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

2025-05-21

AI Summary

This work addresses the grid dependency, scalability, and geometric controllability challenges in travel-time prediction on homogeneous spaces (e.g., Euclidean, spherical, hyperbolic). We propose the first mesh-free, equivariant neural Eikonal solver. Methodologically, we innovatively couple equivariant neural fields (ENFs) with physics-informed neural networks (PINNs), leveraging Lie group representation learning to achieve weight sharing and manifold-agnostic modeling. Crucially, our framework enables explicit geometric control of latent point clouds under group actions. Experiments on 2D/3D seismic benchmark datasets demonstrate that our approach significantly outperforms existing neural-operator-based solvers. It achieves breakthroughs in generalization across geometries and resolutions, computational scalability to high-dimensional domains, adaptability to heterogeneous media, and user-controllable geometric manipulation—without requiring domain discretization or retraining for new manifolds.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Machine Learning: Learning with Manifolds

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks

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 and 3D benchmark datasets. Experimental results demonstrate superior performance, scalability, adaptability, and user controllability compared to existing Neural Operator-based Eikonal solver methods.

Problem

• Grid-free scalable travel-time prediction on homogeneous spaces
• Modeling diverse Eikonal solutions with equivariant neural fields
• Generalizing Eikonal solutions to arbitrary Riemannian manifolds

Innovation

• Equivariant Neural Fields with latent point clouds
• Physics-Informed Neural Networks for Eikonal solutions
• Steerable solutions on Riemannian manifolds