arXiv subjects

Ron Levie

Publications and source records attributed to Ron Levie.

At least 19 recordsLinked to original sources

arXiv2026

Out-of-Distribution Inverse Design of Elastic Networks with Differentiable Graph Neural Network Molecular Dynamics

Machine-learning-based inverse design can accelerate the discovery of materials with targeted properties, but conventional structure--property models often require large training datasets and generalize poorly beyond their training distribution. Here, we present a differentiable inverse design framework based on a graph neural network molecular dynamics simulator. By combining a short dynamical initialization with physics-based refinement during simulation, the framework enables optimization well beyond the conditions represented in the training data. Using disordered elastic networks, we show that a simulator trained only on non-auxetic systems with Poisson's ratios between 0.1 and 0.4 can design strongly auxetic networks with values as low as -0.3. The framework also produces stress--strain responses outside the range observed during training, creates localized mechanical defects absent from the training data, and generalizes across system size, enabling optimization of networks tested up to 5000 nodes despite being trained only on networks with fewer than 200 nodes.

physics.chem-ph↗

arXiv2026

Beyond Oversquashing: Understanding Signal Propagation in GNNs Via Observables

Graph Neural Networks (GNNs) perform computations on graphs by routing the signal between graph regions using a graph shift operator or a message passing scheme. Often, the propagation of the signal leads to a loss of information, where the signal tends to diffuse across the graph instead of being deliberately routed between regions of interest. Two notions that depict this phenomenon are oversmoothing and oversquashing. In this paper, we propose an alternative approach for modeling signal propagation, inspired by quantum mechanics, using the notion of observables. Specifically, we model the place in the graph where the signal lies, how much the signal is concentrated there, and how much of the signal is propagated towards a location of interest when applying a GNN. Using these new concepts, we prove that standard spectral GNNs have poor signal propagation capabilities. We then propose a new type of spectral GNN, termed Schr\"odinger GNN, which we show has a superior capacity to route the signal across the graph.

cs.LG↗

arXiv2026

Graph Learning Should Move Beyond Restrictive Views of Spectral and Message-Passing GNNs

Graph neural networks (GNNs) are commonly divided into message-passing neural networks (MPNNs) and spectral GNNs, reflecting two largely separate research traditions in machine learning and signal processing. While MPNNs have a precise definition, there is no widely accepted criterion for what makes a mapping a spectral GNN. Most existing work restricts spectral GNNs to layered architectures based on linear spectral filters. Under this restriction, we show that spectral and spatial GNNs have largely equivalent expressive power. To promote progress in the field, we propose a precise definition of spectral GNNs based on eigenbasis symmetries, in contrast to the definition of MPNNs via neighborhood permutation symmetries. We further argue that the two perspectives offer complementary strengths. MPNNs provide a natural language for discrete structure and expressivity analysis through tools from logic and graph isomorphism, while the spectral perspective offers principled tools for understanding smoothing, bottlenecks, stability, and community structure. Overall, we argue that progress in graph learning will be accelerated by clarifying the similarities and differences between these perspectives and by moving toward a unified theoretical framework.

cs.LG↗

arXiv2026

A Graphop Analysis of Graph Neural Networks on Sparse Graphs: Generalization and Universal Approximation

Generalization and approximation capabilities of message passing graph neural networks (MPNNs) are often studied by defining a compact metric on a space of input graphs under which MPNNs are equicontinuous. Such analyses are of two varieties: 1) when the metric space includes graphs of unbounded sizes, the theory is only appropriate for dense graphs, and, 2) when studying sparse graphs, the metric space only includes graphs of uniformly bounded size. In this work, we present a unified approach, defining a compact metric on the space of graphs of all sizes, both sparse and dense, under which MPNNs are equicontinuous. This leads to more powerful universal approximation theorems and generalization bounds than previous works. The theory is based on, and extends, a recent approach to graph limit theory called graphop analysis.

cs.LG↗

arXiv2026

Neural Networks With Dense Weights Are Not Universal Approximators

We investigate the approximation capabilities of dense neural networks. While universal approximation theorems establish that sufficiently large architectures can approximate arbitrary continuous functions if there are no restrictions on the weight values, we show that dense neural networks do not possess this universality. Our argument is based on a model compression approach, combining the weak regularity lemma with an interpretation of feedforward networks as message passing graph neural networks. We consider ReLU neural networks subject to natural constraints on weights and input and output dimensions, which model a notion of dense connectivity. Within this setting, we demonstrate the existence of Lipschitz continuous functions that cannot be approximated by such networks. This highlights intrinsic limitations of neural networks with dense layers and motivates the use of sparse connectivity as a necessary ingredient for achieving true universality.

cs.LG↗

arXiv2026

A Non Linear Spectral Graph Neural Network Simulator for More Stable and Accurate Rollouts

Molecular dynamics (MD) simulations are a central tool in science and engineering enabling the study of dynamical behavior and the link between microscopic structure and macroscopic function. Their high computational cost, however, has motivated extensive efforts to develop accelerated alternatives. A promising approach is the use of machine-learning-based simulators that allow for substantially larger time steps than conventional MD. Among these, graph neural network (GNN)-based methods have been found to be especially attractive given that they naturally encode the inductive bias of interacting particle systems, however current architectures remain limited in accuracy and stability. In particular, standard message-passing schemes struggle to efficiently propagate long-range information. Here, we investigate whether spectral-GNN simulators can overcome these limitations by explicitly representing a simulated system in a global eigenmode basis, and therefore better capture long-range and collective behavior. Focusing on disordered elastic networks, model systems for complex materials and biological structures such as proteins, we compare spatial, linear spectral, and nonlinear spectral GNN architectures. We find that while spectral representations alone are not sufficient, nonlinear spectral models substantially outperform alternatives. By learning both the time-dependent dynamics and the mixing of eigenmodes, these models more accurately capture the system's slow, global modes, which dominate macroscopic behavior. This leads to a marked reduction in systematic particle-position error and significantly improved prediction of global physical properties.

physics.chem-ph↗

arXiv2025

Adaptive Canonicalization with Application to Invariant Anisotropic Geometric Networks

Canonicalization is a widely used strategy in equivariant machine learning, enforcing symmetry in neural networks by mapping each input to a standard form. Yet, it often introduces discontinuities that can affect stability during training, limit generalization, and complicate universal approximation theorems. In this paper, we address this by introducing adaptive canonicalization, a general framework in which the canonicalization depends both on the input and the network. Specifically, we present the adaptive canonicalization based on prior maximization, where the standard form of the input is chosen to maximize the predictive confidence of the network. We prove that this construction yields continuous and symmetry-respecting models that admit universal approximation properties. We propose two applications of our setting: (i) resolving eigenbasis ambiguities in spectral graph neural networks, and (ii) handling rotational symmetries in point clouds. We empirically validate our methods on molecular and protein classification, as well as point cloud classification tasks. Our adaptive canonicalization outperforms the three other common solutions to equivariant machine learning: data augmentation, standard canonicalization, and equivariant architectures.

cs.LG↗

arXiv2025

A Note on Graphon-Signal Analysis of Graph Neural Networks

A recent paper, ``A Graphon-Signal Analysis of Graph Neural Networks'', by Levie, analyzed message passing graph neural networks (MPNNs) by embedding the input space of MPNNs, i.e., attributed graphs (graph-signals), to a space of attributed graphons (graphon-signals). Based on extensions of standard results in graphon analysis to graphon-signals, the paper proved a generalization bound and a sampling lemma for MPNNs. However, there are some missing ingredients in that paper, limiting its applicability in practical settings of graph machine learning. In the current paper, we introduce several refinements and extensions to existing results that address these shortcomings. In detail, 1) we extend the main results in the paper to graphon-signals with multidimensional signals (rather than 1D signals), 2) we extend the Lipschitz continuity to MPNNs with readout with respect to cut distance (rather than MPNNs without readout with respect to cut metric), 3) we improve the generalization bound by utilizing robustness-type generalization bounds, and 4) we extend the analysis to non-symmetric graphons and kernels.

cs.LG↗

arXiv2025

Equivariance Everywhere All At Once: A Recipe for Graph Foundation Models

Graph machine learning architectures are typically tailored to specific tasks on specific datasets, which hinders their broader applicability. This has led to a new quest in graph machine learning: how to build graph foundation models capable of generalizing across arbitrary graphs and features? In this work, we present a recipe for designing graph foundation models for node-level tasks from first principles. The key ingredient underpinning our study is a systematic investigation of the symmetries that a graph foundation model must respect. In a nutshell, we argue that label permutation-equivariance alongside feature permutation-invariance are necessary in addition to the common node permutation-equivariance on each local neighborhood of the graph. To this end, we first characterize the space of linear transformations that are equivariant to permutations of nodes and labels, and invariant to permutations of features. We then prove that the resulting network is a universal approximator on multisets that respect the aforementioned symmetries. Our recipe uses such layers on the multiset of features induced by the local neighborhood of the graph to obtain a class of graph foundation models for node property prediction. We validate our approach through extensive experiments on 29 real-world node classification datasets, demonstrating both strong zero-shot empirical performance and consistent improvement as the number of training graphs increases.

cs.LG↗

arXiv2025

Dynamical Data for More Efficient and Generalizable Learning: A Case Study in Disordered Elastic Networks

Machine learning models often require large datasets and struggle to generalize beyond their training distribution. These limitations pose significant challenges in scientific and engineering contexts, where generating exhaustive datasets is often impractical and the goal is frequently to discover novel solutions outside the training domain. In this work, we explore the use of dynamical data through a graph neural network-based simulator to enable efficient system-to-property learning and out-of-distribution prediction in the context of uniaxial compression of two-dimensional disordered elastic networks. We find that the simulator can learn the underlying physical dynamics from a small number of training examples and accurately reproduce the temporal evolution of unseen networks. Notably, the simulator is able to accurately predict emergent properties such as the Poisson's ratio and its dependence on strain, even though it was not explicitly trained for this task. In addition, it generalizes well across variations in system temperature, strain amplitude, and most significantly, Poisson's ratios beyond the training range. These findings suggest that using dynamical data to train machine learning models can support more data efficient and generalizable approaches for materials and molecular design, especially in data-scarce settings.

physics.chem-ph↗

arXiv2025

Efficient Learning on Large Graphs using a Densifying Regularity Lemma

Learning on large graphs presents significant challenges, with traditional Message Passing Neural Networks suffering from computational and memory costs scaling linearly with the number of edges. We introduce the Intersecting Block Graph (IBG), a low-rank factorization of large directed graphs based on combinations of intersecting bipartite components, each consisting of a pair of communities, for source and target nodes. By giving less weight to non-edges, we show how to efficiently approximate any graph, sparse or dense, by a dense IBG. Specifically, we prove a constructive version of the weak regularity lemma, showing that for any chosen accuracy, every graph, regardless of its size or sparsity, can be approximated by a dense IBG whose rank depends only on the accuracy. This dependence of the rank solely on the accuracy, and not on the sparsity level, is in contrast to previous forms of the weak regularity lemma. We present a graph neural network architecture operating on the IBG representation of the graph and demonstrating competitive performance on node classification, spatio-temporal graph analysis, and knowledge graph completion, while having memory and computational complexity linear in the number of nodes rather than edges.

cs.SI↗

arXiv2025

Survey on Generalization Theory for Graph Neural Networks

Message-passing graph neural networks (MPNNs) have emerged as the leading approach for machine learning on graphs, attracting significant attention in recent years. While a large set of works explored the expressivity of MPNNs, i.e., their ability to separate graphs and approximate functions over them, comparatively less attention has been directed toward investigating their generalization abilities, i.e., making meaningful predictions beyond the training data. Here, we systematically review the existing literature on the generalization abilities of MPNNs. We analyze the strengths and limitations of various studies in these domains, providing insights into their methodologies and findings. Furthermore, we identify potential avenues for future research, aiming to deepen our understanding of the generalization abilities of MPNNs.

cs.LG↗

arXiv2024

Covered Forest: Fine-grained generalization analysis of graph neural networks

The expressive power of message-passing graph neural networks (MPNNs) is reasonably well understood, primarily through combinatorial techniques from graph isomorphism testing. However, MPNNs' generalization abilities -- making meaningful predictions beyond the training set -- remain less explored. Current generalization analyses often overlook graph structure, limit the focus to specific aggregation functions, and assume the impractical, hard-to-optimize $0$-$1$ loss function. Here, we extend recent advances in graph similarity theory to assess the influence of graph structure, aggregation, and loss functions on MPNNs' generalization abilities. Our empirical study supports our theoretical insights, improving our understanding of MPNNs' generalization properties.

cs.LG↗

arXiv2024

Generalization, Expressivity, and Universality of Graph Neural Networks on Attributed Graphs

We analyze the universality and generalization of graph neural networks (GNNs) on attributed graphs, i.e., with node attributes. To this end, we propose pseudometrics over the space of all attributed graphs that describe the fine-grained expressivity of GNNs. Namely, GNNs are both Lipschitz continuous with respect to our pseudometrics and can separate attributed graphs that are distant in the metric. Moreover, we prove that the space of all attributed graphs is relatively compact with respect to our metrics. Based on these properties, we prove a universal approximation theorem for GNNs and generalization bounds for GNNs on any data distribution of attributed graphs. The proposed metrics compute the similarity between the structures of attributed graphs via a hierarchical optimal transport between computation trees. Our work extends and unites previous approaches which either derived theory only for graphs with no attributes, derived compact metrics under which GNNs are continuous but without separation power, or derived metrics under which GNNs are continuous and separate points but the space of graphs is not relatively compact, which prevents universal approximation and generalization analysis.

cs.LG↗

arXiv2024

PieClam: A Universal Graph Autoencoder Based on Overlapping Inclusive and Exclusive Communities

We propose PieClam (Prior Inclusive Exclusive Cluster Affiliation Model): a probabilistic graph model for representing any graph as overlapping generalized communities. Our method can be interpreted as a graph autoencoder: nodes are embedded into a code space by an algorithm that maximizes the log-likelihood of the decoded graph, given the input graph. PieClam is a community affiliation model that extends well-known methods like BigClam in two main manners. First, instead of the decoder being defined via pairwise interactions between the nodes in the code space, we also incorporate a learned prior on the distribution of nodes in the code space, turning our method into a graph generative model. Secondly, we generalize the notion of communities by allowing not only sets of nodes with strong connectivity, which we call inclusive communities, but also sets of nodes with strong disconnection, which we call exclusive communities. To model both types of communities, we propose a new type of decoder based the Lorentz inner product, which we prove to be much more expressive than standard decoders based on standard inner products or norm distances. By introducing a new graph similarity measure, that we call the log cut distance, we show that PieClam is a universal autoencoder, able to uniformly approximately reconstruct any graph. Our method is shown to obtain competitive performance in graph anomaly detection benchmarks.

cs.LG↗

arXiv2024

Equivariant Machine Learning on Graphs with Nonlinear Spectral Filters

Equivariant machine learning is an approach for designing deep learning models that respect the symmetries of the problem, with the aim of reducing model complexity and improving generalization. In this paper, we focus on an extension of shift equivariance, which is the basis of convolution networks on images, to general graphs. Unlike images, graphs do not have a natural notion of domain translation. Therefore, we consider the graph functional shifts as the symmetry group: the unitary operators that commute with the graph shift operator. Notably, such symmetries operate in the signal space rather than directly in the spatial space. We remark that each linear filter layer of a standard spectral graph neural network (GNN) commutes with graph functional shifts, but the activation function breaks this symmetry. Instead, we propose nonlinear spectral filters (NLSFs) that are fully equivariant to graph functional shifts and show that they have universal approximation properties. The proposed NLSFs are based on a new form of spectral domain that is transferable between graphs. We demonstrate the superior performance of NLSFs over existing spectral GNNs in node and graph classification benchmarks.

cs.LG↗

arXiv2024

Learning on Large Graphs using Intersecting Communities

Message Passing Neural Networks (MPNNs) are a staple of graph machine learning. MPNNs iteratively update each node's representation in an input graph by aggregating messages from the node's neighbors, which necessitates a memory complexity of the order of the number of graph edges. This complexity might quickly become prohibitive for large graphs provided they are not very sparse. In this paper, we propose a novel approach to alleviate this problem by approximating the input graph as an intersecting community graph (ICG) -- a combination of intersecting cliques. The key insight is that the number of communities required to approximate a graph does not depend on the graph size. We develop a new constructive version of the Weak Graph Regularity Lemma to efficiently construct an approximating ICG for any input graph. We then devise an efficient graph learning algorithm operating directly on ICG in linear memory and time with respect to the number of nodes (rather than edges). This offers a new and fundamentally different pipeline for learning on very large non-sparse graphs, whose applicability is demonstrated empirically on node classification tasks and spatio-temporal data processing.

cs.LG↗

arXiv2024

Generalization Bounds for Message Passing Networks on Mixture of Graphons

We study the generalization capabilities of Message Passing Neural Networks (MPNNs), a prevalent class of Graph Neural Networks (GNN). We derive generalization bounds specifically for MPNNs with normalized sum aggregation and mean aggregation. Our analysis is based on a data generation model incorporating a finite set of template graphons. Each graph within this framework is generated by sampling from one of the graphons with a certain degree of perturbation. In particular, we extend previous MPNN generalization results to a more realistic setting, which includes the following modifications: 1) we analyze simple random graphs with Bernoulli-distributed edges instead of weighted graphs; 2) we sample both graphs and graph signals from perturbed graphons instead of clean graphons; and 3) we analyze sparse graphs instead of dense graphs. In this more realistic and challenging scenario, we provide a generalization bound that decreases as the average number of nodes in the graphs increases. Our results imply that MPNNs with higher complexity than the size of the training set can still generalize effectively, as long as the graphs are sufficiently large.

cs.LG↗