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Erik Bekkers

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3 papers

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NeurIPS Conference 2025 Conference Paper

Equivariant Eikonal Neural Networks: Grid-Free, Scalable Travel-Time Prediction on Homogeneous Spaces

Alejandro García-Castellanos
David Wessels
Nicky J. van den Berg
Remco Duits
Daniel M. Pelt
Erik Bekkers

We introduce Equivariant Neural Eikonal Solvers, a novel framework that integrates Equivariant Neural Fields (ENFs) with Neural Eikonal Solvers. Our approach employs a single neural field where a unified shared backbone is conditioned on signal-specific latent variables – represented as point clouds in a Lie group – to model diverse Eikonal solutions. The ENF integration ensures equivariant mapping from these latent representations to the solution field, delivering three key benefits: enhanced representation efficiency through weight-sharing, robust geometric grounding, and solution steerability. This steerability allows transformations applied to the latent point cloud to induce predictable, geometrically meaningful modifications in the resulting Eikonal solution. By coupling these steerable representations with Physics-Informed Neural Networks (PINNs), our framework accurately models Eikonal travel-time solutions while generalizing to arbitrary Riemannian manifolds with regular group actions. This includes homogeneous spaces such as Euclidean, position–orientation, spherical, and hyperbolic manifolds. We validate our approach through applications in seismic travel-time modeling of 2D, 3D, and spherical benchmark datasets. Experimental results demonstrate superior performance, scalability, adaptability, and user controllability compared to existing Neural Operator-based Eikonal solver methods.

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NeurIPS Conference 2025 Conference Paper

Probing Equivariance and Symmetry Breaking in Convolutional Networks

Sharvaree Vadgama
Mohammad Islam
Domas Buracas
Christian A Shewmake
Artem Moskalev
Erik Bekkers

In this work, we explore the trade-offs of explicit structural priors, particularly group-equivariance. We address this through theoretical analysis and a comprehensive empirical study focusing on point clouds. To enable controlled and fair comparisons, we introduce \texttt{Rapidash}, a unified group convolutional architecture that allows for different variants of equivariant and non-equivariant models. Our results suggest that more constrained equivariant models outperform less constrained alternatives when aligned with the geometry of the task, and increasing representation capacity does not fully eliminate performance gaps. We see improved performance of models with equivariance and symmetry-breaking through tasks like segmentation, regression, and generation across diverse datasets. Explicit \textit{symmetry breaking} via geometric reference frames consistently improves performance, while \textit{breaking equivariance} through geometric input features can be helpful when aligned with task geometry. Our results provide task-specific performance trends that offer a more nuanced way for model selection. Code available at github. com/Sharvaree/EquivarianceStudy

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NeurIPS Conference 2023 Conference Paper

Latent Field Discovery in Interacting Dynamical Systems with Neural Fields

Miltiadis (Miltos) Kofinas
Erik Bekkers
Naveen Nagaraja
Efstratios Gavves

Systems of interacting objects often evolve under the influence of underlying field effects that govern their dynamics, yet previous works have abstracted away from such effects, and assume that systems evolve in a vacuum. In this work, we focus on discovering these fields, and infer them from the observed dynamics alone, without directly observing them. We theorize the presence of latent force fields, and propose neural fields to learn them. Since the observed dynamics constitute the net effect of local object interactions and global field effects, recently popularized equivariant networks are inapplicable, as they fail to capture global information. To address this, we propose to disentangle local object interactions --which are SE(3) equivariant and depend on relative states-- from external global field effects --which depend on absolute states. We model the interactions with equivariant graph networks, and combine them with neural fields in a novel graph network that integrates field forces. Our experiments show that we can accurately discover the underlying fields in charged particles settings, traffic scenes, and gravitational n-body problems, and effectively use them to learn the system and forecast future trajectories.

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