Title: Equivariant Eikonal Neural Networks: Grid-Free, Scalable Travel-Time Prediction on Homogeneous Spaces
Venue: NeurIPS 2025
Accepted: yes
Keywords: Geometric Deep Learning

Neural Fields

Eikonal Equation

Equivariance

Travel-time

Riemannian Geometry
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.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper proposes Equivariant Neural Eikonal Solvers (E-NES), combining Equivariant Neural Fields with Neural Eikonal Solvers for scalable, grid-free travel-time predictions on homogeneous spaces. By conditioning a shared neural field on pose-context point clouds on Lie groups, E-NES enforces equivariance, enabling efficient, interpretable, and controllable solutions. It generalizes across group actions and outperforms neural operator baselines on 2D and 3D seismic benchmarks.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths \n1. The formulation is mathematically explicit, leveraging Lie group theory, and sound arguments for why equivariant conditioning can provide representational and computational advantages \n2. The paper tackles a genuinely relevant challenge—modeling and predicting eikonal travel times in a manner that synergizes geometric equivariance, operator learning, and physics-informed training\n\n\nWeaknesses\n1. The main experiments focus primarily on seismic travel-time datasets and synthetic 2D/3D settings, with broad generalization mentioned more as potential\n2. The only explicit ablation focuses on equivariance v.s. no equivariance. There is little dissection of the importance of other architectural choices  and how these affect learning. The link between theory and actual empirical contribution of steerability is only partially explored.\n3. Though training modes are described in general, more concrete details about resource utilization, failure modes, or hyperparameter sensitivity are omitted from the main text\n\nQUESTIONS:\nCan the authors provide additional experimental results on non-Euclidean (e.g., spherical or hyperbolic) or real-world datasets to support the claim of applicability on arbitrary homogeneous spaces?\n\nLIMITATIONS:\nnon-Euclidean and real-world datasets experiments are missing", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces Neural Eikonal Solvers. The task here is to use Equivariant neural fields to efficiently solve the eikonal equations, such that some geometric symmetries are automatically introduced in the latent space. The key idea here is that the authors try make sure that the latent space that the model learns is equivariant. That is, the group transformations of the input also apply to the latent space. This ensures that certain transformations can be applied to the vector field as the input geometry changes (i.e, the data lies on a different Reimannian manifold) without any need for re-training. The way this equivariance is imposed is by an Invariant cross-attention encoder—-(using the moving frame method).\n\nThe inputs are the velocity fields defined over the domain, the (s, r) pairs for the equation to minimize, and a conditioning variable z. The final loss is imposed in a PINN style setting by minimizing the eikonal equation. It seems like the training is happening in two steps, first the z is being learned for a dataset, and then this z is calculated as a function of the input velocity field, and conditioned for the final loss.\n\nSTRENGTHS AND WEAKNESSES:\nThe paper is interesting, and imposing equivariance to solve the eikonal equation seems to be quite useful. Furthermore, the work is potentially grid free, and therefore can have a lot of potential, unlike previous grid-based methods and solvers. \n\nFurthermore, the idea of embedding all the velocity fields into the same embedding space—by first learning a context variable—and then training a PINN on top is quite interesting.\n\nThe experiments are kind of mixed, while the method outperforms the DeepONet based baseline, the numbers are quite similar. However, the method is consistently faster than the baseline (esp with meta-learning). \n\nThe authors further show the importance of equivariance on the style-B dataset, and show that models that impose equivariance in fact converge faster, and also is more computationally efficient as compared to the Fast Marching method (i.e, a numerical solver) as the grid size grows in three dimensions. \n\n- However, the paper is difficult to follow, and what it is that the authors are trying to precisely solve gets buried in a lot of mathematical details. While I do realize that this detail is necessary, i think that the writing can be improved, and what is the precise thing that is being solve, i.e,. an eikonal equation where the velocities are encoded in a unified space before the PINN loss (something to this form) should be made clear in the introduction.\n- The authors mention that the latent space is steerable, but there are no experiments that show this steerability property.\n- Even though the method can be adapted to multiple manifolds, only the euclidean space is considered here.\n\nQUESTIONS:\nIn equation 5, the authors minimize a version of the eikonal equation but mention something related to the enforcing of the HJB equation. What exactly is happning therre, and can the authors perhaps explain it in more detail (maybe just in words).\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors present Equivariant Neural Eikonal Solvers, a new framework combining Equivariant Neural Fields with Neural Eikonal Solvers. It uses a shared neural backbone conditioned on signal-specific latent variables, modeled as point clouds in a Lie group, to produce diverse Eikonal solutions. The integration of ENFs ensures equivariant mapping, offering benefits like efficient representation, geometric robustness, and solution steerability, which allows intuitive modifications through latent space transformations. Coupled with Physics-Informed Neural Networks, the framework accurately models travel-time solutions across various Riemannian manifolds, including Euclidean, spherical, and hyperbolic spaces. Validated on seismic datasets, the method outperforms existing neural operators in performance, scalability, and user control.\n\nSTRENGTHS AND WEAKNESSES:\nPaper is well written and motivated. One of the key advantages of their approach is the grid-free formulation. The authors show comparable performance to operator methods like DeepONet using their meta-learning strategy. The authors draw heavily upon recent work of Wessels et al., but extend the framework to represent solutions on the eikonal equations. Even the notion of steerability property that the authors espouse relies heavily upon the previous work of Wessels. Authors central contribution is the generalization of Equivariant Neural Fields to functions defined over Riemannian manifolds. Empirical results are impressive, with thorough evaluations in the results section and additional visualizations in the appendix. The paper would be stronger if the author had a special subsection specifically laying out the differences of their approach compared to Wessels, and why the approach of Wessels cannot be blindly applied to this problem. Making this clear from the onset would dramatically improve the readability of the paper. The lack of clarity on this also raises concerns for me over novelty of the proposed approach.\n\nQUESTIONS:\nCan the generalization of the Equivariant Neural Fields be applied to any problems beyond eikonal solvers. If so what are they? If not, why not? A fundamental issue for me raised above is around novelty, if this can be made clearer it would help with the final rating.\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-07-06"}

comments: SUMMARY:
The paper introduces Equivariant Neural Eikonal Solvers (E-NES), a new framework that combines Equivariant Neural Fields (ENFs) with Neural Eikonal Solvers to predict travel times. The core idea is to use a single neural field with a shared backbone conditioned on signal-specific latent variables, represented as point clouds in a Lie group. This integration ensures an equivariant mapping from latent representations to the solution field, leading to improved representation efficiency, geometric grounding, and solution steerability. 
The authors validate their approach using 2D and 3D seismic travel-time modeling benchmarks. Ablation studies show that with equivariance, method achieves much lower losses.

STRENGTHS AND WEAKNESSES:
This paper is backed by theoretical framework and demonstrates superior performance in accuracy and computational efficiency compared to existing Neural Operator-based methods like FC-DeepONet - with experiments in both 2D and 3D 

In term of weakness it requires test-time fitting - which author shows some meta-learning experiments that speeds up the inference. And despite the additional test-time compute required, the performance seems to be mixed for the OpenFWI experiment.

QUESTIONS:
- in table 1, there seems to be mixed performance - any intuition why? If apply same loss/test-time fitting method to FC-DeepONet - would that improve the number for DeepONet further?
- would vanilla PINN be a reasonable baseline for table 1 or I am missing something?

LIMITATIONS:
Yes
TITLE: Official Review
CLARITY: 3
IMPACT: 2
ORIGINALITY: 3
DATE: 2025-07-14