{"id":"cdedf365-0ec3-48d8-ae0f-0b258b90e9b5","arxiv_id":"2505.16035","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"E-NES uses Lie-group point-cloud conditioning and equivariant neural fields to make grid-free eikonal travel-time prediction steerable under rotations and translations, with complete invariant features and competitive OpenFWI benchmark results.","lead":"Equivariant Neural Eikonal Solvers (E-NES) train one neural network to predict travel times for many different velocity fields by conditioning on small point clouds of poses in a Lie group, so rotating or translating the conditioning point cloud predictably rotates or translates the predicted wavefronts.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The latent-fitting map is not constrained to be equivariant, so the claimed automatic group-orbit generalization at inference is not established by the theory or the reported experiments.","rationale":"The paper's theoretical core is internally consistent: the invariant feature construction yields exact equivariance, and Proposition 4.1 correctly transfers a solved eikonal equation along group orbits. The reader's weakest assumption, that finite learned latent codes plus PDE loss identify the correct travel-time solution, is a real concern, and my stress test sharpens it: the equivariance is a property of the network and of the latent transformation, but the per-field fitting procedure that produces latents for new velocity fields is not itself equivariant or uniqueness-guaranteed. This is more specific than a general expressivity worry because it isolates the exact link between the theory and the claimed practical benefit of 'solving one field automatically extends to its group orbit.' The empirical tables are substantial and the code is promised, but they do not test this link: the steerability test in Appendix E.7 transforms a fitted latent and checks the output, which is the forward direction, rather than fitting latents for steered fields and checking consistency. The proposed test settles whether the equivariant structure is actually exploited at inference or merely a controllable latent manipulation. Since this is an unproven but plausible gap, and the paper already has a conditional verdict, I do not change the verdict; I recommend keeping CONDITIONAL and asking the authors to run or report this consistency check.","tokens_in":32801,"tokens_out":18346,"duration_ms":172713,"concrete_test":"On a held-out OpenFWI field v and a grid of g in SE(2) (e.g., rotations 0°, 30°, 90°, 180° and a translation), fit z_v by autodecoding with frozen network weights. Then (a) transform the fitted latent to get g·z_v; (b) fit z_{g·v} from scratch on the steered velocity µ(g,v) using the same protocol; (c) compute the relative L2 difference between T_theta(·,·;z_{g·v}) and T_theta(·,·;g·z_v), and report eikonal residuals for both. Repeat with several random latent initializations. If the two solutions agree within the method's reported RE tolerance for all g, the inference-side equivariance holds; if they diverge or one has markedly larger residual, the automatic group-orbit generalization claim is not realized and steerability should be described as a user-level latent manipulation rather than an inference guarantee.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The equivariance statement in Eq. (2) is exact by construction: with invariant features (g_i^{-1}s, g_i^{-1}r) and an invariant semimetric, any function of these features is steerable, so Proposition 4.1's transfer statement is sound. The load-bearing gap is on the inference side. At test time a new velocity field is encoded not by transforming a known latent but by unconstrained SGD over z (Algorithms 1-2). The map from velocity fields to latents is not constrained to be equivariant: nothing ensures that optimizing the eikonal loss for the steered velocity µ(g,v) returns g·z_v rather than some other local minimum. Hence the paper's headline benefit, 'solving one velocity field automatically extends to its whole group orbit without retraining,' only holds if the optimizer cooperates; no convergence or equivariance-of-fitting result is given. Appendix E.7 checks only the forward direction (transform z, verify output), not whether fitting the steered field recovers the steered latent. This is also the point where the finite N=9 latent code must carry enough information for the PDE loss alone to identify the correct solution; the factored ansatz and symmetry reduce but do not remove the multiplicity of eikonal solutions.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":33071,"tokens_out":12557,"duration_ms":119476,"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":[{"comment":"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.","section":"§4.1, Appendix C.1, Appendix E.7"},{"comment":"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.","section":"Abstract, §5.3, §6"},{"comment":"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.","section":"Tables 1–3, Figures 2–3"},{"comment":"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.","section":"§4.3, Theorem 4.1"}],"minor_comments":[{"comment":"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.","section":"§3.2, Eq. (5)"},{"comment":"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.","section":"Definition 4.1"},{"comment":"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.","section":"Theorem 4.1"},{"comment":"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.","section":"Table 1"},{"comment":"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.","section":"Appendix D.1"},{"comment":"In Figure 3, the abbreviations 'AD' and 'META' are used without definition in the caption; please define them as autodecoding and meta-learning, respectively.","section":"Figure 3"}],"recommendation":"major_revision","confidential_remarks":"The theoretical contribution is sound and likely publishable after revision. The main risks are empirical: the absence of uncertainty quantification, the lack of a non-Euclidean baseline, and the mismatch between the abstract's promised domains and the actual experiments. The inference-time equivariance gap is real but fixable with a targeted experiment or a precise qualification of the claim. I would not recommend rejection, but I would require the above empirical additions before acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nWhat'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.\n\nOn 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.\n\nThe 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.\n\nCitation pattern looks fine: builds on Wessels et al., Olver's moving frames, standard eikonal-PINN literature. No red flags.\n\nRecommendation: 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.\n\nI'd cite it and I'd bring it to a reading group.","headline":"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.","tokens_in":33636,"tokens_out":3568,"would_cite":true,"duration_ms":31786,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"A single network can solve the eikonal equation for many velocity fields, and steering the latent code steers the answer exactly.","keywords":["equivariant neural fields","eikonal equation","travel-time prediction","physics-informed neural networks","moving frame invariants","homogeneous spaces","seismic velocity models","neural eikonal solver"],"falsifier":"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.","tokens_in":32615,"feed_emoji":"🌊","tokens_out":11325,"duration_ms":87781,"temperature":0.7,"pith_summary":"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.","feed_headline":"Grid-free equivariant eikonal solver wins 7 of 10 benchmarks","feed_subtitle":"One shared network fits new velocity fields via tiny latent codes and steers solutions without retraining.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Establishes the equivariant neural field formulation and steerability constraint that this paper extends to products of Riemannian manifolds.","marker":"[Wessels et al., 2024]"},{"why":"Supplies the moving frame method and its completeness/separating property used to justify the joint invariants in Theorem 4.1.","marker":"[Olver, 2001]"},{"why":"Provides the neural eikonal solver design, the Hamiltonian loss, the bounded projection head, and the factored travel-time ansatz.","marker":"[Grubas et al., 2023]"},{"why":"Defines the FC-DeepONet baseline and the OpenFWI evaluation protocol that Table 1 compares against.","marker":"[Mei et al., 2024]"},{"why":"Introduces the Functa conditional neural field baseline and autodecoding procedure used in training.","marker":"[Dupont et al., 2022]"},{"why":"Introduces the EikoNet eikonal PINN and the factored $T=\\tilde d\\,\\tau$ parametrization this paper builds on.","marker":"[Smith et al., 2021]"},{"why":"Provides PINNeik, a reference physics-informed eikonal solver, showing the standard loss-based training approach.","marker":"[Waheed et al., 2021]"},{"why":"Supplies the second-order factored Fast Marching method that generates the ground truth travel times for OpenFWI.","marker":"[Treister and Haber, 2016]"},{"why":"Provides the Hamiltonian fast marching solver used to generate spherical ground-truth travel times.","marker":"[Mirebeau and Portegies, 2019]"}],"fun_headline_variants":["Equivariant eikonal net steers travel-time without retraining","Grid-free eikonal solver wins 7/10 on OpenFWI benchmarks","Steerable neural eikonal solver: one net, many velocity fields","Equivariant eikonal solver beats DeepONet on seven datasets","Travel-time prediction made steerable and grid-free"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Equivariant eikonal net steers travel-time without retraining","Grid-free eikonal solver wins 7/10 on OpenFWI benchmarks","Steerable neural eikonal solver: one net, many velocity fields","Equivariant eikonal solver beats DeepONet on seven datasets","Travel-time prediction made steerable and grid-free"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000244,"raw_usage":{"total_tokens":1601,"prompt_tokens":1084,"completion_tokens":517,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":700,"completion_tokens_details":{"reasoning_tokens":423}},"tokens_in":700,"tokens_out":517,"duration_ms":4748,"temperature":1.0,"reasoning_tokens":423,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T15:09:13.846093+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Neural Eikonal solver: Improving accuracy of physics-informed neural networks for solving eikonal equation in case of caustics","cited_arxiv_id":null,"evidence_quote":"Provides the neural eikonal solver design, the Hamiltonian loss, the bounded projection head, and the factored travel-time ansatz."},{"cited_title":"Fully Convolutional Network - Enhanced DeepONet - Based Surrogate of Predicting the Travel - Time Fields","cited_arxiv_id":null,"evidence_quote":"Defines the FC-DeepONet baseline and the OpenFWI evaluation protocol that Table 1 compares against."},{"cited_title":"Scalars are universal: Equivariant machine learning, structured like classical physics","cited_arxiv_id":null,"evidence_quote":"Introduces the EikoNet eikonal PINN and the factored $T=\\tilde d\\,\\tau$ parametrization this paper builds on."},{"cited_title":"A fast marching algorithm for the factored eikonal equation","cited_arxiv_id":null,"evidence_quote":"Supplies the second-order factored Fast Marching method that generates the ground truth travel times for OpenFWI."},{"cited_title":"Hamiltonian fast marching: A numerical solver for anisotropic and non-holonomic eikonal pdes","cited_arxiv_id":null,"evidence_quote":"Provides the Hamiltonian fast marching solver used to generate spherical ground-truth travel times."}],"review_version":1}