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

TL;DR

This paper presents a novel equivariant neural network framework that models travel-time solutions on various homogeneous spaces, offering improved efficiency, robustness, and control for geophysical applications.

Contribution

It introduces Equivariant Neural Eikonal Solvers that integrate ENFs with PINNs, enabling scalable, geometry-aware travel-time prediction on diverse manifolds with steerability.

Findings

Outperforms existing Neural Operator-based methods in accuracy and scalability.

Effectively models travel times on Euclidean, spherical, and hyperbolic spaces.

Demonstrates robustness and generalization on seismic benchmark datasets.

Abstract

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.