Author name cluster

Robin Walters

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19 papers

1 author row

TMLR Journal 2026 Journal Article

Denoising Hamiltonian Network for Physical Reasoning

Congyue Deng
Brandon Y. Feng
Cecilia Garraffo
Alan Garbarz
Robin Walters
William T. Freeman
Leonidas Guibas
Kaiming He

Machine learning frameworks for physical problems must capture and enforce physical constraints that preserve the structure of dynamical systems. Many existing approaches achieve this by integrating physical operators into neural networks. While these methods offer theoretical guarantees, they face two key limitations: (i) they primarily model local relations between adjacent time steps, overlooking longer-range or higher-level physical interactions, and (ii) they focus on forward simulation while neglecting broader physical reasoning tasks. We propose the Denoising Hamiltonian Network (DHN), a novel framework that generalizes Hamiltonian mechanics operators into more flexible neural operators. DHN captures non-local temporal relationships and mitigates numerical integration errors through a denoising mechanism. DHN also supports multi-system modeling with a global conditioning mechanism. We demonstrate its effectiveness and flexibility across three diverse physical reasoning tasks with distinct inputs and outputs.

TMLR Journal 2026 Journal Article

Discovering Symbolic Differential Equations with Symmetry Invariants

Jianke Yang
Manu Bhat
Bryan Hu
Yadi Cao
Nima Dehmamy
Robin Walters
Rose Yu

Discovering symbolic differential equations from data uncovers fundamental dynamical laws underlying complex systems. However, existing methods often struggle with the vast search space of equations and may produce equations that violate known physical laws. In this work, we address these problems by introducing the concept of \textit{symmetry invariants} in equation discovery. We leverage the fact that differential equations admitting a symmetry group can be expressed in terms of differential invariants of symmetry transformations. Thus, we propose to use these invariants as atomic entities in equation discovery, ensuring the discovered equations satisfy the specified symmetry. Our approach integrates seamlessly with existing equation discovery methods such as sparse regression and genetic programming, improving their accuracy and efficiency. We validate the proposed method through applications to various physical systems, such as Darcy flow and reaction-diffusion, demonstrating its ability to recover parsimonious and interpretable equations that respect the laws of physics.

TMLR Journal 2026 Journal Article

On Uncertainty Calibration for Equivariant Functions

Edward Berman
Jacob Ginesin
Marco Pacini
Robin Walters

Data-sparse settings such as robotic manipulation, molecular physics, and galaxy morphology classification are some of the hardest domains for deep learning. For these problems, equivariant networks can help improve modeling across undersampled parts of the input space, and uncertainty estimation can guard against overconfidence. However, until now, the relationships between equivariance and model confidence, and more generally equivariance and model calibration, has yet to be studied. Since traditional classification and regression error terms show up in the definitions of calibration error, it is natural to suspect that previous work can be used to help understand the relationship between equivariance and calibration error. In this work, we present a theory relating equivariance to uncertainty estimation. By proving lower and upper bounds on uncertainty calibration errors (ECE and ENCE) under various equivariance conditions, we elucidate the generalization limits of equivariant models and illustrate how symmetry mismatch can result in miscalibration in both classification and regression. We complement our theoretical framework with numerical experiments that clarify the relationship between equivariance and uncertainty using a variety of real and simulated datasets, and we comment on trends with symmetry mismatch, group size, and aleatoric and epistemic uncertainties.

TMLR Journal 2026 Journal Article

Symmetry in Neural Network Parameter Spaces

Bo Zhao
Robin Walters
Rose Yu

Modern deep learning models are highly overparameterized, resulting in large sets of parameter configurations that yield the same outputs. A significant portion of this redundancy is explained by symmetries in the parameter space—transformations that leave the network function unchanged. These symmetries shape the loss landscape and constrain learning dynamics, offering a new lens for understanding optimization, generalization, and model complexity that complements existing theory of deep learning. This survey provides an overview of parameter space symmetry. We summarize existing literature, uncover connections between symmetry and learning theory, and identify gaps and opportunities in this emerging field.

NeurIPS Conference 2025 Conference Paper

3D Equivariant Visuomotor Policy Learning via Spherical Projection

Boce Hu
Dian Wang
David Klee
Heng Tian
Xupeng Zhu
Haojie Huang
Robert Platt
Robin Walters

Equivariant models have recently been shown to improve the data efficiency of diffusion policy by a significant margin. However, prior work that explored this direction focused primarily on point cloud inputs generated by multiple cameras fixed in the workspace. This type of point cloud input is not compatible with the now-common setting where the primary input modality is an eye-in-hand RGB camera like a GoPro. This paper closes this gap by incorporating into the diffusion policy model a process that projects features from the 2D RGB camera image onto a sphere. This enables us to reason about symmetries in $\mathrm{SO}(3)$ without explicitly reconstructing a point cloud. We perform extensive experiments in both simulation and the real world that demonstrate that our method consistently outperforms strong baselines in terms of both performance and sample efficiency. Our work, $\textbf{Image-to-Sphere Policy}$ ($\textbf{ISP}$), is the first $\mathrm{SO}(3)$-equivariant policy learning framework for robotic manipulation that works using only monocular RGB inputs.

NeurIPS Conference 2025 Conference Paper

A Practical Guide for Incorporating Symmetry in Diffusion Policy

Dian Wang
Boce Hu
Shuran Song
Robin Walters
Robert Platt

Recently, equivariant neural networks for policy learning have shown promising improvements in sample efficiency and generalization, however, their wide adoption faces substantial barriers due to implementation complexity. Equivariant architectures typically require specialized mathematical formulations and custom network design, posing significant challenges when integrating with modern policy frameworks like diffusion-based models. In this paper, we explore a number of straightforward and practical approaches to incorporate symmetry benefits into diffusion policies without the overhead of full equivariant designs. Specifically, we investigate (i) invariant representations via relative trajectory actions and eye-in-hand perception, (ii) integrating equivariant vision encoders, and (iii) symmetric feature extraction with pretrained encoders using Frame Averaging. We first prove that combining eye-in-hand perception with relative or delta action parameterization yields inherent SE(3)-invariance, thus improving policy generalization. We then perform a systematic experimental study on those design choices for integrating symmetry in diffusion policies, and conclude that an invariant representation with equivariant feature extraction significantly improves the policy performance. Our method achieves performance on par with or exceeding fully equivariant architectures while greatly simplifying implementation.

NeurIPS Conference 2025 Conference Paper

Bridging Equivariant GNNs and Spherical CNNs for Structured Physical Domains

Colin Kohler
Purvik Patel
Nathan Vaska
Justin Goodwin
Matthew Jones
Robert Platt
Rajmonda Caceres
Robin Walters

Many modeling tasks from disparate domains can be framed the same way, computing spherical signals from geometric inputs, for example, computing the radar response of different objects or navigating through an environment. This paper introduces G2Sphere, a general method for mapping object geometries to spherical signals. G2Sphere operates entirely in Fourier space, encoding geometric structure into latent Fourier features using equivariant neural networks and outputting the Fourier coefficients of the continuous target signal, which can be evaluated at any resolution. By utilizing a hybrid GNN-spherical CNN architecture, our method achieves much higher frequency output signal than comparable equivariant GNNs and avoids hand-engineered geometry features used previously by purely spherical methods. We perform experiments on various challenging domains including radar response modeling, aerodynamic drag prediction, and policy learning for manipulation and navigation. We find that G2Sphere outperforms competitive baselines in terms of accuracy and inference time, and we demonstrate that equivariance and Fourier features lead to improved sample efficiency and generalization. The source code is available at: https: //github. com/ColinKohler/geometry2sphere.

TMLR Journal 2024 Journal Article

Fast and Expressive Gesture Recognition using a Combination-Homomorphic Electromyogram Encoder

Niklas Smedemark-Margulies
Yunus Bicer
Elifnur Sunger
Tales Imbiriba
Eugene Tunik
Deniz Erdogmus
Mathew Yarossi
Robin Walters

We study the task of gesture recognition from electromyography (EMG), with the goal of enabling expressive human-computer interaction at high accuracy, while minimizing the time required for new subjects to provide calibration data. To fulfill these goals, we define combination gestures consisting of a direction component and a modifier component. New subjects only demonstrate the single component gestures and we seek to extrapolate from these to all possible single or combination gestures. We extrapolate to unseen combination gestures by combining the feature vectors of real single gestures to produce synthetic training data. This strategy allows us to provide a large and flexible gesture vocabulary, while not requiring new subjects to demonstrate combinatorially many example gestures. We pre-train an encoder and a combination operator using self-supervision, so that we can produce useful synthetic training data for unseen test subjects. To evaluate the proposed method, we collect a real-world EMG dataset, and measure the effect of augmented supervision against two baselines: a partially-supervised model trained with only single gesture data from the unseen subject, and a fully-supervised model trained with real single and real combination gesture data from the unseen subject. We find that the proposed method provides a dramatic improvement over the partially-supervised model, and achieves a useful classification accuracy that in some cases approaches the performance of the fully-supervised model.

NeurIPS Conference 2024 Conference Paper

MatrixNet: Learning over symmetry groups using learned group representations

Lucas Laird
Circe Hsu
Asilata Bapat
Robin Walters

Group theory has been used in machine learning to provide a theoretically grounded approach for incorporating known symmetry transformations in tasks from robotics to protein modeling. In these applications, equivariant neural networks use knownsymmetry groups with predefined representations to learn over geometric input data. We propose MatrixNet, a neural network architecture that learns matrix representations of group element inputs instead of using predefined representations. MatrixNet achieves higher sample efficiency and generalization over several standard baselines in prediction tasks over the several finite groups and the Artin braid group. We also show that MatrixNet respects group relations allowing generalization to group elements of greater word length than in the training set. Our code is available at https: //github. com/lucas-laird/MatrixNet.

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NeurIPS Conference 2024 Conference Paper

Symmetry-Informed Governing Equation Discovery

Jianke Yang
Wang Rao
Nima Dehmamy
Robin Walters
Rose Yu

Despite the advancements in learning governing differential equations from observations of dynamical systems, data-driven methods are often unaware of fundamental physical laws, such as frame invariance. As a result, these algorithms may search an unnecessarily large space and discover less accurate or overly complex equations. In this paper, we propose to leverage symmetry in automated equation discovery to compress the equation search space and improve the accuracy and simplicity of the learned equations. Specifically, we derive equivariance constraints from the time-independent symmetries of ODEs. Depending on the types of symmetries, we develop a pipeline for incorporating symmetry constraints into various equation discovery algorithms, including sparse regression and genetic programming. In experiments across diverse dynamical systems, our approach demonstrates better robustness against noise and recovers governing equations with significantly higher probability than baselines without symmetry.

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NeurIPS Conference 2024 Conference Paper

The Empirical Impact of Neural Parameter Symmetries, or Lack Thereof

Derek Lim
Theo Putterman
Robin Walters
Haggai Maron
Stefanie Jegelka

Many algorithms and observed phenomena in deep learning appear to be affected by parameter symmetries --- transformations of neural network parameters that do not change the underlying neural network function. These include linear mode connectivity, model merging, Bayesian neural network inference, metanetworks, and several other characteristics of optimization or loss-landscapes. However, theoretical analysis of the relationship between parameter space symmetries and these phenonmena is difficult. In this work, we empirically investigate the impact of neural parameter symmetries by introducing new neural network architectures that have reduced parameter space symmetries. We develop two methods, with some provable guarantees, of modifying standard neural networks to reduce parameter space symmetries. With these new methods, we conduct a comprehensive experimental study consisting of multiple tasks aimed at assessing the effect of removing parameter symmetries. Our experiments reveal several interesting observations on the empirical impact of parameter symmetries; for instance, we observe linear mode connectivity between our networks without alignment of weight spaces, and we find that our networks allow for faster and more effective Bayesian neural network training.

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JMLR Journal 2024 Journal Article

TopoX: A Suite of Python Packages for Machine Learning on Topological Domains

Mustafa Hajij
Mathilde Papillon
Florian Frantzen
Jens Agerberg
Ibrahem AlJabea
Rubén Ballester
Claudio Battiloro
Guillermo Bernárdez

We introduce TopoX, a Python software suite that provides reliable and user-friendly building blocks for computing and machine learning on topological domains that extend graphs: hypergraphs, simplicial, cellular, path and combinatorial complexes. TopoX consists of three packages: TopoNetX facilitates constructing and computing on these domains, including working with nodes, edges and higher-order cells; TopoEmbedX provides methods to embed topological domains into vector spaces, akin to popular graph-based embedding algorithms such as node2vec; TopoModelX is built on top of PyTorch and offers a comprehensive toolbox of higher-order message passing functions for neural networks on topological domains. The extensively documented and unit-tested source code of TopoX is available under MIT license at https://pyt-team.github.io. [abs] [ pdf ][ bib ] [ code ] &copy JMLR 2024. ( edit, beta )

NeurIPS Conference 2023 Conference Paper

A General Theory of Correct, Incorrect, and Extrinsic Equivariance

Dian Wang
Xupeng Zhu
Jung Yeon Park
Mingxi Jia
Guanang Su
Robert Platt
Robin Walters

Although equivariant machine learning has proven effective at many tasks, success depends heavily on the assumption that the ground truth function is symmetric over the entire domain matching the symmetry in an equivariant neural network. A missing piece in the equivariant learning literature is the analysis of equivariant networks when symmetry exists only partially in the domain. In this work, we present a general theory for such a situation. We propose pointwise definitions of correct, incorrect, and extrinsic equivariance, which allow us to quantify continuously the degree of each type of equivariance a function displays. We then study the impact of various degrees of incorrect or extrinsic symmetry on model error. We prove error lower bounds for invariant or equivariant networks in classification or regression settings with partially incorrect symmetry. We also analyze the potentially harmful effects of extrinsic equivariance. Experiments validate these results in three different environments.

NeurIPS Conference 2023 Conference Paper

Equivariant Single View Pose Prediction Via Induced and Restriction Representations

Owen Howell
David Klee
Ondrej Biza
Linfeng Zhao
Robin Walters

Learning about the three-dimensional world from two-dimensional images is a fundamental problem in computer vision. An ideal neural network architecture for such tasks would leverage the fact that objects can be rotated and translated in three dimensions to make predictions about novel images. However, imposing $SO(3)$-equivariance on two-dimensional inputs is difficult because the group of three-dimensional rotations does not have a natural action on the two-dimensional plane. Specifically, it is possible that an element of $SO(3)$ will rotate an image out of plane. We show that an algorithm that learns a three-dimensional representation of the world from two dimensional images must satisfy certain consistency properties which we formulate as $SO(2)$-equivariance constraints. We use the induced representation of $SO(2)$ on $SO(3)$ to construct and classify architectures that have two-dimensional inputs and which satisfy these consistency constraints. We prove that any architecture which respects said consistency constraints can be realized as an instance of our construction. We show that three previously proposed neural architectures for 3D pose prediction are special cases of our construction. We propose a new algorithm that is a learnable generalization of previously considered methods. We test our architecture on three pose predictions task and achieve SOTA results on both the PASCAL3D+ and SYMSOL pose estimation tasks.

NeurIPS Conference 2023 Conference Paper

Modeling Dynamics over Meshes with Gauge Equivariant Nonlinear Message Passing

Jung Yeon Park
Lawson Wong
Robin Walters

Data over non-Euclidean manifolds, often discretized as surface meshes, naturally arise in computer graphics and biological and physical systems. In particular, solutions to partial differential equations (PDEs) over manifolds depend critically on the underlying geometry. While graph neural networks have been successfully applied to PDEs, they do not incorporate surface geometry and do not consider local gauge symmetries of the manifold. Alternatively, recent works on gauge equivariant convolutional and attentional architectures on meshes leverage the underlying geometry but underperform in modeling surface PDEs with complex nonlinear dynamics. To address these issues, we introduce a new gauge equivariant architecture using nonlinear message passing. Our novel architecture achieves higher performance than either convolutional or attentional networks on domains with highly complex and nonlinear dynamics. However, similar to the non-mesh case, design trade-offs favor convolutional, attentional, or message passing networks for different tasks; we investigate in which circumstances our message passing method provides the most benefit.

NeurIPS Conference 2023 Conference Paper

Topological Obstructions and How to Avoid Them

Babak Esmaeili
Robin Walters
Heiko Zimmermann
Jan-Willem van de Meent

Incorporating geometric inductive biases into models can aid interpretability and generalization, but encoding to a specific geometric structure can be challenging due to the imposed topological constraints. In this paper, we theoretically and empirically characterize obstructions to training encoders with geometric latent spaces. We show that local optima can arise due to singularities (e. g. self-intersection) or due to an incorrect degree or winding number. We then discuss how normalizing flows can potentially circumvent these obstructions by defining multimodal variational distributions. Inspired by this observation, we propose a new flow-based model that maps data points to multimodal distributions over geometric spaces and empirically evaluate our model on 2 domains. We observe improved stability during training and a higher chance of converging to a homeomorphic encoder.

NeurIPS Conference 2022 Conference Paper

Meta-Learning Dynamics Forecasting Using Task Inference

Rui Wang
Robin Walters
Rose Yu

Current deep learning models for dynamics forecasting struggle with generalization. They can only forecast in a specific domain and fail when applied to systems with different parameters, external forces, or boundary conditions. We propose a model-based meta-learning method called DyAd which can generalize across heterogeneous domains by partitioning them into different tasks. DyAd has two parts: an encoder that infers the time-invariant hidden features of the task with weak supervision, and a forecaster which learns the shared dynamics of the entire domain. The encoder adapts and controls the forecaster during inference using adaptive instance normalization and adaptive padding. Theoretically, we prove that the generalization error of such a procedure is related to the task relatedness in the source domain, as well as the domain differences between source and target. Experimentally, we demonstrate that our model outperforms state-of-the-art approaches on forecasting complex physical dynamics including turbulent flow, real-world sea surface temperature, and ocean currents.

NeurIPS Conference 2022 Conference Paper

Symmetry Teleportation for Accelerated Optimization

Bo Zhao
Nima Dehmamy
Robin Walters
Rose Yu

Existing gradient-based optimization methods update parameters locally, in a direction that minimizes the loss function. We study a different approach, symmetry teleportation, that allows parameters to travel a large distance on the loss level set, in order to improve the convergence speed in subsequent steps. Teleportation exploits symmetries in the loss landscape of optimization problems. We derive loss-invariant group actions for test functions in optimization and multi-layer neural networks, and prove a necessary condition for teleportation to improve convergence rate. We also show that our algorithm is closely related to second order methods. Experimentally, we show that teleportation improves the convergence speed of gradient descent and AdaGrad for several optimization problems including test functions, multi-layer regressions, and MNIST classification.

NeurIPS Conference 2021 Conference Paper

Automatic Symmetry Discovery with Lie Algebra Convolutional Network

Nima Dehmamy
Robin Walters
Yanchen Liu
Dashun Wang
Rose Yu

Existing equivariant neural networks require prior knowledge of the symmetry group and discretization for continuous groups. We propose to work with Lie algebras (infinitesimal generators) instead of Lie groups. Our model, the Lie algebra convolutional network (L-conv) can automatically discover symmetries and does not require discretization of the group. We show that L-conv can serve as a building block to construct any group equivariant feedforward architecture. Both CNNs and Graph Convolutional Networks can be expressed as L-conv with appropriate groups. We discover direct connections between L-conv and physics: (1) group invariant loss generalizes field theory (2) Euler-Lagrange equation measures the robustness, and (3) equivariance leads to conservation laws and Noether current. These connections open up new avenues for designing more general equivariant networks and applying them to important problems in physical sciences.