Equivariant GNNs

GNN: Graph Neural Network

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4 papers in the last four weeks, against 1 the four weeks before. 0.0% of all new papers.

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Latest papers 24

Oct 5, 2026cs.LG

FlashCart: Fast Cartesian Tensor Products for Equivariant Interatomic Potentials

Machine-learned interatomic potentials extend atomistic simulations beyond the length- and timescales accessible to electronic-structure methods. However, the computational cost of equivariant architectures limits the local correlations they can represent in practice and therefore their achievable accuracy. Here we introduce FlashCart, which makes higher-order correlations affordable by combining generated GPU kernels with an architecture that recursively builds equivariant features and compresses them to a fixed width at each step. We express tensor products in independent Cartesian components and symbolically simplify them and their derivatives, producing fused kernels that often outperform optimized spherical counterparts. We then show that increasing correlation order improves accuracy more efficiently than increasing width, depth, or tensor rank. On SPICE-MACE-OFF, FlashCart models advance the measured accuracy-efficiency frontier: a model with

5.6

million parameters achieves lower energy and force errors and

10\times

faster inference than a transformer with

189

million parameters.

Viktor Zaverkin, Payman Goodarzi, Sergey V. Sukhomlinov +4

INM – Leibniz Institute for New Materials · Saarland University · German Research Center for Artificial Intelligence (DFKI) +2

Machine Learning Interatomic Potentials GPU Kernel Optimization Equivariant GNNs

Oct 1, 2026cs.LG

CrossGMN: Graph Metanetworks for Cross-Architecture Weight-Space Transformations

Weight-space networks operate directly on parameters of other neural networks, enabling tasks such as predicting model properties, editing trained models, and generating weights. Weight-space symmetries such as neuron permutations make equivariance a key design principle. However, existing equivariant weight-space architectures have primarily been studied for transformations that preserve the network architecture. In contrast, many practical transformations, including model compression and upscaling, map a trained source network into a target network with a different architecture. In this setting, the source and target permutation symmetries act on different parameter spaces, making equivariance less straightforward to formulate. Our key idea for addressing this mismatch is to reformulate cross-architecture operators with two inputs: a trained source network and an initialization of the target network. This lets us define equivariant cross-architecture operators that refine the initialization of the target network using information from the source network, while being invariant to source-network permutations and equivariant to target-network permutations. Based on this formulation, we introduce CrossGMN, a graph metanetwork that jointly processes both networks through symmetry-preserving cross-network message passing. We prove CrossGMN is universal for continuous cross-architecture operators on compact sets under a general-position assumption. We evaluate CrossGMN for model compression, predicting a smaller network's parameters to accelerate subsequent knowledge distillation. Across 2-D and 3-D INRs and image classification with MLPs, CNNs, and Vision Transformers, CrossGMN speeds up distillation by up to 8.89x, transfers across datasets without retraining (3.78x), and a single model can accelerate compression from heterogeneous source architectures into a common target architecture.

Adir Dayan, Yam Eitan, Haggai Maron

Technion – Israel Institute of Technology · NVIDIA Research

Model Compression Hypernetworks Weight-Space Analysis Equivariant GNNs

Oct 1, 2026cs.LG

Clifford Sheaf Neural Networks

We introduce the Clifford Sheaf Neural Network (CSNN), an equivariant sheaf neural network for geometric graphs that places a Clifford algebra on each stalk of a cellular sheaf and transports multivector features along edges. The canonical choice of restriction map for sheaves with algebra-valued stalks is algebra homomorphism. Adding the constraint of equivariance, the naive choice becomes versor conjugation. However, versor conjugation is expressively weak, so we drop algebra homomorphism and arrive at the K-term sandwich. The resulting sheaf Laplacian is positive semidefinite by construction, needs no versor constraint, and still mixes grades. Our main contribution characterizes the resulting family of restriction maps along three axes: which grades a map couples, how much of the endomorphism space it reaches, and how well it is conditioned. The K-term sandwich spans half of the endomorphism space, and in Cl(3, 0, 0) it corresponds to the maps that commute with the central pseudoscalar. The number of terms controls expressivity. CSNN is the reversion member, a first-order model by construction and the grade-mixing corner of this family, developed as a sheaf construction for graph-level equivariant regression.

Kotaro Kamiya, Joel Nicholls

SyntheticGestalt KK

Sheaf Neural Networks Equivariant GNNs Sheaf Theory

Sep 28, 2026cs.LG

Multi-Attractor GNNs: Set-Valued Expressivity Beyond Unique Equilibria

Recurrent and equilibrium graph neural networks (GNNs) often enforce a unique fixed point or use one training target per graph. Yet many combinatorial and scientific problems admit multiple valid solutions, with no preferred one. A designated target can then impose an arbitrary selection rule. For tasks invariant to node relabeling, a symmetric graph may have a symmetric solution set but no symmetric solution. We show that multiple equilibria enable one weight-tied message-passing GNN to represent set-valued equivariant maps: different initializations approach different valid solutions. Under stated regularity assumptions, we first construct globally Lipschitz, permutation-equivariant dynamics that converge almost surely to valid solutions and reach every solution branch with positive probability. We then establish approximate realization by recurrent message passing with continuous component maps, with arbitrarily small update and limiting errors and arbitrarily high probability. This goes beyond standard universality arguments: although message passing alone cannot distinguish symmetric nodes, the evolving state keeps nodes distinguishable at every finite step without auxiliary node identifiers. Such dynamics can be learned without solution labels using problem-specific energies. On Ising ground states, structural module detection in protein graphs, and chemical reaction steady states, the learned updates produce multiple high-quality predictions with high numerical convergence rates. They achieve better average solution quality than the tested unique-equilibrium, single-target, and feedforward baselines, while remaining competitive with much larger diffusion-based solvers.

Jialin Liu

GNN Expressivity Combinatorial Optimization Equivariant GNNs

Sep 28, 2026cs.LG

Edge-Level Automorphism in GNNs: A Quantitative Framework and Effective Designs For Link Prediction

Graph Neural Networks (GNNs) are effective for learning node and link embeddings through permutation-equivariant aggregation. However, standard GNNs collapse automorphic nodes, i.e., those with identical structural roles (or orbits) into indistinguishable representations, leading to the node automorphism problem. This collapse limits their expressive power and degrades link prediction performance. Existing approaches to characterize GNN expressiveness rely primarily on Weisfeiler-Lehman (WL) analyses, but these methods are typically qualitative and often misaligned with empirical results. To address this gap, we begin by introducing a novel quantitative framework to assess GNN expressiveness for link prediction. We first formalize edge-level automorphism through edge orbits, which capture the set of structural role pairs for nodes that share a link. Then, we introduce the edge automorphism ratio (EAR), a scalar metric that quantifies a GNN's ability to distinguish links in a given graph. We empirically demonstrate that EAR correlates strongly with performance, validating its practical benefit. Building on this insight, we design EDGE-ORBIT EQUIVARIANT GRAPH NEURAL NETWORK (EO-GNN), a GNN architecture that addresses automorphism collapse while preserving equivariance and incurring minimal computational overhead. EO-GNN accomplishes this through two core designs combined with WL-based node hashes: (i) automorphism-aware dropouts and (ii) subgraph orbit-biased aggregation. Empirical evaluations on synthetic and real graphs show improvements of up to 42.36% and 28.44%, respectively, in predicting links in scenarios with high automorphism.

Chen Shao, Donald Loveland, Tobias Käfer +1

Karlsruhe Institute of Technology Karlsruhe, Germany · Department of Computer Science University of Michigan Ann Arbor, MI, USA · Computer Science & Engineering University of Michigan Ann Arbor, MI, USA

Link Prediction GNN Expressivity Equivariant GNNs

Aug 28, 2026cs.LG

Equivariant Sheaf Neural Networks: Learning Geometric Transport on Graphs

Equivariant graph neural networks provide a principled way to model geometric systems, but efficient first-order architectures remain limited in how vector information can be transformed as it moves across a graph. We introduce \textsc{ESNN}, an Equivariant Sheaf Neural Network that enriches this interaction by learning directed, matrix-valued transport between neighboring vector features while preserving exact Euclidean equivariance. Rather than increasing the order of the representation, ESNN keeps scalar and vector features first-order and places the additional geometric flexibility in the edge transport itself. We characterize this transport theoretically, showing that when relative displacement is the only covariant geometric input, every linear

O(n)

-equivariant map decomposes into independent radial and tangential components, while learned covariant features enable richer feature-conditioned transformations. We also introduce controlled symmetry relaxation for systems with a preferred ambient direction, which may be prescribed or inferred from data while recovering full

E(n)

-equivariance when the directional pathway is inactive. Across particle dynamics, mesh-based simulation, point-cloud classification, and molecular property prediction, ESNN improves dynamics prediction, recovers the gravity axis when symmetry is broken, yields substantial gains on selected mesh tasks and long-horizon rollouts, and remains robust to unseen rotations. These results show that learning how geometric information is transported across edges offers a complementary route to expressive equivariant message passing without requiring higher-order representations.

Alessio Borgi, Mario Severino, Fabrizio Silvestri +1

Department of Computer Science and Technology, University of Cambridge · Department of Computer, Control and Management Engineering, Sapienza University of Rome · Department of Information Engineering, University of Padua

Equivariant Neural Networks Sheaf Neural Networks Equivariant GNNs Equivariant Representation Learning

Jul 23, 2026cs.LG

EnsembleEGNN: Set-Based Graph Learning for Thermodynamic Ensembles of Cyclic Peptides

Molecular graph encoding often relies on a single, static structure, ignoring the thermodynamic ensemble of molecules that are present in solution. Here, we introduce EnsembleEGNN, a foundation model that encodes structural ensembles by processing individual conformers through shared equivariant graph neural network layers, pooled with a set attention block, to make property predictions from the whole ensemble. Pretrained on the CREMP cyclic peptide dataset using multi-task self-supervision, the model is trained to encode the conformational variability of each molecule. When predicting membrane permeability from the CycPeptMPDB benchmark, EnsembleEGNN achieves an

R^2

of

0.477

under random cross-validation, outperforming a sequence-only BERT baseline (

R^2=0.439

). This representation advantage persists under rigorous out-of-distribution Butina splits (

R^2=0.401

versus

0.354

). Finally, a hybrid architecture co-training EnsembleEGNN with the BERT model achieves the highest overall accuracy across both random (

R^2=0.538

) and structural holdout evaluations (

R^2=0.444

). These results demonstrate that encoding conformational ensembles into latent representations improves predictions for properties governed by thermodynamics.

Aaron L. Feller, Kris Deibler, Maxim Secor

Interdisciplinary Life Sciences, University of Texas at Austin, Austin, TX. · Molecular AI, Novo Nordisk, Lexington, MA.

Equivariant GNNs Molecular Property Prediction Molecular Representation Learning

Jul 21, 2026cs.LG

GEqTrain: A Configuration-Driven Framework for Retargeting Equivariant Graph Neural Networks Across 3D Scientific Tasks

Equivariant graph neural networks provide a powerful modeling language for three-dimensional scientific data, but their reuse is often limited by implementations tied to specific tasks, outputs, and training regimes. We present GEqTrain, a configuration-driven framework that separates dataset semantics, model composition, and training objectives. Raw data are mapped to typed node-, edge-, and graph-level fields, while model stacks, losses, and training workflows are assembled declaratively through Hydra configurations. A shared equivariant backbone and training infrastructure can therefore be retargeted to a new task primarily through configuration. We demonstrate this flexibility on three different problems handled within one software stack: coarse-grained-to-atomistic backmapping of biomolecular systems, prediction of NMR chemical shifts in molecular solids, and equivariant generative modeling. Our aim is not to surpass individually optimized task-specific systems, but to show that a shared representation and training infrastructure can achieve competitive accuracy across qualitatively different tasks at the cost of a configuration change. We further introduce GEqDiff, a generative extension based on equivariant flow matching. GEqDiff treats user-defined equivariant fields as first-class generation targets, jointly transporting Cartesian positions and non-scalar node fields spanning representations up to l=3 within a single equivariant flow. We validate this capability on a controlled synthetic benchmark inspired by protein secondary-structure motifs, showing that fields with heterogeneous transformation properties can be reconstructed jointly and with high fidelity. By reducing the software overhead of moving between predictive and generative, scalar and tensorial settings, GEqTrain aims to make equivariant modeling more reproducible, extensible, and reusable.

Daniele Angioletti, Marco Nobile, Vittorio Limongelli

Faculty of Biomedical Sciences, Euler Institute, Universit´a della Svizzera italiana, 6900 Lugano, Switzerland

Flow Matching Generative Modeling Equivariant GNNs Equivariant Representation Learning

Jul 20, 2026cs.LG

Sobek: Streaming Equivariant Tensor Product Convolutions

Equivariant graph neural networks repeatedly apply edge-conditioned tensor-product convolutions over graph edges. Conventional implementations materialize edge-specific weights, messages, and adjoints, causing tensor-product workspace and memory traffic to grow rapidly with graph size and operator width. This limits feasible workloads and can prevent larger problems from fully utilizing the GPU. We show that these edge-sized intermediates are artifacts of the execution schedule, not requirements of the equivariant operator. By reassociating radial projection, spherical-harmonic coupling, and graph aggregation, edge-local products can be consumed directly into bounded receiver-side state. The resulting streaming formulation preserves fully connected multiplicity mixing and extends through forward, backward, and double backward. We implement this formulation in Sobek, a generated-CUDA backend, and evaluate it across edge-scaling regimes and varied feature structures. Across two operator families and all three differentiation orders, Sobek is faster in all 75 capacity-matched comparisons, with speedups ranging from

1.2\times

to

49.7\times

, and reduces peak allocated memory by up to 99%. It also executes workloads up to two orders of magnitude beyond OpenEquivariance's capacity while retaining near-peak throughput. These results show that edge-scaled tensor-product workspace is a property of the conventional schedule, not of equivariant convolution itself.

Vladimir Chorošajev, Cédric Bény

Cortex Discovery

GPU Kernel Optimization GPU Acceleration Equivariant Neural Networks Equivariant GNNs

Jul 8, 2026eess.SP

Stability of Flow Models for Graph Signals

Generating signals on graphs requires permutation-equivariant models that exhibit stability with respect to relative structural perturbations. While favorable stability properties of Graph Neural Networks (GNNs) have been well documented, it is unclear how structural errors propagate through the dynamics of continuous generative flow models that are gaining traction for graph signal generation. In this paper, we analyze continuous normalized flow models parameterized by GNNs and show that permutation equivariance is preserved for both the resulting continuous-time ordinary differential equations and their discrete numerical approximations used as graph signal samplers. Our primary contribution is to derive explicit stability bounds on the generated probability distributions, which quantify how relative graph perturbations affect the final sampled signals. Motivated by these theoretical bounds, we introduce a stability-promoting regularized flow matching strategy that actively penalizes the spatial Lipschitz constant of the vector field during model training. Experiments using synthetic smooth signals on stochastic block model graphs and real-world fMRI signals on brain connectomes demonstrate that this bound-oriented approach yields generative models that are more robust to structural noise, without sacrificing output quality.

Martin Schmidt, Gonzalo Mateos

Dept. of Electrical and Computer Engineering, University of Rochester, Rochester, NY 14627, USA

Graph Signal Processing Flow Matching Continuous Normalizing Flows Equivariant GNNs Flow-Based Generative Modeling

Jul 1, 2026cs.LG

Spin-Weighted Spherical Harmonics Enable Complete and Scalable $\mathrm{E}(3)$ -Equivariant Networks

\mathrm{E}(3)

-equivariant networks are promising for 3D atomistic system modeling, yet their scalability is limited by the

O(L^6)

complexity of the Clebsch-Gordan Tensor Product (CGTP). The recently proposed Gaunt Tensor Product (GTP) reduces the complexity but is unable to capture the antisymmetric paths, resulting in incomplete expressivity. In this work, we present SpinGTP, an approach to overcome the GTP incompleteness by generalizing from scalar functions to Spin-Weighted Spherical Harmonics (SWSH). By relying on the algebraic properties of SWSH, SpinGTP recovers the missing antisymmetric interactions while maintaining the asymptotic efficiency of GTP. It also allows for a more expressive equivariant basis that naturally accounts for the parity-odd components of tensor products. We evaluate SpinGTP across diverse benchmarks, including Tetris, 3BPA, SPICE-MACE-OFF, and OC20. Our results show that SpinGTP achieves accuracies comparable to full CGTP. Notably, by explicitly capturing antisymmetric paths, SpinGTP exhibits superior performance in tasks involving chiral materials and non-centrosymmetric geometries. This work provides a complete, scalable, and mathematically rigorous path toward high-order equivariance in large-scale 3D atomistic system simulations.

Chenxing Liang, Yuchao Lin, Andrii Kryvenko +5

Department of Computer Science and Engineering, Texas A&M University · Lambda, Inc. · Department of Materials Science and Engineering, Texas A&M University +3

Machine Learning Interatomic Potentials Equivariant Neural Networks Equivariant GNNs Spherical Harmonics

Jun 28, 2026physics.chem-ph

Geometric Algebra Meets Cartesian Tensors: Higher-Order Equivariance for Interatomic Potentials

\mathrm{Cl}(3,0)

interatomic potentials, despite their algebraic elegance, predict force magnitudes accurately but force directions poorly. Across ten rMD17 molecules, every

L \leq 1

baseline in our twelve-model study attains aggregate force-cosine similarity below

0.25

. The cause is structural. The geometric product of two vectors in

\mathbb{R}^3

realises only the

L=0

and

L=1

components of its irreducible representation content, leaving the symmetric-traceless rank-2 component absent from the per-edge bilinear that drives each message-passing layer. We address this with CliffordSTF, which couples the Clifford multivector to closed-form symmetric-traceless tensor tracks at ranks two and three through bilinear cross-track contractions, using a single learned bilinear and no Clebsch--Gordan tables, Wigner-

D

matrices, or e3nn calls. On rMD17, CliffordSTF raises aggregate force-cosine similarity from

0.055

(base Clifford) to

0.551

, an order-of-magnitude relative directional gain, alongside improved magnitude accuracy (force MAE

15.8\%

lower; energy MAE

10.9\%

lower). It outperforms all CG-free or body-ordered baselines in our study (all

\leq 0.17

). On catalysis benchmarks, CliffordSTF achieves the best out-of-distribution S2EF energy MAE on OC22 in our experiments, and the best in-distribution energy MAE among

L \geq 2

methods on OC22 IS2RE. An eleven-variant ablation shows the two tracks are complementary: neither alone matches the combined model.

Can Polat, Erchin Serpedin, Mustafa Kurban +1

Dept. of Electrical & Computer Eng. Texas A&M Univ. College Station, TX, USA · Dept. of Prosthetics & Orthotics College of Science & Eng. Ankara Univ., Ankara, Turkey · Hamad Bin Khalifa Univ. and Texas A&M Univ. at Qatar Doha, Qatar

Machine Learning Interatomic Potentials Equivariant GNNs

Jun 17, 2026physics.optics

Equivariant Graph Neural Networks Improve Optical Spectra Prediction for Materials Screening

Scalable prediction of optical spectra is a critical component of high-throughput materials screening for optoelectronic applications such as solar cells. Existing surrogate models are trained on spectra computed from lower levels of theory or rely on rotation-invariant scalar features, limiting their geometric expressiveness. We explore the use of equivariant graph neural networks for optical spectra prediction, adapting GotenNet to this task and evaluating it on multiple datasets including a recently published collection of 10,533 structures with spectra computed at the level of the random phase approximation (RPA). The proposed model outperforms the current state of the art, with the largest gains in the 0-8 eV range and on predicting the static real permittivity, both of particular relevance for thin-film optics.

Kasper Helverskov Petersen, François R J Cornet, Martin Ovesen +3

Department of Applied Mathematics and Computer Science, Technical University of Denmark, Kongens Lyngby, Denmark · Department of Physics, Technical University of Denmark, Kongens Lyngby, Denmark

Materials Property Prediction Equivariant GNNs Computational Materials Science

Jun 14, 2026cs.LG

Scalar-pathway fidelity improves physical accuracy in short-range equivariant interatomic potentials

Accurate interatomic potentials enable molecular dynamics of materials, molecules, and interfaces beyond density-functional-theory length and time scales. Equivariant neural network potentials have improved the representation of local geometry. However, their deployable energy surfaces ultimately manifest through invariant scalar channels, whose aggregation and spectral resolution remain comparatively underexamined. Here we use Physics-Aware Neighborhood (PAN) pooling and Physics-Guided Spectral (PGS) mixers as controlled scalar-pathway probes: lightweight, symmetry-preserving modifications that act only on

\ell=0

channels while leaving the equivariant tensor backbone unchanged. Using MACE as a high-body-order mechanistic scaffold, PAN adds coordination-sensitive amplitude modulation, whereas PGS augments edge and readout scalar features with radial and tapered spectral bases. Across metallic Ag, covalent Si, a short-range ionic LiF/Li--F subset, and MD17/rMD17 molecules, this scalar-pathway correction reduces MACE force errors by 22--27% and energy errors by 19--22%; on systems with stress labels, stress errors decrease by 27--28%, at approximately 5% additional inference-FLOPs cost. Directionally consistent gains in Allegro and NequIP further indicate that the correction is portable across distinct short-range equivariant backbones, although effect sizes remain architecture-dependent. These results identify scalar-pathway fidelity as a practical design dimension for short-range equivariant interatomic potentials.

Jia Bi, Alin Marin Elena, Samuel Pinilla

Science and Technology Facilities Council, Harwell Campus, Didcot, OX11 0QX, Oxfordshire, United Kingdom. · Science and Technology Facilities Council, Keckwick Lane, Daresbury, WA4 4AD, Cheshire, United Kingdom. · Diamond Light Source, Harwell Science and Innovation Campus,2026 Didcot, OX11 0QX, Oxfordshire, United Kingdom.

Machine Learning Interatomic Potentials Equivariant GNNs Computational Materials Science

May 20, 2026cs.LG

EvoStruct: Bridging Evolutionary and Structural Priors for Antibody CDR Design via Protein Language Model Adaptation

Equivariant graph neural network (GNN) methods for antibody complementarity-determining region (CDR) design achieve the highest sequence recovery but suffer from severe vocabulary collapse. The current best GNN methods over-predict very few amino acids, such as tyrosine and glycine, while ignoring functionally important residues. We trace this failure to GNN encoders learning amino acid distributions de novo from limited structural data, discarding substitution patterns encoded in evolutionary databases. To resolve this, we propose EvoStruct, which bridges a frozen protein language model (PLM) with 3D structural context from an E(3)-equivariant GNN via a cross-attention adapter. Unlike prior PLM-structure adapters for general protein design, EvoStruct targets the vocabulary collapse problem specific to CDR design through progressive PLM unfreezing and R-Drop consistency regularization. On the CHIMERA-Bench dataset, EvoStruct achieves the highest amino acid recovery and lowest perplexity among several antibody design methods, improving sequence recovery by 16% and reducing perplexity by 43% relative to the best GNN baselines, while recovering 2.3x greater amino acid diversity and the highest binding-pair correlation with ground truth.

Mansoor Ahmed, Sujin Lee, Umar Khayaz +1

Georgia State University, Atlanta, USA · Georgia Institute of Technology, Atlanta, USA

Protein Language Models Equivariant GNNs Molecular Generation Protein Design

May 20, 2026astro-ph.CO

Velocityformer: Broken-Symmetry-Matched Equivariant Graph Transformers for Cosmological Velocity Reconstruction

Precise measurement of the kinematic Sunyaev-Zel'dovich (kSZ) effect - a probe of the large-scale distribution of baryonic matter, a key observable for cosmological inference - requires accurate reconstruction of galaxy velocities from spectroscopic surveys. The signal-to-noise ratio (SNR) of kSZ measurements scales directly with the correlation coefficient

r

between reconstructed and true velocities. We introduce Velocityformer, an equivariant graph transformer architecture designed to match the specific symmetry of the observational data. While the underlying physics is equivariant with respect to translations and rotations, observational effects break this symmetry due to the preferred line-of-sight direction. Matching the model's inductive bias to the data's broken symmetry consistently improves performance across all model sizes and training volumes, with Velocityformer improving

r

by 35% over the standard linear theory baseline and outperforming ML baselines at every data volume. By matching the model's inductive bias to the data and conditioning on the physics-based long-wavelength solution, Velocityformer is highly data-efficient, training to high accuracy on as few as 4 low-fidelity simulations, and generalises zero-shot across input geometry, cosmological parameters, and galaxy sample. On high-fidelity simulated galaxy catalogues, this yields a 30% improvement in

r

over the physical baseline, directly translating to the same SNR gain on observational data.

Tilman Tröster, David Mirkovic, Veronika Oehl +1

Department of Physics ETH Zürich

Equivariant Neural Networks Equivariant GNNs Graph Transformer

May 16, 2026cs.LG

Tensor Channel Equivariant Graph Neural Networks for Molecular Polarizability Prediction

We introduce a tensor-channel equivariant graph neural network for direct prediction of molecular polarizability tensors. Building on the efficient PaiNN architecture, we augment the hidden representation with explicit symmetric rank-2 tensor channels aligned with the decomposition of polarizability into isotropic and anisotropic components. In contrast to approaches that construct tensor outputs only at readout, our model propagates tensor structure throughout message passing using geometrically motivated tensor bases. This yields a target-aligned architecture for tensor-valued molecular prediction. On optimized QM7-X geometries, the proposed model achieves lower full-tensor and anisotropic error than both a PaiNN-style readout baseline and a dielectric MACE baseline under matched training conditions and at nearly identical parameter count. In this controlled setting, it also outperforms MACE while remaining substantially faster at inference. Ablation studies show that the gain does not arise from increased capacity alone, but from the combination of explicit tensor propagation and a traceless target parameterization matched to the anisotropic part of the polarizability tensor. Among the tensor bases considered, the strongest results are obtained from interactions between learned directional features, indicating that these are particularly effective for modeling molecular polarizability. Rotational equivariance tests further confirm that all compared models are numerically equivariant, so the observed improvements are attributable to better learning of the target tensor itself. Overall, our results show that for structured tensor-valued targets, propagating target-aligned tensor features can outperform both readout-only tensor construction and a more general higher-order equivariant model in the present training setting.

Jean Philip Filling, Daniel Franzen, Michael Wand

Institute for Computer Science, Johannes Gutenberg University Mainz, Germany

Equivariant GNNs Molecular Property Prediction

May 11, 2026cs.LG

QT-Net: Rethinking Evaluation of AI Models in Atomic Chemical Space

Atomic properties such as partial charges or multipoles encode chemically meaningful information that can inform downstream molecular property prediction, but their evaluation as machine learning targets has been complicated by the absence of a principled out-of-distribution evaluation protocol at the atomic level. In this work, we propose a held-out evaluation protocol that clusters atomic environments by SOAP descriptors and computes metrics accounting only for cluster labels unseen during training. Following this procedure, we use 5

\times

5 cross-validation and Tukey's HSD to run a statistically rigorous comparison of E(3)-equivariant against non-equivariant, rotationally augmented models for predicting electron populations and multipoles of H, C, N, and O atoms. Building on our results, we introduce the Quantum Topological Neural Network (QT-Net), a rotationally augmented, non-equivariant graph neural network. We show that QT-Net can be used to infer properties of atoms in molecules from QM9 outside our training set, and that these inferred properties can yield improvement when used as input features for downstream molecular property prediction. To further validate the framework, molecular dipole moments computed from QT-Net's per-atom outputs recover the ground-truth values reported in QM9. We release all code and data, including a JAX implementation of QT-Net, to support the broader use of learned QTA properties as inductive biases for atomic-scale molecular machine learning.

Pablo Martínez Crespo, Stefano Ribes, Martin Rahm +6

Department of Computer Science and Engineering Chalmers University of Technology and University of Gothenburg · Department of Chemistry and Chemical Engineering Chalmers University of Technology · Technology Research Intel Corporation +3

Graph Neural Networks Quantum Chemistry OOD Generalization Equivariant GNNs

May 9, 2026cs.LG

Compact SO(3) Equivariant Atomistic Foundation Models via Structural Pruning

SO(3) equivariant graph neural networks have become the dominant paradigm for atomistic foundation models, achieving high accuracy and data efficiency by building rotational symmetry directly into the architecture. Yet the computational cost of their higher-order tensor operations creates a tough trade-off between model accuracy and inference efficiency. In this paper, we propose a structural pruning method for SO(3) equivariant atomistic foundation models to bridge this accuracy-efficiency gap. The pruning is applied along the channel and order dimensions, with each irreducible representation kept or removed as a complete block, thereby retaining SO(3) equivariance. Starting from a large checkpoint, the pruned model substantially reduces the inference cost while retaining higher accuracy than an independently trained small model. The pruned MACE-MP model outperforms the official from-scratch trained small model on 7 of 9 metrics on the Matbench Discovery leaderboard. In terms of efficiency, compressed MACE-MP and MACE-OFF models contain 1.5

\times

to 4

\times

fewer parameters and require 2.5

\times

to 4

\times

less pre-training compute than training a small model from scratch. For downstream applications, fine-tuning the pruned model reduces energy and force errors by 70.1% and 34.4% compared to training task-specific models from scratch across eight representative downstream datasets. We demonstrate that the method generalizes to other SO(3) equivariant architectures (SevenNet, eSCN) and can be combined with quantization and knowledge distillation for further gains.

Chen Wang, Siyu Hu, Guangming Tan +1

State Key Lab of Processors, Institute of Computing Technology, Chinese Academy of Sciences · University of Chinese Academy of Sciences

Equivariant GNNs Structured Pruning Neural Network Compression Neural Network Pruning

May 4, 2026cs.RO

Learning Equivariant Neural-Augmented Object Dynamics From Few Interactions

Learning data-efficient object dynamics models for robotic manipulation remains challenging, especially for deformable objects. A popular approach is to model objects as sets of 3D particles and learn their motion using graph neural networks. In practice, this is not enough to maintain physical feasibility over long horizons and may require large amounts of interaction data to learn. We introduce PIEGraph, a novel approach to combining analytical physics and data-driven models to capture object dynamics for both rigid and deformable bodies using limited real-world interaction data. PIEGraph consists of two components: (1) a \textbf{P}hysically \textbf{I}nformed particle-based analytical model (implemented as a spring--mass system) to enforce physically feasible motion, and (2) an \textbf{E}quivariant \textbf{Graph} Neural Network with a novel action representation that exploits symmetries in particle interactions to guide the analytical model. We evaluate PIEGraph in simulation and on robot hardware for reorientation and repositioning tasks with ropes, cloth, stuffed animals and rigid objects. We show that our method enables accurate dynamics prediction and reliable downstream robotic manipulation planning, which outperforms state of the art baselines.

Sergio Orozco, Tushar Kusnur, Brandon May +2

Brown University, Providence, RI · Robotics and AI (RAI) Institute, Cambridge, MA · General Motors

Deformable Object Manipulation Robotic Manipulation Equivariant GNNs Physics-Informed ML Physics-Informed GNNs Deformable Object Simulation

May 2, 2026cs.LG

PRIME: Protein Representation via Physics-Informed Multiscale Equivariant Hierarchies

Proteins are inherently multiscale physical systems whose functional properties emerge from coordinated structural organization across multiple spatial resolutions, ranging from atomic interactions to global fold topology. However, existing protein representation learning methods typically operate at a single structural level or treat different sources of structural information as parallel modalities, without explicitly modeling their hierarchical relationships. We introduce PRIME (Protein Representation via Physics-Informed Multiscale Equivariant Hierarchies), a unified framework that models proteins as a nested family of five physically grounded structural graphs spanning surface, atomic, residue, secondary-structure, and protein levels. Adjacent levels are connected through deterministic, physics-informed assignment operators, enabling bidirectional information exchange via bottom-up aggregation and top-down contextual refinement. Experiments on standard protein representation learning benchmarks demonstrate strong and competitive performance across diverse tasks, with particularly notable gains on the Fold Classification benchmark, where PRIME outperforms the strongest geometric GNN baseline by margins of 13.80 and 18.30 points on the harder Superfamily and Fold splits, and achieves a state-of-the-art accuracy of 84.10% on Reaction Class prediction, surpassing all baseline methods, including ESM. Ablation studies confirm that each structural level contributes complementary and non-redundant information, and adaptive cross-attention analysis reveals that PRIME autonomously identifies the most task-relevant structural resolutions at prediction time. Our source code is publicly available at https://github.com/HySonLab/PRIME

Viet Thanh Duy Nguyen, John K. Johnstone, Truong-Son Hy

University of Alabama at Birmingham Birmingham, Alabama, USA

Graph Representation Learning Hierarchical Representation Learning Equivariant GNNs Protein Representation Learning Molecular Representation Learning

Apr 22, 2026cond-mat.str-el

Gauge-Equivariant Graph Neural Networks for Lattice Gauge Theories

Local gauge symmetry underlies fundamental interactions and strongly correlated quantum matter, yet existing machine-learning approaches lack a general, principled framework for learning under site-dependent symmetries, particularly for intrinsically nonlocal observables. Here we introduce a gauge-equivariant graph neural network that embeds non-Abelian symmetry directly into message passing via matrix-valued, gauge-covariant features and symmetry-compatible updates, extending equivariant learning from global to fully local symmetries. In this formulation, message passing implements gauge-covariant transport across the lattice, allowing nonlocal correlations and loop-like structures to emerge naturally from local operations. We validate the approach across pure gauge, gauge-matter, and dynamical regimes, establishing gauge-equivariant message passing as a general paradigm for learning in systems governed by local symmetry.

Ali Rayat, Yaohang Li, Gia-Wei Chern

Department of Physics, University of Virginia, Charlottesville, VA 22904, USA · Department of Computer Science, Old Dominion University, Norfolk, VA 22904, USA.

Message Passing Neural Networks High-Energy Physics Equivariant GNNs

Nov 30, 2025cs.RO

Beyond Topology: A Morphological Symmetry Graph Representation for Locomotion Policy Learning

Reinforcement learning has enabled impressive locomotion skills on articulated robots, but common policy representations remain only weakly aligned with robot physics. Generic networks ignore kinematic structure, while graph-based policies encode connectivity without specifying how physical quantities transform across symmetric body parts. We introduce a morphological symmetry graph representation for locomotion policy learning and instantiate it in MS-PPO. Starting from the robot's topological graph, our representation augments each observation and action space with the permutation and sign transformations induced by morphological symmetry. This yields a symmetry-equivariant graph actor and a symmetry-invariant graph critic, enforcing the desired policy and value constraints by construction rather than through reward shaping or data augmentation. We evaluate MS-PPO on a variety of locomotion tasks using both Unitree Go2 quadruped and Unitree G1 humanoid, including command tracking, asymmetric joint failures, out-of-distribution command generalization, and zero-shot sim-to-real deployment. Experiments show improved symmetry generalization, robustness, sample efficiency, and model efficiency over topology- and symmetry-aware baselines.

Sizhe Wei, Xulin Chen, Fengze Xie +3

Georgia Institute of Technology · Syracuse University · California Institute of Technology

RL for Legged Locomotion Equivariant GNNs Robot Policy Generalization

Oct 28, 2021cs.LG

Roto-translated Local Coordinate Frames For Interacting Dynamical Systems

Modelling interactions is critical in learning complex dynamical systems, namely systems of interacting objects with highly non-linear and time-dependent behaviour. A large class of such systems can be formalized as

\textit{geometric graphs}

,

\textit{i.e.}

, graphs with nodes positioned in the Euclidean space given an

\textit{arbitrarily}

chosen global coordinate system, for instance vehicles in a traffic scene. Notwithstanding the arbitrary global coordinate system, the governing dynamics of the respective dynamical systems are invariant to rotations and translations, also known as

\textit{Galilean invariance}

. As ignoring these invariances leads to worse generalization, in this work we propose local coordinate frames per node-object to induce roto-translation invariance to the geometric graph of the interacting dynamical system. Further, the local coordinate frames allow for a natural definition of anisotropic filtering in graph neural networks. Experiments in traffic scenes, 3D motion capture, and colliding particles demonstrate that the proposed approach comfortably outperforms the recent state-of-the-art.

Miltiadis Kofinas, Naveen Shankar Nagaraja, Efstratios Gavves

University of Amsterdam · BMW Group

Dynamical Systems Equivariant GNNs Spatiotemporal GNNs