreover, we introduce a unified graph contrastive learning algorithm for graphs across diverse domains, which adaptively removes unimportant edges to augment graphs, thereby significantly improving generalization performance. Extensive experiments demonstrate that our model achieves superior performance against fourteen effective baselines across twenty-seven graph datasets, establishing it as a robust and generalizable model for graph-level tasks.

This paper introduces a new model to learn graph neural networks equivariant to rotations, translations, reflections and permutations called E(n)-Equivariant Graph Neural Networks (EGNNs). In contrast with existing methods, our work does not require computationally expensive higher-order representations in intermediate layers while it still achieves competitive or better performance. In addition, whereas existing methods are limited to equivariance on 3 dimensional spaces, our model is easily scaled to higher-dimensional spaces. We demonstrate the effectiveness of our method on dynamical systems modelling, representation learning in graph autoencoders and predicting molecular properties.

Generalization, the ability to perform well beyond the training context, is a hallmark of biological and artificial intelligence, yet anticipating unseen failures remains a central challenge. Conventional approaches often take a ``bottom-up'' mechanistic route by reverse-engineering interpretable features or circuits to build explanatory models. While insightful, these methods often struggle to provide the high-level, predictive signals for anticipating failure in real-world deployment. Here, we propose using a ``top-down'' approach to studying generalization failures inspired by medical biomarkers: identifying system-level measurements that serve as robust indicators of a model's future performance. Rather than mapping out detailed internal mechanisms, we systematically design and test network markers to probe structure, function links, identify prognostic indicators, and validate predictions in real-world settings. In image classification, we find that task-relevant geometric properties of in-distribution (ID) object manifolds consistently forecast poor out-of-distribution (OOD) generalization. In particular, reductions in two geometric measures, effective manifold dimensionality and utility, predict weaker OOD performance across diverse architectures, optimizers, and datasets. We apply this finding to transfer learning with ImageNet-pretrained models. We consistently find that the same geometric patterns predict OOD transfer performance more reliably than ID accuracy. This work demonstrates that representational geometry can expose hidden vulnerabilities, offering more robust guidance for model selection and AI interpretability.

Despite the tremendous success, existing machine learning models still fall short of human-like systematic generalization -- learning compositional rules from limited data and applying them to unseen combinations in various domains. We propose Neural-Symbolic Recursive Machine (NSR) to tackle this deficiency. The core representation of NSR is a Grounded Symbol System (GSS) with combinatorial syntax and semantics, which entirely emerges from training data. Akin to the neuroscience studies suggesting separate brain systems for perceptual, syntactic, and semantic processing, NSR implements analogous separate modules of neural perception, syntactic parsing, and semantic reasoning, which are jointly learned by a deduction-abduction algorithm. We prove that NSR is expressive enough to model various sequence-to-sequence tasks. Superior systematic generalization is achieved via the inductive biases of equivariance and recursiveness embedded in NSR. In experiments, NSR achieves state-of-the-art performance in three benchmarks from different domains: SCAN for semantic parsing, PCFG for string manipulation, and HINT for arithmetic reasoning. Specifically, NSR achieves 100% generalization accuracy on SCAN and PCFG and outperforms state-of-the-art models on HINT by about 23%. Our NSR demonstrates stronger generalization than pure neural networks due to its symbolic representation and inductive biases. NSR also demonstrates better transferability than existing neural-symbolic approaches due to less domain-specific knowledge required.

Graph neural networks that model 3D data, such as point clouds or atoms, are typically desired to be SO(3) equivariant, i.e., equivariant to 3D rotations. Unfortunately equivariant convolutions, which are a fundamental operation for equivariant networks, increase significantly in computational complexity as higher-order tensors are used. In this paper, we address this issue by reducing the SO(3) convolutions or tensor products to mathematically equivalent convolutions in SO(2) . This is accomplished by aligning the node embeddings' primary axis with the edge vectors, which sparsifies the tensor product and reduces the computational complexity from O(L^6) to O(L^3), where L is the degree of the representation. We demonstrate the potential implications of this improvement by proposing the Equivariant Spherical Channel Network (eSCN), a graph neural network utilizing our novel approach to equivariant convolutions, which achieves state-of-the-art results on the large-scale OC-20 and OC-22 datasets.

Much research in machine learning involves finding appropriate inductive biases (e.g. convolutional neural networks, momentum-based optimizers, transformers) to promote generalization on tasks. However, quantification of the amount of inductive bias associated with these architectures and hyperparameters has been limited. We propose a novel method for efficiently computing the inductive bias required for generalization on a task with a fixed training data budget; formally, this corresponds to the amount of information required to specify well-generalizing models within a specific hypothesis space of models. Our approach involves modeling the loss distribution of random hypotheses drawn from a hypothesis space to estimate the required inductive bias for a task relative to these hypotheses. Unlike prior work, our method provides a direct estimate of inductive bias without using bounds and is applicable to diverse hypothesis spaces. Moreover, we derive approximation error bounds for our estimation approach in terms of the number of sampled hypotheses. Consistent with prior results, our empirical results demonstrate that higher dimensional tasks require greater inductive bias. We show that relative to other expressive model classes, neural networks as a model class encode large amounts of inductive bias. Furthermore, our measure quantifies the relative difference in inductive bias between different neural network architectures. Our proposed inductive bias metric provides an information-theoretic interpretation of the benefits of specific model architectures for certain tasks and provides a quantitative guide to developing tasks requiring greater inductive bias, thereby encouraging the development of more powerful inductive biases.

Catalyst discovery and optimization is key to solving many societal and energy challenges including solar fuels synthesis, long-term energy storage, and renewable fertilizer production. Despite considerable effort by the catalysis community to apply machine learning models to the computational catalyst discovery process, it remains an open challenge to build models that can generalize across both elemental compositions of surfaces and adsorbate identity/configurations, perhaps because datasets have been smaller in catalysis than related fields. To address this we developed the OC20 dataset, consisting of 1,281,040 Density Functional Theory (DFT) relaxations (~264,890,000 single point evaluations) across a wide swath of materials, surfaces, and adsorbates (nitrogen, carbon, and oxygen chemistries). We supplemented this dataset with randomly perturbed structures, short timescale molecular dynamics, and electronic structure analyses. The dataset comprises three central tasks indicative of day-to-day catalyst modeling and comes with pre-defined train/validation/test splits to facilitate direct comparisons with future model development efforts. We applied three state-of-the-art graph neural network models (CGCNN, SchNet, Dimenet++) to each of these tasks as baseline demonstrations for the community to build on. In almost every task, no upper limit on model size was identified, suggesting that even larger models are likely to improve on initial results. The dataset and baseline models are both provided as open resources, as well as a public leader board to encourage community contributions to solve these important tasks.

Improving the generalization of deep networks is an important open challenge, particularly in domains without plentiful data. The mixup algorithm improves generalization by linearly interpolating a pair of examples and their corresponding labels. These interpolated examples augment the original training set. Mixup has shown promising results in various classification tasks, but systematic analysis of mixup in regression remains underexplored. Using mixup directly on regression labels can result in arbitrarily incorrect labels. In this paper, we propose a simple yet powerful algorithm, C-Mixup, to improve generalization on regression tasks. In contrast with vanilla mixup, which picks training examples for mixing with uniform probability, C-Mixup adjusts the sampling probability based on the similarity of the labels. Our theoretical analysis confirms that C-Mixup with label similarity obtains a smaller mean square error in supervised regression and meta-regression than vanilla mixup and using feature similarity. Another benefit of C-Mixup is that it can improve out-of-distribution robustness, where the test distribution is different from the training distribution. By selectively interpolating examples with similar labels, it mitigates the effects of domain-associated information and yields domain-invariant representations. We evaluate C-Mixup on eleven datasets, ranging from tabular to video data. Compared to the best prior approach, C-Mixup achieves 6.56%, 4.76%, 5.82% improvements in in-distribution generalization, task generalization, and out-of-distribution robustness, respectively. Code is released at https://github.com/huaxiuyao/C-Mixup.

Multi-center positron emission tomography (PET) image synthesis aims at recovering low-dose PET images from multiple different centers. The generalizability of existing methods can still be suboptimal for a multi-center study due to domain shifts, which result from non-identical data distribution among centers with different imaging systems/protocols. While some approaches address domain shifts by training specialized models for each center, they are parameter inefficient and do not well exploit the shared knowledge across centers. To address this, we develop a generalist model that shares architecture and parameters across centers to utilize the shared knowledge. However, the generalist model can suffer from the center interference issue, i.e. the gradient directions of different centers can be inconsistent or even opposite owing to the non-identical data distribution. To mitigate such interference, we introduce a novel dynamic routing strategy with cross-layer connections that routes data from different centers to different experts. Experiments show that our generalist model with dynamic routing (DRMC) exhibits excellent generalizability across centers. Code and data are available at: https://github.com/Yaziwel/Multi-Center-PET-Image-Synthesis.

Design of neural networks that incorporate symmetry is crucial for geometric deep learning. Central to this effort is the development of invariant and equivariant operations. This works presents a systematic method for constructing valid invariant and equivariant operations. It can handle inputs and outputs in the form of Cartesian tensors with different rank, as well as spherical tensors with different types. In addition, our method features a graphical representation utilizing the symmetric tensor network, which simplifies both the proofs and constructions related to invariant and equivariant functions. We also apply this approach to design the equivariant interaction message for the geometry graph neural network, and equivariant machine learning model to learn the constitutive law of materials.

Symmetry learning has proven to be an effective approach for extracting the hidden structure of data, with the concept of equivariance relation playing the central role. However, most of the current studies are built on architectural theory and corresponding assumptions on the form of data. We propose Neural Fourier Transform (NFT), a general framework of learning the latent linear action of the group without assuming explicit knowledge of how the group acts on data. We present the theoretical foundations of NFT and show that the existence of a linear equivariant feature, which has been assumed ubiquitously in equivariance learning, is equivalent to the existence of a group invariant kernel on the dataspace. We also provide experimental results to demonstrate the application of NFT in typical scenarios with varying levels of knowledge about the acting group.

The measure of a machine learning algorithm is the difficulty of the tasks it can perform, and sufficiently difficult tasks are critical drivers of strong machine learning models. However, quantifying the generalization difficulty of machine learning benchmarks has remained challenging. We propose what is to our knowledge the first model-agnostic measure of the inherent generalization difficulty of tasks. Our inductive bias complexity measure quantifies the total information required to generalize well on a task minus the information provided by the data. It does so by measuring the fractional volume occupied by hypotheses that generalize on a task given that they fit the training data. It scales exponentially with the intrinsic dimensionality of the space over which the model must generalize but only polynomially in resolution per dimension, showing that tasks which require generalizing over many dimensions are drastically more difficult than tasks involving more detail in fewer dimensions. Our measure can be applied to compute and compare supervised learning, reinforcement learning and meta-learning generalization difficulties against each other. We show that applied empirically, it formally quantifies intuitively expected trends, e.g. that in terms of required inductive bias, MNIST < CIFAR10 < Imagenet and fully observable Markov decision processes (MDPs) < partially observable MDPs. Further, we show that classification of complex images < few-shot meta-learning with simple images. Our measure provides a quantitative metric to guide the construction of more complex tasks requiring greater inductive bias, and thereby encourages the development of more sophisticated architectures and learning algorithms with more powerful generalization capabilities.

Neural networks trained by gradient descent (GD) have exhibited a number of surprising generalization behaviors. First, they can achieve a perfect fit to noisy training data and still generalize near-optimally, showing that overfitting can sometimes be benign. Second, they can undergo a period of classical, harmful overfitting -- achieving a perfect fit to training data with near-random performance on test data -- before transitioning ("grokking") to near-optimal generalization later in training. In this work, we show that both of these phenomena provably occur in two-layer ReLU networks trained by GD on XOR cluster data where a constant fraction of the training labels are flipped. In this setting, we show that after the first step of GD, the network achieves 100% training accuracy, perfectly fitting the noisy labels in the training data, but achieves near-random test accuracy. At a later training step, the network achieves near-optimal test accuracy while still fitting the random labels in the training data, exhibiting a "grokking" phenomenon. This provides the first theoretical result of benign overfitting in neural network classification when the data distribution is not linearly separable. Our proofs rely on analyzing the feature learning process under GD, which reveals that the network implements a non-generalizable linear classifier after one step and gradually learns generalizable features in later steps.

Neural networks that process the parameters of other neural networks find applications in domains as diverse as classifying implicit neural representations, generating neural network weights, and predicting generalization errors. However, existing approaches either overlook the inherent permutation symmetry in the neural network or rely on intricate weight-sharing patterns to achieve equivariance, while ignoring the impact of the network architecture itself. In this work, we propose to represent neural networks as computational graphs of parameters, which allows us to harness powerful graph neural networks and transformers that preserve permutation symmetry. Consequently, our approach enables a single model to encode neural computational graphs with diverse architectures. We showcase the effectiveness of our method on a wide range of tasks, including classification and editing of implicit neural representations, predicting generalization performance, and learning to optimize, while consistently outperforming state-of-the-art methods. The source code is open-sourced at https://github.com/mkofinas/neural-graphs.

Distribution shift presents a significant challenge in machine learning, where models often underperform during the test stage when faced with a different distribution than the one they were trained on. This paper focuses on domain shifts, which occur when the model is applied to new domains that are different from the ones it was trained on, and propose a new approach called D^3G. Unlike previous methods that aim to learn a single model that is domain invariant, D^3G leverages domain similarities based on domain metadata to learn domain-specific models. Concretely, D^3G learns a set of training-domain-specific functions during the training stage and reweights them based on domain relations during the test stage. These domain relations can be directly obtained and learned from domain metadata. Under mild assumptions, we theoretically prove that using domain relations to reweight training-domain-specific functions achieves stronger out-of-domain generalization compared to the conventional averaging approach. Empirically, we evaluate the effectiveness of D^3G using real-world datasets for tasks such as temperature regression, land use classification, and molecule-protein binding affinity prediction. Our results show that D^3G consistently outperforms state-of-the-art methods.

Auxiliary-Task Learning (ATL) aims to improve the performance of the target task by leveraging the knowledge obtained from related tasks. Occasionally, learning multiple tasks simultaneously results in lower accuracy than learning only the target task, which is known as negative transfer. This problem is often attributed to the gradient conflicts among tasks, and is frequently tackled by coordinating the task gradients in previous works. However, these optimization-based methods largely overlook the auxiliary-target generalization capability. To better understand the root cause of negative transfer, we experimentally investigate it from both optimization and generalization perspectives. Based on our findings, we introduce ForkMerge, a novel approach that periodically forks the model into multiple branches, automatically searches the varying task weights by minimizing target validation errors, and dynamically merges all branches to filter out detrimental task-parameter updates. On a series of auxiliary-task learning benchmarks, ForkMerge outperforms existing methods and effectively mitigates negative transfer.

The big empirical success of group equivariant networks has led in recent years to the sprouting of a great variety of equivariant network architectures. A particular focus has thereby been on rotation and reflection equivariant CNNs for planar images. Here we give a general description of E(2)-equivariant convolutions in the framework of Steerable CNNs. The theory of Steerable CNNs thereby yields constraints on the convolution kernels which depend on group representations describing the transformation laws of feature spaces. We show that these constraints for arbitrary group representations can be reduced to constraints under irreducible representations. A general solution of the kernel space constraint is given for arbitrary representations of the Euclidean group E(2) and its subgroups. We implement a wide range of previously proposed and entirely new equivariant network architectures and extensively compare their performances. E(2)-steerable convolutions are further shown to yield remarkable gains on CIFAR-10, CIFAR-100 and STL-10 when used as a drop-in replacement for non-equivariant convolutions.

We present Clifford-Steerable Convolutional Neural Networks (CS-CNNs), a novel class of E(p, q)-equivariant CNNs. CS-CNNs process multivector fields on pseudo-Euclidean spaces R^{p,q}. They cover, for instance, E(3)-equivariance on R^3 and Poincaré-equivariance on Minkowski spacetime R^{1,3}. Our approach is based on an implicit parametrization of O(p,q)-steerable kernels via Clifford group equivariant neural networks. We significantly and consistently outperform baseline methods on fluid dynamics as well as relativistic electrodynamics forecasting tasks.

We introduce Clifford Group Equivariant Simplicial Message Passing Networks, a method for steerable E(n)-equivariant message passing on simplicial complexes. Our method integrates the expressivity of Clifford group-equivariant layers with simplicial message passing, which is topologically more intricate than regular graph message passing. Clifford algebras include higher-order objects such as bivectors and trivectors, which express geometric features (e.g., areas, volumes) derived from vectors. Using this knowledge, we represent simplex features through geometric products of their vertices. To achieve efficient simplicial message passing, we share the parameters of the message network across different dimensions. Additionally, we restrict the final message to an aggregation of the incoming messages from different dimensions, leading to what we term shared simplicial message passing. Experimental results show that our method is able to outperform both equivariant and simplicial graph neural networks on a variety of geometric tasks.

Graph neural networks have recently achieved great successes in predicting quantum mechanical properties of molecules. These models represent a molecule as a graph using only the distance between atoms (nodes). They do not, however, consider the spatial direction from one atom to another, despite directional information playing a central role in empirical potentials for molecules, e.g. in angular potentials. To alleviate this limitation we propose directional message passing, in which we embed the messages passed between atoms instead of the atoms themselves. Each message is associated with a direction in coordinate space. These directional message embeddings are rotationally equivariant since the associated directions rotate with the molecule. We propose a message passing scheme analogous to belief propagation, which uses the directional information by transforming messages based on the angle between them. Additionally, we use spherical Bessel functions and spherical harmonics to construct theoretically well-founded, orthogonal representations that achieve better performance than the currently prevalent Gaussian radial basis representations while using fewer than 1/4 of the parameters. We leverage these innovations to construct the directional message passing neural network (DimeNet). DimeNet outperforms previous GNNs on average by 76% on MD17 and by 31% on QM9. Our implementation is available online.

Learning features from data is one of the defining characteristics of deep learning, but our theoretical understanding of the role features play in deep learning is still rudimentary. To address this gap, we introduce a new tool, the interaction tensor, for empirically analyzing the interaction between data and model through features. With the interaction tensor, we make several key observations about how features are distributed in data and how models with different random seeds learn different features. Based on these observations, we propose a conceptual framework for feature learning. Under this framework, the expected accuracy for a single hypothesis and agreement for a pair of hypotheses can both be derived in closed-form. We demonstrate that the proposed framework can explain empirically observed phenomena, including the recently discovered Generalization Disagreement Equality (GDE) that allows for estimating the generalization error with only unlabeled data. Further, our theory also provides explicit construction of natural data distributions that break the GDE. Thus, we believe this work provides valuable new insight into our understanding of feature learning.

The learnable, linear neural network layers between tensor power spaces of R^{n} that are equivariant to the orthogonal group, O(n), the special orthogonal group, SO(n), and the symplectic group, Sp(n), were characterised in arXiv:2212.08630. We present an algorithm for multiplying a vector by any weight matrix for each of these groups, using category theoretic constructions to implement the procedure. We achieve a significant reduction in computational cost compared with a naive implementation by making use of Kronecker product matrices to perform the multiplication. We show that our approach extends to the symmetric group, S_n, recovering the algorithm of arXiv:2303.06208 in the process.

E(3)-equivariant neural networks have demonstrated success across a wide range of 3D modelling tasks. A fundamental operation in these networks is the tensor product, which interacts two geometric features in an equivariant manner to create new features. Due to the high computational complexity of the tensor product, significant effort has been invested to optimize the runtime of this operation. For example, Luo et al. (2024) recently proposed the Gaunt tensor product (GTP) which promises a significant speedup. In this work, we provide a careful, systematic analysis of a number of tensor product operations. In particular, we emphasize that different tensor products are not performing the same operation. The reported speedups typically come at the cost of expressivity. We introduce measures of expressivity and interactability to characterize these differences. In addition, we realized the original implementation of GTP can be greatly simplified by directly using a spherical grid at no cost in asymptotic runtime. This spherical grid approach is faster on our benchmarks and in actual training of the MACE interatomic potential by 30%. Finally, we provide the first systematic microbenchmarks of the various tensor product operations. We find that the theoretical runtime guarantees can differ wildly from empirical performance, demonstrating the need for careful application-specific benchmarking. Code is available at https://github.com/atomicarchitects/PriceofFreedom.

Our work presents extensive empirical evidence that layer rotation, i.e. the evolution across training of the cosine distance between each layer's weight vector and its initialization, constitutes an impressively consistent indicator of generalization performance. In particular, larger cosine distances between final and initial weights of each layer consistently translate into better generalization performance of the final model. Interestingly, this relation admits a network independent optimum: training procedures during which all layers' weights reach a cosine distance of 1 from their initialization consistently outperform other configurations -by up to 30% test accuracy. Moreover, we show that layer rotations are easily monitored and controlled (helpful for hyperparameter tuning) and potentially provide a unified framework to explain the impact of learning rate tuning, weight decay, learning rate warmups and adaptive gradient methods on generalization and training speed. In an attempt to explain the surprising properties of layer rotation, we show on a 1-layer MLP trained on MNIST that layer rotation correlates with the degree to which features of intermediate layers have been trained.

Domain generalization (DG) aims to learn domain-generalizable models from one or multiple source domains that can perform well in unseen target domains. Despite its recent progress, most existing work suffers from the misalignment between the difficulty level of training samples and the capability of contemporarily trained models, leading to over-fitting or under-fitting in the trained generalization model. We design MoDify, a Momentum Difficulty framework that tackles the misalignment by balancing the seesaw between the model's capability and the samples' difficulties along the training process. MoDify consists of two novel designs that collaborate to fight against the misalignment while learning domain-generalizable models. The first is MoDify-based Data Augmentation which exploits an RGB Shuffle technique to generate difficulty-aware training samples on the fly. The second is MoDify-based Network Optimization which dynamically schedules the training samples for balanced and smooth learning with appropriate difficulty. Without bells and whistles, a simple implementation of MoDify achieves superior performance across multiple benchmarks. In addition, MoDify can complement existing methods as a plug-in, and it is generic and can work for different visual recognition tasks.

Overparametrization often helps improve the generalization performance. This paper proposes a dual view of overparametrization suggesting that downsampling may also help generalize. Motivated by this dual view, we characterize two out-of-sample prediction risks of the sketched ridgeless least square estimator in the proportional regime masymp n asymp p, where m is the sketching size, n the sample size, and p the feature dimensionality. Our results reveal the statistical role of downsampling. Specifically, downsampling does not always hurt the generalization performance, and may actually help improve it in some cases. We identify the optimal sketching sizes that minimize the out-of-sample prediction risks, and find that the optimally sketched estimator has stabler risk curves that eliminates the peaks of those for the full-sample estimator. We then propose a practical procedure to empirically identify the optimal sketching size. Finally, we extend our results to cover central limit theorems and misspecified models. Numerical studies strongly support our theory.

Graph Neural Networks (GNNs) extend basic Neural Networks (NNs) by using graph structures based on the relational inductive bias (homophily assumption). While GNNs have been commonly believed to outperform NNs in real-world tasks, recent work has identified a non-trivial set of datasets where their performance compared to NNs is not satisfactory. Heterophily has been considered the main cause of this empirical observation and numerous works have been put forward to address it. In this paper, we first revisit the widely used homophily metrics and point out that their consideration of only graph-label consistency is a shortcoming. Then, we study heterophily from the perspective of post-aggregation node similarity and define new homophily metrics, which are potentially advantageous compared to existing ones. Based on this investigation, we prove that some harmful cases of heterophily can be effectively addressed by local diversification operation. Then, we propose the Adaptive Channel Mixing (ACM), a framework to adaptively exploit aggregation, diversification and identity channels node-wisely to extract richer localized information for diverse node heterophily situations. ACM is more powerful than the commonly used uni-channel framework for node classification tasks on heterophilic graphs and is easy to be implemented in baseline GNN layers. When evaluated on 10 benchmark node classification tasks, ACM-augmented baselines consistently achieve significant performance gain, exceeding state-of-the-art GNNs on most tasks without incurring significant computational burden.

Integrating a notion of symmetry into point cloud neural networks is a provably effective way to improve their generalization capability. Of particular interest are E(3) equivariant point cloud networks where Euclidean transformations applied to the inputs are preserved in the outputs. Recent efforts aim to extend networks that are E(3) equivariant, to accommodate inputs made of multiple parts, each of which exhibits local E(3) symmetry. In practical settings, however, the partitioning into individually transforming regions is unknown a priori. Errors in the partition prediction would unavoidably map to errors in respecting the true input symmetry. Past works have proposed different ways to predict the partition, which may exhibit uncontrolled errors in their ability to maintain equivariance to the actual partition. To this end, we introduce APEN: a general framework for constructing approximate piecewise-E(3) equivariant point networks. Our primary insight is that functions that are equivariant with respect to a finer partition will also maintain equivariance in relation to the true partition. Leveraging this observation, we propose a design where the equivariance approximation error at each layers can be bounded solely in terms of (i) uncertainty quantification of the partition prediction, and (ii) bounds on the probability of failing to suggest a proper subpartition of the ground truth one. We demonstrate the effectiveness of APEN using two data types exemplifying part-based symmetry: (i) real-world scans of room scenes containing multiple furniture-type objects; and, (ii) human motions, characterized by articulated parts exhibiting rigid movement. Our empirical results demonstrate the advantage of integrating piecewise E(3) symmetry into network design, showing a distinct improvement in generalization compared to prior works for both classification and segmentation tasks.

Deep ensembles (DE) have been successful in improving model performance by learning diverse members via the stochasticity of random initialization. While recent works have attempted to promote further diversity in DE via hyperparameters or regularizing loss functions, these methods primarily still rely on a stochastic approach to explore the hypothesis space. In this work, we present Multi-Symmetry Ensembles (MSE), a framework for constructing diverse ensembles by capturing the multiplicity of hypotheses along symmetry axes, which explore the hypothesis space beyond stochastic perturbations of model weights and hyperparameters. We leverage recent advances in contrastive representation learning to create models that separately capture opposing hypotheses of invariant and equivariant functional classes and present a simple ensembling approach to efficiently combine appropriate hypotheses for a given task. We show that MSE effectively captures the multiplicity of conflicting hypotheses that is often required in large, diverse datasets like ImageNet. As a result of their inherent diversity, MSE improves classification performance, uncertainty quantification, and generalization across a series of transfer tasks.

Optimization and generalization are two essential aspects of statistical machine learning. In this paper, we propose a framework to connect optimization with generalization by analyzing the generalization error based on the optimization trajectory under the gradient flow algorithm. The key ingredient of this framework is the Uniform-LGI, a property that is generally satisfied when training machine learning models. Leveraging the Uniform-LGI, we first derive convergence rates for gradient flow algorithm, then we give generalization bounds for a large class of machine learning models. We further apply our framework to three distinct machine learning models: linear regression, kernel regression, and two-layer neural networks. Through our approach, we obtain generalization estimates that match or extend previous results.

The design of novel protein structures remains a challenge in protein engineering for applications across biomedicine and chemistry. In this line of work, a diffusion model over rigid bodies in 3D (referred to as frames) has shown success in generating novel, functional protein backbones that have not been observed in nature. However, there exists no principled methodological framework for diffusion on SE(3), the space of orientation preserving rigid motions in R3, that operates on frames and confers the group invariance. We address these shortcomings by developing theoretical foundations of SE(3) invariant diffusion models on multiple frames followed by a novel framework, FrameDiff, for learning the SE(3) equivariant score over multiple frames. We apply FrameDiff on monomer backbone generation and find it can generate designable monomers up to 500 amino acids without relying on a pretrained protein structure prediction network that has been integral to previous methods. We find our samples are capable of generalizing beyond any known protein structure.

Generalist robot policies trained on large-scale datasets such as Open X-Embodiment (OXE) demonstrate strong performance across a wide range of tasks. However, they often struggle to generalize beyond the distribution of their training data. In this paper, we investigate the underlying cause of this limited generalization capability. We identify shortcut learning -- the reliance on task-irrelevant features -- as a key impediment to generalization. Through comprehensive theoretical and empirical analysis, we uncover two primary contributors to shortcut learning: (1) limited diversity within individual sub-datasets, and (2) significant distributional disparities across sub-datasets, leading to dataset fragmentation. These issues arise from the inherent structure of large-scale datasets like OXE, which are typically composed of multiple sub-datasets collected independently across varied environments and embodiments. Our findings provide critical insights into dataset collection strategies that can reduce shortcut learning and enhance the generalization ability of generalist robot policies. Moreover, in scenarios where acquiring new large-scale data is impractical, we demonstrate that carefully selected robotic data augmentation strategies can effectively reduce shortcut learning in existing offline datasets, thereby improving generalization capabilities of generalist robot policies, e.g., pi_0, in both simulation and real-world environments. More information at https://lucky-light-sun.github.io/proj/shortcut-learning-in-grps/.

For vision-and-language reasoning tasks, both fully connectionist, end-to-end methods and hybrid, neuro-symbolic methods have achieved high in-distribution performance. In which out-of-distribution settings does each paradigm excel? We investigate this question on both single-image and multi-image visual question-answering through four types of generalization tests: a novel segment-combine test for multi-image queries, contrast set, compositional generalization, and cross-benchmark transfer. Vision-and-language end-to-end trained systems exhibit sizeable performance drops across all these tests. Neuro-symbolic methods suffer even more on cross-benchmark transfer from GQA to VQA, but they show smaller accuracy drops on the other generalization tests and their performance quickly improves by few-shot training. Overall, our results demonstrate the complementary benefits of these two paradigms, and emphasize the importance of using a diverse suite of generalization tests to fully characterize model robustness to distribution shift.

Machine Learning Force Fields (MLFFs) are a promising alternative to expensive ab initio quantum mechanical molecular simulations. Given the diversity of chemical spaces that are of interest and the cost of generating new data, it is important to understand how MLFFs generalize beyond their training distributions. In order to characterize and better understand distribution shifts in MLFFs, we conduct diagnostic experiments on chemical datasets, revealing common shifts that pose significant challenges, even for large foundation models trained on extensive data. Based on these observations, we hypothesize that current supervised training methods inadequately regularize MLFFs, resulting in overfitting and learning poor representations of out-of-distribution systems. We then propose two new methods as initial steps for mitigating distribution shifts for MLFFs. Our methods focus on test-time refinement strategies that incur minimal computational cost and do not use expensive ab initio reference labels. The first strategy, based on spectral graph theory, modifies the edges of test graphs to align with graph structures seen during training. Our second strategy improves representations for out-of-distribution systems at test-time by taking gradient steps using an auxiliary objective, such as a cheap physical prior. Our test-time refinement strategies significantly reduce errors on out-of-distribution systems, suggesting that MLFFs are capable of and can move towards modeling diverse chemical spaces, but are not being effectively trained to do so. Our experiments establish clear benchmarks for evaluating the generalization capabilities of the next generation of MLFFs. Our code is available at https://tkreiman.github.io/projects/mlff_distribution_shifts/.

Clifford-Steerable CNNs (CSCNNs) provide a unified framework that allows incorporating equivariance to arbitrary pseudo-Euclidean groups, including isometries of Euclidean space and Minkowski spacetime. In this work, we demonstrate that the kernel basis of CSCNNs is not complete, thus limiting the model expressivity. To address this issue, we propose Conditional Clifford-Steerable Kernels, which augment the kernels with equivariant representations computed from the input feature field. We derive the equivariance constraint for these input-dependent kernels and show how it can be solved efficiently via implicit parameterization. We empirically demonstrate an improved expressivity of the resulting framework on multiple PDE forecasting tasks, including fluid dynamics and relativistic electrodynamics, where our method consistently outperforms baseline methods.

Graph neural networks (GNNs) have become compelling models designed to perform learning and inference on graph-structured data. However, little work has been done to understand the fundamental limitations of GNNs for scaling to larger graphs and generalizing to out-of-distribution (OOD) inputs. In this paper, we use a random graph generator to systematically investigate how the graph size and structural properties affect the predictive performance of GNNs. We present specific evidence that the average node degree is a key feature in determining whether GNNs can generalize to unseen graphs, and that the use of multiple node update functions can improve the generalization performance of GNNs when dealing with graphs of multimodal degree distributions. Accordingly, we propose a multi-module GNN framework that allows the network to adapt flexibly to new graphs by generalizing a single canonical nonlinear transformation over aggregated inputs. Our results show that the multi-module GNNs improve the OOD generalization on a variety of inference tasks in the direction of diverse structural features.

Invariance and equivariance to geometrical transformations have proven to be very useful inductive biases when training (convolutional) neural network models, especially in the low-data regime. Much work has focused on the case where the symmetry group employed is compact or abelian, or both. Recent work has explored enlarging the class of transformations used to the case of Lie groups, principally through the use of their Lie algebra, as well as the group exponential and logarithm maps. The applicability of such methods to larger transformation groups is limited by the fact that depending on the group of interest G, the exponential map may not be surjective. Further limitations are encountered when G is neither compact nor abelian. Using the structure and geometry of Lie groups and their homogeneous spaces, we present a framework by which it is possible to work with such groups primarily focusing on the Lie groups G = GL^{+}(n, R) and G = SL(n, R), as well as their representation as affine transformations R^{n} rtimes G. Invariant integration as well as a global parametrization is realized by decomposing the `larger` groups into subgroups and submanifolds which can be handled individually. Under this framework, we show how convolution kernels can be parametrized to build models equivariant with respect to affine transformations. We evaluate the robustness and out-of-distribution generalisation capability of our model on the standard affine-invariant benchmark classification task, where we outperform all previous equivariant models as well as all Capsule Network proposals.

We introduce `dualGNN', an autoregressive message-passing GNN for sampling fine, regular triangulations (FRTs) of convex polytopes. dualGNN operates on a generalization of the dual graph of a triangulation, with edges labeled by `signed circuits' -- combinatorial invariants from oriented matroid theory which we show are both necessary and sufficient for exposing regularity. The model is independent of the number of points in the polytope and invariant under the polytope's orientation-preserving symmetries (SL(d,Z) ltimes Z^d). When implemented with a certain masking procedure, one can also guarantee that every rollout produces a fine triangulation (in 2D). On unseen polygons with N_pts leq 40, dualGNN is the most uniform FRT sampler we tested, and even a model trained on a single polygon generalizes well to other polygons. The model is small (sim92k parameters), trains in sim7.5 hours on a single consumer GPU, and runs without modification on an M1 MacBook Pro. We apply dualGNN to string theory, uniformly sampling Calabi-Yau threefolds at h^{1,1}=86 and consistent with uniformity at h^{1,1}=128. This is an order of magnitude beyond previous learned methods with a model sim1000times smaller. Code, training scripts, and pretrained models are available at https://github.com/natemacfadden/dualGNN .

As an alternative to classical numerical solvers for partial differential equations (PDEs) subject to boundary value constraints, there has been a surge of interest in investigating neural networks that can solve such problems efficiently. In this work, we design a general solution operator for two different time-independent PDEs using graph neural networks (GNNs) and spectral graph convolutions. We train the networks on simulated data from a finite elements solver on a variety of shapes and inhomogeneities. In contrast to previous works, we focus on the ability of the trained operator to generalize to previously unseen scenarios. Specifically, we test generalization to meshes with different shapes and superposition of solutions for a different number of inhomogeneities. We find that training on a diverse dataset with lots of variation in the finite element meshes is a key ingredient for achieving good generalization results in all cases. With this, we believe that GNNs can be used to learn solution operators that generalize over a range of properties and produce solutions much faster than a generic solver. Our dataset, which we make publicly available, can be used and extended to verify the robustness of these models under varying conditions.

A fundamental question in modern machine learning is why large, over-parameterized models, such as deep neural networks and transformers, tend to generalize well, even when their number of parameters far exceeds the number of training samples. We investigate this phenomenon through the lens of information theory, grounded in universal learning theory. Specifically, we study a Bayesian mixture learner with log-loss and (almost) uniform prior over an expansive hypothesis class. Our key result shows that the learner's regret is not determined by the overall size of the hypothesis class, but rather by the cumulative probability of all models that are close, in Kullback-Leibler divergence distance, to the true data-generating process. We refer to this cumulative probability as the weight of the hypothesis. This leads to a natural notion of model simplicity: simple models are those with large weight and thus require fewer samples to generalize, while complex models have small weight and need more data. This perspective provides a rigorous and intuitive explanation for why over-parameterized models often avoid overfitting: the presence of simple hypotheses allows the posterior to concentrate on them when supported by the data. We further bridge theory and practice by recalling that stochastic gradient descent with Langevin dynamics samples from the correct posterior distribution, enabling our theoretical learner to be approximated using standard machine learning methods combined with ensemble learning. Our analysis yields non-uniform regret bounds and aligns with key practical concepts such as flat minima and model distillation. The results apply broadly across online, batch, and supervised learning settings, offering a unified and principled understanding of the generalization behavior of modern AI systems.

Underlying data structures, such as symmetries or invariances to transformations, are often exploited to improve the solution of learning tasks. However, embedding these properties in models or learning algorithms can be challenging and computationally intensive. Data augmentation, on the other hand, induces these symmetries during training by applying multiple transformations to the input data. Despite its ubiquity, its effectiveness depends on the choices of which transformations to apply, when to do so, and how often. In fact, there is both empirical and theoretical evidence that the indiscriminate use of data augmentation can introduce biases that outweigh its benefits. This work tackles these issues by automatically adapting the data augmentation while solving the learning task. To do so, it formulates data augmentation as an invariance-constrained learning problem and leverages Monte Carlo Markov Chain (MCMC) sampling to solve it. The result is a practical algorithm that not only does away with a priori searches for augmentation distributions, but also dynamically controls if and when data augmentation is applied. Our experiments illustrate the performance of this method, which achieves state-of-the-art results in automatic data augmentation benchmarks for CIFAR datasets. Furthermore, this approach can be used to gather insights on the actual symmetries underlying a learning task.

Fundamental tasks in computational chemistry, from transition state search to vibrational analysis, rely on molecular Hessians, which are the second derivatives of the potential energy. Yet, Hessians are computationally expensive to calculate and scale poorly with system size, with both quantum mechanical methods and neural networks. In this work, we demonstrate that Hessians can be predicted directly from a deep learning model, without relying on automatic differentiation or finite differences. We observe that one can construct SE(3)-equivariant, symmetric Hessians from irreducible representations (irrep) features up to degree l=2 computed during message passing in graph neural networks. This makes HIP Hessians one to two orders of magnitude faster, more accurate, more memory efficient, easier to train, and enables more favorable scaling with system size. We validate our predictions across a wide range of downstream tasks, demonstrating consistently superior performance for transition state search, accelerated geometry optimization, zero-point energy corrections, and vibrational analysis benchmarks. We open-source the HIP codebase and model weights to enable further development of the direct prediction of Hessians at https://github.com/BurgerAndreas/hip

Rigorous performance evaluation is essential for developing robust algorithms for high-throughput computational chemistry. Traditional benchmarking, however, often struggles to account for system-specific variability, making it difficult to form actionable conclusions. We present a Bayesian hierarchical modeling framework that rigorously quantifies performance metrics and their uncertainty, enabling a nuanced comparison of algorithmic strategies. We apply this framework to analyze the Dimer method, comparing Conjugate Gradient (CG) and L-BFGS rotation optimizers, with and without the removal of external rotations, across a benchmark of 500 molecular systems. Our analysis confirms that CG offers higher overall robustness than L-BFGS in this context. While the theoretically-motivated removal of external rotations led to higher computational cost (>40% more energy and force calls) for most systems in this set, our models also reveal a subtle interplay, hinting that this feature may improve the reliability of the L-BFGS optimizer. Rather than identifying a single superior method, our findings support the design of adaptive "chain of methods" workflows. This work showcases how a robust statistical paradigm can move beyond simple performance rankings to inform the intelligent, context-dependent application of computational chemistry methods.

In state-of-the-art self-supervised learning (SSL) pre-training produces semantically good representations by encouraging them to be invariant under meaningful transformations prescribed from human knowledge. In fact, the property of invariance is a trivial instance of a broader class called equivariance, which can be intuitively understood as the property that representations transform according to the way the inputs transform. Here, we show that rather than using only invariance, pre-training that encourages non-trivial equivariance to some transformations, while maintaining invariance to other transformations, can be used to improve the semantic quality of representations. Specifically, we extend popular SSL methods to a more general framework which we name Equivariant Self-Supervised Learning (E-SSL). In E-SSL, a simple additional pre-training objective encourages equivariance by predicting the transformations applied to the input. We demonstrate E-SSL's effectiveness empirically on several popular computer vision benchmarks, e.g. improving SimCLR to 72.5% linear probe accuracy on ImageNet. Furthermore, we demonstrate usefulness of E-SSL for applications beyond computer vision; in particular, we show its utility on regression problems in photonics science. Our code, datasets and pre-trained models are available at https://github.com/rdangovs/essl to aid further research in E-SSL.

Creating a single unified interatomic potential capable of attaining ab initio accuracy across all chemistry remains a long-standing challenge in computational chemistry and materials science. This work introduces a training protocol for foundation machine-learning interatomic potentials (MLIPs) that bridge molecular, surface, and materials chemistry through cross-domain learning. First, we introduce enhancements to the MACE architecture that improve its performance on chemically diverse databases by increasing weight sharing across chemical elements and introducing non-linear factors into the tensor decomposition of the product basis. Second, we develop a multi-head replay post-training methodology that enables efficient knowledge transfer across diverse chemical domains. By fine-tuning on datasets at different levels of electronic structure theory, including inorganic crystals, molecular systems, surface chemistry, and reactive organic chemistry, we demonstrate that a single unified model achieves state-of-the-art performance across several chemical domains. Comprehensive benchmarking reveals superior cross-domain transferability compared with existing specialised and multi-task models, with notable improvements in molecular and surface properties while maintaining state-of-the-art performance in materials-property prediction.

Recent findings suggest that consecutive layers of neural networks with the ReLU activation function fold the input space during the learning process. While many works hint at this phenomenon, an approach to quantify the folding was only recently proposed by means of a space folding measure based on Hamming distance in the ReLU activation space. We generalize this measure to a wider class of activation functions through introduction of equivalence classes of input data, analyse its mathematical and computational properties and come up with an efficient sampling strategy for its implementation. Moreover, it has been observed that space folding values increase with network depth when the generalization error is low, but decrease when the error increases. This underpins that learned symmetries in the data manifold (e.g., invariance under reflection) become visible in terms of space folds, contributing to the network's generalization capacity. Inspired by these findings, we outline a novel regularization scheme that encourages the network to seek solutions characterized by higher folding values.

Molecule generation is advancing rapidly in chemical discovery and drug design. Flow matching methods have recently set the state of the art (SOTA) in unconditional molecule generation, surpassing score-based diffusion models. However, diffusion models still lead in property-guided generation. In this work, we introduce PropMolFlow, a novel approach for property-guided molecule generation based on geometry-complete SE(3)-equivariant flow matching. Integrating five different property embedding methods with a Gaussian expansion of scalar properties, PropMolFlow outperforms previous SOTA diffusion models in conditional molecule generation across various properties while preserving the stability and validity of the generated molecules, consistent with its unconditional counterpart. Additionally, it enables faster inference with significantly fewer time steps compared to baseline models. We highlight the importance of validating the properties of generated molecules through DFT calculations performed at the same level of theory as the training data. Specifically, our analysis identifies properties that require DFT validation and others where a pretrained SE(3) geometric vector perceptron regressors provide sufficiently accurate predictions on generated molecules. Furthermore, we introduce a new property metric designed to assess the model's ability to propose molecules with underrepresented property values, assessing its capacity for out-of-distribution generalization. Our findings reveal shortcomings in existing structural metrics, which mistakenly validate open-shell molecules or molecules with invalid valence-charge configurations, underscoring the need for improved evaluation frameworks. Overall, this work paves the way for developing targeted property-guided generation methods, enhancing the design of molecular generative models for diverse applications.

Applications of machine learning techniques for materials modeling typically involve functions known to be equivariant or invariant to specific symmetries. While graph neural networks (GNNs) have proven successful in such tasks, they enforce symmetries via the model architecture, which often reduces their expressivity, scalability and comprehensibility. In this paper, we introduce (1) a flexible framework relying on stochastic frame-averaging (SFA) to make any model E(3)-equivariant or invariant through data transformations. (2) FAENet: a simple, fast and expressive GNN, optimized for SFA, that processes geometric information without any symmetrypreserving design constraints. We prove the validity of our method theoretically and empirically demonstrate its superior accuracy and computational scalability in materials modeling on the OC20 dataset (S2EF, IS2RE) as well as common molecular modeling tasks (QM9, QM7-X). A package implementation is available at https://faenet.readthedocs.io.

In diffraction-based crystal structure analysis, thermal ellipsoids, quantified via Anisotropic Displacement Parameters (ADPs), are critical yet challenging to determine. ADPs capture atomic vibrations, reflecting thermal and structural properties, but traditional computation is often expensive. This paper introduces CartNet, a novel graph neural network (GNN) for efficiently predicting crystal properties by encoding atomic geometry into Cartesian coordinates alongside the crystal temperature. CartNet integrates a neighbour equalization technique to emphasize covalent and contact interactions, and a Cholesky-based head to ensure valid ADP predictions. We also propose a rotational SO(3) data augmentation strategy during training to handle unseen orientations. An ADP dataset with over 200,000 experimental crystal structures from the Cambridge Structural Database (CSD) was curated to validate the approach. CartNet significantly reduces computational costs and outperforms existing methods in ADP prediction by 10.87%, while delivering a 34.77% improvement over theoretical approaches. We further evaluated CartNet on other datasets covering formation energy, band gap, total energy, energy above the convex hull, bulk moduli, and shear moduli, achieving 7.71% better results on the Jarvis Dataset and 13.16% on the Materials Project Dataset. These gains establish CartNet as a state-of-the-art solution for diverse crystal property predictions. Project website and online demo: https://www.ee.ub.edu/cartnet

Zero-shot Neural Architecture Search (NAS) typically optimises the architecture search process by exploiting the network or gradient properties at initialisation through zero-cost proxies. The existing proxies often rely on labelled data, which is usually unavailable in real-world settings. Furthermore, the majority of the current methods focus either on optimising the convergence and generalisation attributes or solely on the expressivity of the network architectures. To address both limitations, we first demonstrate how channel collinearity affects the convergence and generalisation properties of a neural network. Then, by incorporating the convergence, generalisation and expressivity in one approach, we propose a zero-cost proxy that omits the requirement of labelled data for its computation. In particular, we leverage the Singular Value Decomposition (SVD) of the neural network layer features and the extrinsic curvature of the network output to design our proxy. %As a result, the proposed proxy is formulated as the simplified harmonic mean of the logarithms of two key components: the sum of the inverse of the feature condition number and the extrinsic curvature of the network output. Our approach enables accurate prediction of network performance on test data using only a single label-free data sample. Our extensive evaluation includes a total of six experiments, including the Convolutional Neural Network (CNN) search space, i.e. DARTS and the Transformer search space, i.e. AutoFormer. The proposed proxy demonstrates a superior performance on multiple correlation benchmarks, including NAS-Bench-101, NAS-Bench-201, and TransNAS-Bench-101-micro; as well as on the NAS task within the DARTS and the AutoFormer search space, all while being notably efficient. The code is available at https://github.com/rohanasthana/Dextr.

Cancer is the second leading cause of death, with chemotherapy as one of the primary forms of treatment. As a result, researchers are turning to drug combination therapy to decrease drug resistance and increase efficacy. Current methods of drug combination screening, such as in vivo and in vitro, are inefficient due to stark time and monetary costs. In silico methods have become increasingly important for screening drugs, but current methods are inaccurate and generalize poorly to unseen anticancer drugs. In this paper, I employ a geometric deep-learning model utilizing a graph attention network that is equivariant to 3D rotations, translations, and reflections with structural motifs. Additionally, the gene expression of cancer cell lines is utilized to classify synergistic drug combinations specific to each cell line. I compared the proposed geometric deep learning framework to current state-of-the-art (SOTA) methods, and the proposed model architecture achieved greater performance on all 12 benchmark tasks performed on the DrugComb dataset. Specifically, the proposed framework outperformed other SOTA methods by an accuracy difference greater than 28%. Based on these results, I believe that the equivariant graph attention network's capability of learning geometric data accounts for the large performance improvements. The model's ability to generalize to foreign drugs is thought to be due to the structural motifs providing a better representation of the molecule. Overall, I believe that the proposed equivariant geometric deep learning framework serves as an effective tool for virtually screening anticancer drug combinations for further validation in a wet lab environment. The code for this work is made available online at: https://github.com/WeToTheMoon/EGAT_DrugSynergy.

We present Symphony, an E(3)-equivariant autoregressive generative model for 3D molecular geometries that iteratively builds a molecule from molecular fragments. Existing autoregressive models such as G-SchNet and G-SphereNet for molecules utilize rotationally invariant features to respect the 3D symmetries of molecules. In contrast, Symphony uses message-passing with higher-degree E(3)-equivariant features. This allows a novel representation of probability distributions via spherical harmonic signals to efficiently model the 3D geometry of molecules. We show that Symphony is able to accurately generate small molecules from the QM9 dataset, outperforming existing autoregressive models and approaching the performance of diffusion models.

There often is a dilemma between ease of optimization and robust out-of-distribution (OoD) generalization. For instance, many OoD methods rely on penalty terms whose optimization is challenging. They are either too strong to optimize reliably or too weak to achieve their goals. We propose to initialize the networks with a rich representation containing a palette of potentially useful features, ready to be used by even simple models. On the one hand, a rich representation provides a good initialization for the optimizer. On the other hand, it also provides an inductive bias that helps OoD generalization. Such a representation is constructed with the Rich Feature Construction (RFC) algorithm, also called the Bonsai algorithm, which consists of a succession of training episodes. During discovery episodes, we craft a multi-objective optimization criterion and its associated datasets in a manner that prevents the network from using the features constructed in the previous iterations. During synthesis episodes, we use knowledge distillation to force the network to simultaneously represent all the previously discovered features. Initializing the networks with Bonsai representations consistently helps six OoD methods achieve top performance on ColoredMNIST benchmark. The same technique substantially outperforms comparable results on the Wilds Camelyon17 task, eliminates the high result variance that plagues other methods, and makes hyperparameter tuning and model selection more reliable.

We generalize the Hamiltonian Monte Carlo algorithm with a stack of neural network layers and evaluate its ability to sample from different topologies in a two dimensional lattice gauge theory. We demonstrate that our model is able to successfully mix between modes of different topologies, significantly reducing the computational cost required to generated independent gauge field configurations. Our implementation is available at https://github.com/saforem2/l2hmc-qcd .

Machine learning techniques are essential tools to compute efficient, yet accurate, force fields for atomistic simulations. This approach has recently been extended to incorporate quantum computational methods, making use of variational quantum learning models to predict potential energy surfaces and atomic forces from ab initio training data. However, the trainability and scalability of such models are still limited, due to both theoretical and practical barriers. Inspired by recent developments in geometric classical and quantum machine learning, here we design quantum neural networks that explicitly incorporate, as a data-inspired prior, an extensive set of physically relevant symmetries. We find that our invariant quantum learning models outperform their more generic counterparts on individual molecules of growing complexity. Furthermore, we study a water dimer as a minimal example of a system with multiple components, showcasing the versatility of our proposed approach and opening the way towards larger simulations. Our results suggest that molecular force fields generation can significantly profit from leveraging the framework of geometric quantum machine learning, and that chemical systems represent, in fact, an interesting and rich playground for the development and application of advanced quantum machine learning tools.

Recent advances in large language models (LLMs) have led to models that tackle diverse molecular tasks, such as chemical reaction prediction and molecular property prediction. Large-scale molecular instruction-tuning datasets have enabled sequence-only (e.g., SMILES or SELFIES) generalist molecular LLMs, and researchers are now exploring multimodal approaches that incorporate molecular structural information for further gains. However, a genuinely multimodal, generalist LLM that covers a broad spectrum of molecular tasks has yet to be fully investigated. We observe that naive next token prediction training ignores graph-structural information, limiting an LLM's ability to exploit molecular graphs. To address this, we propose (i) Molecular structure Preference Optimization (MolPO), which facilitates graph usage by optimizing preferences between pairs of correct and perturbed molecular structures, and (ii) an advanced graph encoder with a tailored pre-training strategy to improve the effect of graph utilization by MolPO. Building on these contributions, we introduce Mol-LLM, the first multimodal generalist model that (a) handles a broad spectrum of molecular tasks among molecular LLMs, (b) explicitly leverages molecular-structure information, and (c) takes advantage of extensive instruction tuning. Mol-LLM attains state-of-the-art or comparable results across the most comprehensive molecular-LLM benchmark-even on out-of-distribution datasets for reaction and property prediction, where it surpasses prior generalist molecular LLMs by a large margin.

Recently, sophisticated deep learning-based approaches have been developed for generating efficient initial guesses to accelerate the convergence of density functional theory (DFT) calculations. While the actual initial guesses are often density matrices (DM), quantities that can convert into density matrices also qualify as alternative forms of initial guesses. Hence, existing works mostly rely on the prediction of the Hamiltonian matrix for obtaining high-quality initial guesses. However, the Hamiltonian matrix is both numerically difficult to predict and intrinsically non-transferable, hindering the application of such models in real scenarios. In light of this, we propose a method that constructs DFT initial guesses by predicting the electron density in a compact auxiliary basis representation using E(3)-equivariant neural networks. Trained on small molecules with up to 20 atoms, our model is able to achieve an average 33.3% self-consistent field (SCF) step reduction on systems up to 60 atoms, substantially outperforming Hamiltonian-centric and DM-centric models. Critically, this acceleration remains nearly constant with increasing system sizes and exhibits strong transferring behaviors across orbital basis sets and exchange-correlation (XC) functionals. To the best of our knowledge, this work represents the first and robust candidate for a universally transferable DFT acceleration method. We are also releasing the SCFbench dataset and its accompanying code to facilitate future research in this promising direction.

Invariance learning algorithms that conditionally filter out domain-specific random variables as distractors, do so based only on the data semantics, and not the target domain under evaluation. We show that a provably optimal and sample-efficient way of learning conditional invariances is by relaxing the invariance criterion to be non-commutatively directed towards the target domain. Under domain asymmetry, i.e., when the target domain contains semantically relevant information absent in the source, the risk of the encoder varphi^* that is optimal on average across domains is strictly lower-bounded by the risk of the target-specific optimal encoder Phi^*_tau. We prove that non-commutativity steers the optimization towards Phi^*_tau instead of varphi^*, bringing the H-divergence between domains down to zero, leading to a stricter bound on the target risk. Both our theory and experiments demonstrate that non-commutative invariance (NCI) can leverage source domain samples to meet the sample complexity needs of learning Phi^*_tau, surpassing SOTA invariance learning algorithms for domain adaptation, at times by over 2%, approaching the performance of an oracle. Implementation is available at https://github.com/abhrac/nci.

This paper proposes a new optimization algorithm called Entropy-SGD for training deep neural networks that is motivated by the local geometry of the energy landscape. Local extrema with low generalization error have a large proportion of almost-zero eigenvalues in the Hessian with very few positive or negative eigenvalues. We leverage upon this observation to construct a local-entropy-based objective function that favors well-generalizable solutions lying in large flat regions of the energy landscape, while avoiding poorly-generalizable solutions located in the sharp valleys. Conceptually, our algorithm resembles two nested loops of SGD where we use Langevin dynamics in the inner loop to compute the gradient of the local entropy before each update of the weights. We show that the new objective has a smoother energy landscape and show improved generalization over SGD using uniform stability, under certain assumptions. Our experiments on convolutional and recurrent networks demonstrate that Entropy-SGD compares favorably to state-of-the-art techniques in terms of generalization error and training time.

We systematically study antithetic initial noise in diffusion models, discovering that pairing each noise sample with its negation consistently produces strong negative correlation. This universal phenomenon holds across datasets, model architectures, conditional and unconditional sampling, and even other generative models such as VAEs and Normalizing Flows. To explain it, we combine experiments and theory and propose a symmetry conjecture that the learned score function is approximately affine antisymmetric (odd symmetry up to a constant shift), supported by empirical evidence. This negative correlation leads to substantially more reliable uncertainty quantification with up to 90% narrower confidence intervals. We demonstrate these gains on tasks including estimating pixel-wise statistics and evaluating diffusion inverse solvers. We also provide extensions with randomized quasi-Monte Carlo noise designs for uncertainty quantification, and explore additional applications of the antithetic noise design to improve image editing and generation diversity. Our framework is training-free, model-agnostic, and adds no runtime overhead. Code is available at https://github.com/jjia131/Antithetic-Noise-in-Diffusion-Models-page.

Symmetry-based neural networks often constrain the architecture in order to achieve invariance or equivariance to a group of transformations. In this paper, we propose an alternative that avoids this architectural constraint by learning to produce a canonical representation of the data. These canonicalization functions can readily be plugged into non-equivariant backbone architectures. We offer explicit ways to implement them for many groups of interest. We show that this approach enjoys universality while providing interpretable insights. Our main hypothesis is that learning a neural network to perform canonicalization is better than using predefined heuristics. Our results show that learning the canonicalization function indeed leads to better results and that the approach achieves excellent performance in practice.

Single-source domain generalization attempts to learn a model on a source domain and deploy it to unseen target domains. Limiting access only to source domain data imposes two key challenges - how to train a model that can generalize and how to verify that it does. The standard practice of validation on the training distribution does not accurately reflect the model's generalization ability, while validation on the test distribution is a malpractice to avoid. In this work, we construct an independent validation set by transforming source domain images with a comprehensive list of augmentations, covering a broad spectrum of potential distribution shifts in target domains. We demonstrate a high correlation between validation and test performance for multiple methods and across various datasets. The proposed validation achieves a relative accuracy improvement over the standard validation equal to 15.4% or 1.6% when used for method selection or learning rate tuning, respectively. Furthermore, we introduce a novel family of methods that increase the shape bias through enhanced edge maps. To benefit from the augmentations during training and preserve the independence of the validation set, a k-fold validation process is designed to separate the augmentation types used in training and validation. The method that achieves the best performance on the augmented validation is selected from the proposed family. It achieves state-of-the-art performance on various standard benchmarks. Code at: https://github.com/NikosEfth/crafting-shifts

Foundation models have been transformational in machine learning fields such as natural language processing and computer vision. Similar success in atomic property prediction has been limited due to the challenges of training effective models across multiple chemical domains. To address this, we introduce Joint Multi-domain Pre-training (JMP), a supervised pre-training strategy that simultaneously trains on multiple datasets from different chemical domains, treating each dataset as a unique pre-training task within a multi-task framework. Our combined training dataset consists of sim120M systems from OC20, OC22, ANI-1x, and Transition-1x. We evaluate performance and generalization by fine-tuning over a diverse set of downstream tasks and datasets including: QM9, rMD17, MatBench, QMOF, SPICE, and MD22. JMP demonstrates an average improvement of 59% over training from scratch, and matches or sets state-of-the-art on 34 out of 40 tasks. Our work highlights the potential of pre-training strategies that utilize diverse data to advance property prediction across chemical domains, especially for low-data tasks.