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

Alejandro Garc\'ia-Castellanos; Dani\"el M. Pelt; David R. Wessels; Erik J. Bekkers; Nicky J. van den Berg; Remco Duits

REVIEW 4 major objections 6 minor 62 references

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

T0 review · 4 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A single network can solve the eikonal equation for many velocity fields, and steering the latent code steers the answer exactly.

desk verdict A sound theory paper with a useful equivariant eikonal solver, but the abstract oversells validation on spaces that are only future work, and the empirical claims need error bars before fully trusting the numbers. read the letter →

arxiv 2505.16035 v3 pith:VYNG6NLW submitted 2025-05-21 cs.LG cs.AI

Alejandro García-Castellanos , David R. Wessels , Nicky J. van den Berg , Remco Duits , Daniël M. Pelt , Erik J. Bekkers This is my paper · ORCID

classification cs.LGcs.AI

keywords equivariantneuralfieldseikonalequationtravel-timepredictionphysics-informednetworksmovingframeinvariantshomogeneousspacesseismicvelocitymodelssolver

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper introduces Equivariant Neural Eikonal Solvers (E-NES), a way to learn travel-time solutions of the eikonal equation for many different velocity fields with a single neural network. The paper's central claim is that a network built from the invariants $(g_i^{-1}s, g_i^{-1}r)$ for source $s$, receiver $r$, and latent pose $g_i$ is exactly equivariant, so transforming the learned latent code for a velocity field transforms the predicted travel-time field---and the correspondingly steered velocity field---without retraining. If this claim is right, one shared, grid-free model can replace per-field numerical solvers or per-field networks, with a new velocity field represented by a small latent point cloud fitted by a physics-based loss alone. Empirically, converged autodecoding E-NES beats the FC-DeepONet operator baseline on seven of ten OpenFWI benchmark datasets, and meta-learning fits new fields two orders of magnitude faster.

What carries the argument

The load-bearing machinery is the pose-context point cloud $z=\{(g_i,c_i)\}$ combined with an invariant cross-attention encoder. For each source--receiver pair and each latent pose $g_i$, the encoder consumes the joint invariants $(g_i^{-1}s,\,g_i^{-1}r)$, produced by the moving-frame method, a systematic procedure that constructs a complete set of functionally independent invariants of a group action. Theorem 4.1 restores the moving-frame method to non-free group actions by adjoining the latent pose itself to the product manifold, making the action free and guaranteeing the invariants separate orbits. The architecture symmetrizes over source and receiver with a Reynolds average, factors the travel time as $T=\tilde d\,\tau$ with an invariant semimetric $\tilde d$, trains with the eikonal Hamiltonian loss on both gradients, and fits per-field latents by autodecoding or meta-learning.

What would settle it

Fit a held-out velocity field from a family far outside the training set, starting its latent from scratch and driving the eikonal residual (5) to a small value, then compare predicted travel times against second-order factored fast marching ground truth. If the loss can be driven near zero while the travel-time error stays large, the PDE loss plus a finite latent code is not uniquely identifying solutions. A targeted version on the sphere would check whether the factorized ansatz with Euclidean chordal distance can satisfy the eikonal equation in a strongly bent, high-contrast velocity field.

Watch

Extended reading notes

Core claim

The discovery is that equivariant conditioning on a Lie-group point cloud makes travel-time prediction steerable without losing expressivity. For a latent $z=\{(g_i,c_i)\}$ of pose-context pairs and inputs $(s,r)$, the features $\mathrm{Inv}(s,r,g_i)=(g_i^{-1}s,\,g_i^{-1}r)$ form a complete, functionally independent set of invariants for the diagonal group action on the augmented product manifold (Theorem 4.1); the network $T_\theta(s,r;z)=\tilde d(s,r)\,\tau_\theta(s,r;z)$ therefore loses no information by using them and is exactly equivariant, $T_\theta(s,r;g\cdot z)=T_\theta(g^{-1}s,\,g^{-1}r;z)$. Proposition 4.1 then shows that if $T_\theta(\cdot,\cdot;z_l)$ solves the two-point eikonal equation for velocity $v_l$, the steered network $T_\theta(\cdot,\cdot;g\cdot z_l)$ solves it for the group-transformed velocity $\mu(g,v_l)$; for isometric actions this is simply $v_l(g^{-1}\cdot s)$. Training uses only the eikonal Hamiltonian residual (Equation 5) and never ground-truth travel times, keeping the solver grid-free and allowing extension to 3D and spherical domains, where it still recovers correct wavefronts. On the OpenFWI benchmark, converged autodecoding outperforms FC-DeepONet on seven of ten datasets.

Load-bearing premise

The premise that must hold is that a small learned code (nine geometric-context pairs in 2D) together with the eikonal equation as the only loss contains enough information to single out the correct travel-time solution for a new velocity field, and that factoring the solution as a distance-like semimetric times a correction does not exclude valid wavefronts.

Editorial extensions

If this is right

Once one velocity field in a group orbit is encoded, every transformed field is solved for free by applying the same transformation to the latent pose set.
Storing a field costs a few hundred latent parameters (315 in the 2D setup) independent of grid resolution, rather than a discretized field or a dedicated trained network.
Meta-learning reduces per-batch fitting time for new fields from roughly 1000 seconds to under 6 seconds, making near-real-time travel-time prediction plausible.
The same architecture handles non-Euclidean homogeneous spaces, predicting travel times on the sphere and recovering geodesic paths by bidirectional gradient backtracking.
Because the loss is the eikonal equation itself, the method inherits grid-free, resolution-independent evaluation.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Beyond the paper's benchmarks, the same invariant construction should transfer to other two-point physical quantities such as wavefields, Green's functions, or distance transforms whenever the domain carries a Lie-group action.
A testable consequence of the equivariance claim: training on one orbit of transformed velocity fields should already generalize to unseen fields in that orbit, so any errors on such fields would localize the failure in the latent code rather than in the equivariant backbone.
The factored ansatz $T=\tilde d\,\tau$ with a chordal-distance semimetric is justified empirically here; a stress test on high-contrast spherical fields with strongly bending geodesics would show whether a geodesic-aware semimetric is ever needed.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: this is a genuinely useful paper that does most of what it claims, but the abstract and framing run ahead of the experiments in one or two places. The core theory is sound; the soft spots are empirical and presentational, not mathematical.

What's actually new: the latent-pose extension theorem (4.1) giving complete invariants for non-free actions is a real improvement over the heuristic invariant choices in previous ENF work. The steered eikonal group action (Prop 4.1/Cor 4.1) is clean, correctly derived from the equivariance condition, and makes the weight-sharing and 'no retraining needed for transformed velocities' story precise. The first ENF-to-eikonal application with OpenFWI results is sensible and shows a competitive method, not just a toy.

On the stress-test concern: I think it's overstated as a logical flaw. The paper's claim is that if you transform the latent, you get the transformed solution. That's exactly what the equivariance property guarantees, and Appendix E.7 verifies it. The map from velocity fields to latents does not need to be equivariant for this to hold; you don't need the optimizer to recover g·z when fitting a new field. What the paper should do is be clearer that the orbit generalization is a property of the model given the transformed latent, not a guarantee about what SGD will find when fitting a new velocity from scratch. So the concern is about framing, not a hole in the proof.

The real soft spots are empirical: all results are single-run, no error bars or multiple seeds, which matters when you claim superiority over DeepONet. The spherical benchmark has no baseline at all, so 'strong performance' is uncalibrated. And the abstract says the framework covers hyperbolic and SE(2) spaces, but those appear only in future work; the experiments are Euclidean and spherical. That's a clear over-claim and should be fixed.

Citation pattern looks fine: builds on Wessels et al., Olver's moving frames, standard eikonal-PINN literature. No red flags.

Recommendation: this deserves a serious referee. It's a solid, careful piece with a real theoretical contribution and a working implementation. Ask for a revision that tones down the abstract, adds seeds/error bars, and gives the spherical results at least one baseline, and it should be acceptable.

I'd cite it and I'd bring it to a reading group.

Referee Report

4 major / 6 minor

Summary. The paper introduces E-NES, a conditional neural field for solving the two-point eikonal equation on Riemannian manifolds. The travel-time is factored as T(s,r)=d̃(s,r)τ(s,r), where d̃ is an invariant semimetric and τ is produced by a network conditioned on a latent point cloud of pose-context pairs. The network is made steerable by using the invariant features (g_i^{-1}s, g_i^{-1}r) inside a cross-attention encoder. The authors prove that if the network solves the eikonal equation for a velocity field v_l with latent z_l, then the same network with latent g·z_l solves the eikonal equation for a g-steered velocity field (Proposition 4.1), and they give a moving-frame construction of the invariants (Theorem 4.1). Training is done purely with a Hamiltonian PDE loss in autodecoding and meta-learning modes. Experiments cover 2D and 3D OpenFWI seismic benchmarks and spherical travel-time problems, with comparisons to FC-DeepONet and Functa.

Significance. The theoretical core is genuinely useful: Proposition 4.1 gives a clean, exact steerability statement, and Theorem 4.1 provides a principled way to obtain complete invariants by augmenting the product space with a latent pose. The paper also ships reproducible code, contains detailed appendix proofs, and includes a falsifiable steerability experiment. If the empirical claims are confirmed, E-NES would be a valuable resolution-invariant, geometry-aware alternative to operator-based eikonal solvers. However, the empirical support is currently too thin for the advertised breadth: all comparisons are single runs without uncertainty, the spherical benchmark has no baseline, and the abstract promises domains that are not tested.

major comments (4)

[§4.1, Appendix C.1, Appendix E.7] The headline orbit-generalization claim is only verified in the forward direction. Proposition 4.1 shows that if z_l solves for v_l, then g·z_l solves for µ(g,v_l). But at inference a new velocity field is encoded by unconstrained SGD over z (Algorithms 1 and 2), and nothing ensures that fitting the eikonal loss for µ(g,v_l) recovers g·z_l (or an equivalent latent). Appendix E.7 transforms a fitted latent and checks the output; it does not fit latents for steered fields and compare them to the steered latents. Thus the statement in §4.1 that 'solving the eikonal equation for one velocity field automatically extends to its entire family under group actions' is not established end-to-end. Please add an experiment that fits latents for µ(g,v) for several g and reports both the distance between the fitted latent and g·z_v and the resulting travel-time error, or explicitly qualify the claim as a property of the representation rather than of the inference procedure.
[Abstract, §5.3, §6] The abstract states that the framework includes 'position-orientation, spherical, and hyperbolic manifolds,' but the experiments cover only Euclidean domains and the 2-sphere; Section 6 explicitly lists position-orientation spaces and hyperbolic spaces as future work. This mismatch should be corrected. Moreover, Table 3 reports only E-NES's own errors on the sphere with no baseline comparison, so the claim of superior generalization to non-Euclidean domains is not actually supported. Please add at least one baseline on the sphere (e.g., a non-equivariant conditional neural field or a conventional eikonal solver) or temper the abstract and discussion accordingly.
[Tables 1–3, Figures 2–3] All empirical comparisons appear to be single runs with no error bars, confidence intervals, or multiple seeds. Since the central performance claim is that E-NES 'outperforms' FC-DeepONet on 7 of 10 datasets, and some differences are modest (e.g., Table 1, FlatVel-A: E-NES RE 0.00506 vs FC-DeepONet 0.00277, where E-NES is worse), the reader cannot assess whether the differences are meaningful. Please provide at least three seeds with means and standard deviations for the headline comparisons, or clearly state that the results are single-run and avoid unqualified 'superior performance' wording.
[§4.3, Theorem 4.1] The completeness claim is under-specified for the actual architecture. Theorem 4.1 proves completeness of the invariants for the augmented space Π×G with a single auxiliary group element, but the network uses N latent poses and only the per-latent invariants (g_i^{-1}s, g_i^{-1}r) in Eq. (4). The paper does not prove that this collection of invariants, together with the contexts c_i, is complete for the joint action on the full multi-latent cloud, nor does it discuss the role of relative poses g_i^{-1}g_j. If the intended claim is completeness for each latent individually, the text should say so; if the claim is that the multi-latent architecture is maximally expressive, a proof is needed.

minor comments (6)

[§3.2, Eq. (5)] The loss in Eq. (5) does not explicitly include the boundary condition T(s,s)=0 or the symmetry condition, although symmetry is enforced in the architecture and the factorization with d̃ is intended to handle the singularity. Please clarify how T(s,s)=0 is enforced during optimization.
[Definition 4.1] In Definition 4.1, the notation G^g_p ... := G_{gp}(...) should be written as G_{g·p}(...) to avoid ambiguity between the group element g and the point g·p.
[Theorem 4.1] In the statement of Theorem 4.1, 'of the action µ' should presumably read 'of the action δ', since µ is not defined in that theorem.
[Table 1] The FC-DeepONet fitting time is reported as '~0.615' without a definition; please specify whether this is GPU/CPU forward-pass encoding time and whether it is measured under conditions comparable to the E-NES fitting times.
[Appendix D.1] The dataset description is confusing: 'we sample 600 velocity fields for training and 100 for validation. We further divide the training set into 500 fields for training and 100 fields for testing.' Please clarify the total split and how the 100 testing fields relate to the 600 training and 100 validation fields.
[Figure 3] In Figure 3, the abbreviations 'AD' and 'META' are used without definition in the caption; please define them as autodecoding and meta-learning, respectively.

Circularity Check

1 steps flagged · score 4.0 of 10

General orbit-transfer in Prop. 4.1 is built into the definition of the steered velocity; SE(2)/SO(2) results remain non-circular.

self definitional [Section 4.1, Definition 4.1 / Eq. (3) and Proposition 4.1 (proof, 'By Eq. (3)...')]
"The map µ:G×(M →R∗+)→(M →R∗+) defined by µ(g, vl)(s) := ||grad_{g−1s} Tl(g−1 ·s, g−1 ·r)||^{-1}_{G^g} ... By Eq. (3), we now get ||grad_s T^g_l(s, r)||_G = 1/µ(g, vl)(s), i.e., T^g_l solves the eikonal equation with velocity µ(g, vl)."

The 'steered velocity' µ(g,v_l) is defined as the reciprocal of the gradient norm of the transformed solution T_l(g^{-1}s, g^{-1}r). The proposition's conclusion that T_l^g satisfies the eikonal equation with velocity µ is therefore an identity: the last line of the proof simply invokes Eq. (3). For a general non-isometric action, this makes the advertised consequence — 'solving the eikonal equation for one velocity field automatically extends to its entire family under group actions' — true by construction of the family rather than by any property of the network or equation. The substantive, non-tautological content is confined to Corollary 4.1, where isometric or conformal actions express µ in terms of v alone; those are exactly the cases used in the SE(2) and SO(2) experiments.

full rationale

The core equivariant construction is largely self-contained: Eq. (2) holds because the features are the joint invariants (g_i^{-1}s, g_i^{-1}r), and Theorem 4.1's completeness claim is proven via Olver's moving-frame theorems rather than by appeal to prior work. The minor self-citation of Wessels et al. (two shared authors) is for the architecture and fitting modes, not for the load-bearing completeness proof, so it does not by itself raise the score. Test-time predictions are not fitted to travel-time labels: latents are optimized against the eikonal Hamiltonian loss (Eq. 5), and the FMM ground truth is used only for evaluation, so those predictions are not statistically forced. The one definitional step is Proposition 4.1: for general non-isometric actions, µ is defined as the reciprocal gradient norm of the transformed solution, making the transfer conclusion true by construction. The non-tautological and empirically used cases are isometric/conformal (Corollary 4.1), giving µ(g,v_l)(s)=v_l(g^{-1}s) or a conformally scaled version, which is an independent geometric statement. The skeptic's concern that unconstrained latent fitting may not recover the steered latent is a robustness/correctness gap, not circularity.

Assumptions & free parameters 5 free parameters · 4 assumptions · 0 invented entities

The central derivation rests on standard invariant theory plus a regularity assumption on group actions; the only truly ad hoc ingredients are the finite latent point cloud and the factored ansatz. No new physical entities are postulated.

free parameters (5)

Per-field latent code z_l (poses and contexts) = Optimized per velocity field; N=9, d=32 in 2D
The conditioning mechanism that lets one shared network represent many velocity fields; at test time autodecoding or 5 inner SGD steps re-fit it, so the method's empirical claims depend on it.
Latent point count and dimension (N,d) = 9x32 for 2D; 8x32 for 3D; 4x16 or 9x32 for sphere
Chosen by hand; ablation in Table 6 shows this trade-off materially changes error from 0.021 to 0.078.
RFF frequency initializations for invariant embeddings = 0.05 (query), 0.2 (value)
Hyperparameters taken from Wessels et al. 2024; no principled derivation.
Projection temperature alpha_0 = learned
Scalar in bounded projection head; trained with the network.
Meta-learning inner-loop SGD settings = LR 30 context, 2 pose, 5 steps
Meta-learning accuracy and speed depend on these values; chosen empirically in Appendix D.

assumptions (4)

standard math Olver's moving frame existence theorem (Theorem B.1) and cross-section construction apply to the augmented G-action.
Used in Appendix B to derive complete invariants for Theorem 4.1; this is accepted background from Olver 2001 and 2011.
domain assumption The group actions on each factor manifold M_i are regular, and regularity is preserved on the product and on the augmented space P x G.
Theorem 4.1 assumes regular actions; for the S^2/SO(2) and Euclidean tests this is true, but for the claimed hyperbolic and SE(2) cases it is not demonstrated.
ad hoc to paper A finite latent point cloud of N pose-context pairs is expressive enough to encode arbitrary velocity fields after optimization.
No expressivity theorem is provided; Section E.4 only ablates N and d on one dataset, so sufficiency is empirical.
domain assumption The factored ansatz T=d_tilde*tau with an invariant semimetric (e.g., chordal distance on S^2) is a valid representation for eikonal solutions.
Borrowed from prior eikonal PINN works such as Smith et al. 2021 and Grubas et al. 2023; on non-isometric group actions the suggested discrete semimetric is not tested.

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Cite this review

Pith. "Pith review of Equivariant Eikonal Neural Networks: Grid-Free, Scalable Travel-Time Prediction on Homogeneous Spaces." pith.science (2026). https://pith.science/paper/VYNG6NLW

@misc{pith2026250516035,
  author       = {Pith},
  title        = {Pith review of: Equivariant Eikonal Neural Networks: Grid-Free, Scalable Travel-Time Prediction on Homogeneous Spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VYNG6NLW}},
  note         = {Machine review of arXiv:2505.16035}
}

read the original 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

Figures reproduced from arXiv: 2505.16035 by the authors.

**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 zl , induces a corresponding transformation on the traveltime function Tθ(·, ·; zl) through the group’s left regular representation, and on the associated velocity field vl via the non-linear group action µ, as formalized in Section 4.1. R… view at source ↗

**Figure 2.** Comparative analysis of equivariant con [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**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, ×10−2 ) across increasing grid dimensions. (b) E-NES demonstrates computational advantages (seconds ×103 ) over FMM even at minimal grid sizes, with efficiency gains amplifying as dimensions increase. Note that meta-learning fitting times (approximately 3 seconds) are b… view at source ↗

Figures from the paper (8 more)

**Figure 4.** Figure 4: Geodesic path planning on the sphere using gradient integration of the travel-time function [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]

**Figure 5.** Figure 5: Example of conformal group action on v(s) = e −s 2 . As an example of an isometric action, consider 2D rotations with G = SO(2) acting on M = R 2 . In [PITH_FULL_IMAGE:figures/full_fig_p019_5.png]

**Figure 6.** Figure 6: Comparative analysis of meta-learning convergence with pretrained versus random initial [PITH_FULL_IMAGE:figures/full_fig_p025_6.png]

**Figure 7.** Figure 7: Comparative analysis of Relative Error versus total fitting time for all 100 velocity fields in [PITH_FULL_IMAGE:figures/full_fig_p028_7.png]

**Figure 8.** Figure 8: Comparative analysis of Relative Error versus total fitting time for all 100 velocity fields in [PITH_FULL_IMAGE:figures/full_fig_p029_8.png]

**Figure 9.** Figure 9: Comparative visualization of E-NES predicted travel-times against reference solutions for OpenFWI type B datasets. Each panel displays the velocity field (left), ground-truth travel-time surface (center-left), E-NES predicted travel-time surface (center-right), and rel…

**Figure 10.** Figure 10: Comparative visualization of E-NES predicted travel-times against reference solutions for OpenFWI type A datasets. Each panel displays the velocity field (left), ground-truth travel-time surface (center-left), E-NES predicted travel-time surface (center-right), and re…

**Figure 11.** Figure 11: Steerability test demonstrating SE(2) equivariance on the FlatVel-A dataset. When the latent conditioning point cloud is transformed by elements of SE(2) (rotations and translations), the predicted travel-time field and recovered velocity field undergo corresponding g…

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