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

TL;DR

The paper tackles travel-time prediction under the Eikonal equation on Riemannian manifolds, introducing the grid-free Equivariant Neural Eikonal Solver (E-NES). E-NES fuses Equivariant Neural Fields with physics-informed learning, conditioning a shared backbone on a latent pose-context cloud $z=igl\{(g_i,\mathbf{c}_i)\bigr\}_{i=1}^N$ and enforcing steerability via $T_{ heta}(s,r;g\cdot z)=T_{ heta}(g^{-1}s,g^{-1}r;z)$, while transforming velocity fields through a group action $\mu(g,v_l)$. The approach extends ENFs to products of manifolds with regular group actions, provides a complete invariant set via moving frames, and deploys an invariant cross-attention encoder with a bounded velocity projection to yield grid-free, scalable travel-time fields across Euclidean, position-orientation, spherical, and hyperbolic spaces. Empirical results on 2D and 3D OpenFWI benchmarks and spherical geometries show competitive or superior accuracy to neural-operator baselines, with strong memory efficiency and flexible test-time trade-offs between autodecoding and meta-learning. The framework enables accurate travel-time modeling and downstream geodesic tasks in diverse domains, offering practical impact for seismic modeling, robotics, and geometric vision while paving the way for broader homogeneous-space applications.

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.

Figures (13)

• Figure 1: Steerability in the Equivariant Neural Eikonal Solver (E-NES) enables weight sharing across the entire group orbit: applying a group transformation to the conditioning variable $z_l$, induces a corresponding transformation on the travel-time function $T_{\theta}(\cdot,\cdot; z_l)$ through the group’s left regular representation, and on the associated velocity field $v_l$ via the non-linear group action $\mu$, as formalized in Section \ref{['sec:theory']}.
• Figure 2: Comparative analysis of equivariant conditioning variables on the Style-B dataset. For non-equivariant models $\mathcal{Z}\cong\mathbb{R}^c$, while equivariant models use $\mathcal{Z}=SE(2)\times \mathbb{R}^c$.
• Figure 2: Performance comparison on OpenFWI B-type datasets against Functa. Fitting time represents the total computational time required to fit the latent conditioning variables for all 100 testing velocity fields. Here both methods perform 100 epochs of autodecoding to fit the latents.
• Figure 3: Scaling analysis of E-NES versus FMM on 3D OpenFWI data. (a) Both autodecoding and meta-learning maintain consistent error metrics (RE and RMAE, $\times 10^{-2}$) across increasing grid dimensions. (b) E-NES demonstrates computational advantages (seconds $\times 10^3$) over FMM even at minimal grid sizes, with efficiency gains amplifying as dimensions increase. Note that meta-learning fitting times (approximately 3 seconds) are barely visible in (b) due to their minimal magnitude relative to other displayed times.
• Figure 3: Performance of our method on Eikonal solvers over the 2-sphere.

Theorems & Definitions (17)

• Definition 4.1: $g$-steered metric
• Proposition 4.1: Steered Eikonal Solution
• Corollary 4.1
• Theorem 4.1: Canonicalization via latent-pose extension
• proof : Sketch of proof (full at Appendix, Section \ref{['app:inv']})
• Lemma A.1: Gradient Equivariance
• proof
• proof
• proof
• Remark A.1: Implementation in Euclidean Coordinates