Equivariant Eikonal Neural Networks: Grid-Free, Scalable Travel-Time Prediction on Homogeneous Spaces
Authors: Alejandro García-Castellanos
David Wessels
Nicky J. van den Berg
Remco Duits
Daniel Pelt
Erik J Bekkers
Venue: NeurIPS 2025 poster ()
Keywords: Geometric Deep Learning
Neural Fields
Eikonal Equation
Equivariance
Travel-time
Riemannian Geometry
Primary area: deep_learning
TL;DR: We introduce a geometry-aware, equivariant neural field framework for solving Eikonal equations on arbitrary manifolds using steerable latent representations.

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.

Equivariant Eikonal Neural Networks: Grid-Free, Scalable Travel-Time Prediction on Homogeneous Spaces
Alejandro García-Castelle Casteelaasllllnoss 1+
Nicky J. van den Berg" Danië M. Pelt Erik Bekkers'
David R. Wessels 1.2
Remeo Duits
Amsterdam Machine Learning Lab (AMLab), University of Amsterdam New Theory
Department of Mathematics and Computer Science, Eindhoven University of Technology 4 "Leiden Institute Advanced Computer Science, Universiteit Leiden
Abstract
We introduce Equivariant Neural Eikonal Solvers, novel framework that integrates Equivariant Neural Fields (ENFs) with Neural Eikonal Solvers. Our approach em- ploys 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 map- ping from these latent representations to the solution field, delivering three key benefits: enhanced representation efficiency through weight-sharing. robust geo- metric grounding, and solution steerability. This steerability allows transformations applied to the latent point cloud to induce predictable, geometrically meaning- ful modifications in the resulting Eikonal solution. By coupling these steerable representations with Physics-Informet 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 bpssd Eikonal solver methods.
Introduction
The eikonal equation is first-order nonlinear partial differential equation (PDE) that plays central role in wide range of scientific and engineering applications. Serving as the high-frequency approximation to the wave equation [Noack and Clark, (017]) its solution represents the shortest arrival time from source point to any receiver point within a specified scalar velocity field [Sethian 1996). This formulation underpins numerous applications: in Computer Vision, it is integral to the computation of Signed Distance Functions (SDFs) [Jones et al., (006) and geodesic-based image segmentation [Chen and Cohen 2019); in Robotics, facilitates optimal motion planning and inverse kinematics Ni and Qureshi 2023] Li et al.) 2024b); and in Geophysics, it models seismic wave propagation through heterogencous media, enabling critical travel-time estimations [Abgrall and Benamou 1999 Rawlinson et al. 2010 Schuster and Quintus-Boo 1993).
Conventional numerical solvers, such as the Fast Marching Method (FMM) [Sethian 1996 and the Fast Sweeping Method (FSM) [Zaoo] 2004), have historically been used to compute solutions the eikonal equation. However, these approaches are heavily dependent on spatial discretization, leading
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39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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to challenging trade-off: higher resolution is required for complex velocity models, which in turn dramatically increases computational and memory demands [Grubas al. 2023 Song et al. 2024 Mei et al. 2024 Smith et al. 2021 Waheed et al. 2021]. This issue is exacerbated in scenarios involving complex input geometries, such as Riemannian manifolds, which are prevalent in both
computer vision [Bekkers et al. 2015 and robotics applications Li et al. 2024b].
Recent advances in scientific machine learning have introduced neural network-based solvers as promising alternatives. Cyysics-Informed Neu- ral Networks (PINNs) integrate the PDE con- straints into the training loss, offering a grid-free approximation to the eikonal equation and alle- viating the discretization issues inherent in tradi- tional numerical methods ([Smith et al.) 2021
Latents
Travel-Time
Velocity Field
To
Solution to
g 24
Tolg 8, r:))
p(g.vi)
Waheed et al.] 2021 Ni and Qureshi 2023
Grubas et al. 2023 Li al, 2022bb. However, a significant limitation of PINN-like approaches is their requirement to train a new network for each distinct velocity field, which hampers their applicabilliy in real-time scenarios.
Tp
Solution
Figure 1: Stecrability in the Equivariant Neural Eikonal Solver (E-NES) enables weight sharing across the entire group orbit: applying group transformation to the conditioning variable 2,, in- duces a corresponding transformation on the travel- time function (= through the group's 's left regular representation, and the associated ve- locity field dy via the non-linear group action as formalized in Section
Neural Operators overcome this constraint by learning mappings between function spaces specifically, from velocity fields to their corre- sponding travel-time solutions. Unlike PINNs, neural operators utilize a shared backbone and incorporate conditioning variables to handle dif- ferent velocity profiles [Song et al., 2024 Mei et al. 2024]. Current approaches leveraging sim- ple architectures such as DeepONet [Mei et al.)
2024] and Fourier Neural Operators [Song et al.)
2024 have demonstrated promising results, yet there remain several avenues for improvement, as discussed in Section
As stated in Wang et al. 2024b), one promising direction is to recognize that Neural Operators belong to broader class models known as Conditional Neural Fields. These models, which have been popularized within the Computer Vision community, explore advanced conditioning techniques to enhance expressivity, adaptability, and controllability [Dupont al.) [022]] Wessels et al. 2024 Wang et al. (224b). In this work, we focus the recently introduced Equivariant Neural Fields, which ground these conditioning variables geometric principles, leading to improved representation quality and steerability ensuring that transformations in the latent space correspond directly to transformations in the solution space (nigge et al.) 2024].
Our key contributions are follows:
We introduce a novel, expressive generalization of Equivariant Neural Fields to functions defined over products of Riemannian manifolds with regular group actions, including homogeneous spaces associated with linear Lie groups.
We implement this framework through our Equivariant Neural Eikonal Solver (E-NES), to efficiently solve eikonal equations by leveraging geometric symmetries, enabling generaliza- tion across group transformations without explicit data augmentation (see Figure [
We validate our approach through comprehensive experiments on 2D, 3D, and spheri- cal seismic travel-time benchmarks, achieving superior scalability, adaptability, and user controllability compared to existing methods in grid-free manner.
Our code, including scripts to generate the results, is provided at: https://gi thub com/ ACarciaCast/E-MBS
2
Related work
Neural eikonal solvers. Initial neural approaches for solving the eikonal equation have primarily relied on Informed Neural Networks (PINNs), which incorporate the PDE directly into

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the loss function [mmihh et al., 2021, Waheed et al. 2021 Ni and Qureshi 2023 Grubas et al. 2023 Li al., 2024b Kelshaw and Magni 2024). These models are trained individually for each velocity field, achieving high-accuracy reconstructions at the expense of significant computational and memory overhead. Moreover, this per-instance training lacks cross-instance generalization, limiting its practicality for large-scale or real-time applications.
To enable generalization across velocity fields, operator learning methods have been proposed. DeepONet variants (Lu et al. 2021 Mei et al. [024] learn mappings between function spaces, but typically require discretization
either the source or receiver points. This limits resolution invariance and complicates applications requiring continuous evaluations, such as geodesic path planning. Moreover, some methods, such as Mei al. [2024), rely on the supervision of numerical solver, which can be beneficial in some simple scenarios but is bottleneck in complex ones. On the hand, other operator learning approaches such as Fourier Neural Operators [Song et al., 2024] offer solver-free alternatives and incorporate the PDE into the loss, but still rely on partial discretization, inheriting similar resolution constraints.
In contrast, our method avoids discretization entirely by representing both inputs continuously and training solely with PDE supervision. We adopt the Conditional Neural Field (CNF) framework as our backbone, enabling scalable conditioning on velocity fields while preserving grid-free inference and resolution independence.
Conditional and equivariant neural fields. Conditional Neural Fields (CNFs) are coordinate- based networks that reconstruct continuous signals from discrete observations. Formally, a CNF fo M x 2 - Rd maps coordinates p E .M and latent codes = E 2 to outputs fo =) approxi- mating target signals. Given a dataset 'D = [fi] a single network can represent all signals via instance-specific latents: fo 2,) f((p).
Early CNFs used global latent vectors for conditioning [Dupont et al., 2022]. Subsequent work demonstrated that learnable point cloud latents {z, Im C Z significantly enhance expressivity and
reconstruction quality [Bauer et al., 2023 Luijmes et al. 2025 Kazerouni et al., 2025 Wessels et al., 2024). Equivariant Neural Fields (ENFs) further impose symmetry priors through the steerability property: fg p; {z}) = fa {g- ) for all g E G. This equivariant design improves sample
efficiency and generalization (Wessels et al. 2024 Chen et al. 2022 Chatzipantazis et al. 2023 by
encoding geometric structure directly into the latent space.
In this work, we extend ENFs to signals defined products of Riemannian manifolds, i.e., f M, X x.. X M, + Rd, enabling complex inter-input interactions while preserving equivari- ance. This generalization substantially broadens the scope of addressable problems beyond prior single-point formulations, with implications extending beyond eikonal solving (see Appendix
Invariant function learning. Steerability in ENFs requires invariance to joint transformations:
f(g-p: (g [g- fp: {2+}) for all g E G [Wessels et al. 22224 Chen et al. 2022 Chatzipantazis al. 2023). Constructing expressive invariant functions is thus central to our Iramework.
We provide a formal expressivity analysis establishing a complete and maximally expressive set of independent invariants, guaranteeing zero information loss during canonicalization. This addresses a critical gap in prior work (Wessels et al.] 2024,, Knigge et al., [202], where invariants were selected heuristically without completeness guarantees. Our analysis extends arbitrary (possibly non-transitive) Lie group actions on product manifolds.
From Invariant Theory, a complete set of fundamental invariants must: (1) express any invariant func- tion, and (2) separate
(p) = (g) for all invariants I, if and only if p and q share the same orbit [Olver, ] While several computational approaches exist- iniiit-iinuuinn Weyl's theorem
[Wey]] 1946 Villar et al. 2021]. infinitesimal methods [Andreassen, 2020), moving frames [llerr
2001]. and differential invariants Olver 1995 Sangalii et al. 2022 et al. 2024a -we employ the
moving frame technique [Olver 2001 for its conceptual clarity and natural connection to modern
canonicalization methods [Shumaylov et al. 2024 (see Appendix B for detailed comparisons).
Moreover, rather than relying on global canonicalization
produces a single canonical rep- resentation for an entire point cloud- we adopt a local canonicalization strategy (Hu et al.) 2024
Wessels et al. 2024, Chen et al., [022 Du et al., 2022 Wang et al. 2024a] Zhang et al. 2019]. 9 By canonicalizing small patches, our approach is better able to capture relevant local information and
3

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facilitates the use of transformer based architectures, as opposed to the DeepSet-basee based architectures commonly employed in (BBum-Smiii et al. 2024). Dym and Gortler (2023), Villar et al. [202T]
3 Background
In this section, we present the necessary mathematical foundations and formulate the eikonal equation problem. For comprehensive treatment of these topics, we refer the reader to Lee 20188 Olver [1955].
3.1
Mathematical Preliminaries
Differential Geometry. A Riemannian manifold is defined as pair (M.9), where M is a smooth manifold and & is a Riemannian metric tensor field on M [Lee 2018 The metric tensor Gp : T..MM x T.M + R assigns to each p E . positive-definite inner product on the tangent space T..M Specifically, for any tangent vectors pu, P2 E Tp.M, the inner product is given by 9p(01. (p))..2). and the corresponding norm is defined as | واا = V (p(1),))..
The Riemannian distance between two points p and in connected manifold M, denoted as de (p. q). is defined as the infimum of the length of all smooth curves joining them (Lee) 2018).
Curves that achieve this infimum while traveling at constant speed are known as geodesics.
For smooth scalar field/ f :M- + I. the Riemannian gradient grad f is the unique vector field reciprocal to the differential d.f : TMM + R. meaning (grad f, )) df. Given a smooth function
)=d] é : M + M, we can also define the adjoint of a differential do(p) .p..- + To at a point p E . the map (dø(p))* M + T.MM such that for every p E M, q To have that (do())))))
Gp(2, (do(p))) (4]) [Lezcano-Casado 2019
Note that when M I² and 9p I, for all p R", all Riemannian notions reduce to their Euclidean counterparts.
Group Theory. A Lie group G is a smooth manifold with group operations that are smooth. A (left) group action on set X is map μ G X + X satisfying x(e, = and p(g, p(h, x)) 2) for all x E X, g, h E G. When μ is clear by the context we write g x. The orbit space X/G is the quotient space obtained by identifying points in X that lie in the same orbit under the G-action. Formally, X/G = (Orb(z) x E X} consists of all distinct orbits, where each orbit {g x 9 E G} represents equivalence class under the relation G such that y = g x. These equivalence classes partition X into mutually disjoint subsets whose union equals X, yielding canonical decomposition that reflects the underlying symmetry of the group action.
Orb(x)
y + By
In this work, we will focus on the Special Euclidean group SO(n) = =[R= E R RRRT In, det(R) = 1} and the Special Euclidean group 5E(n) = R" x (O(n), representing roto- translations. For (t, R) E SE(n), the group product is g g (t, () R²) Rt',,', RR)).
The isotropy subgroup (or stabilizer) at x E X is G2 = {g E G g. x = x}. group G acts freely if G2 = {e} for all x E XX meaning no non con-identyty element fixes any point. An r-dimensional Lie group acts freely on manifold M if and only if its orbits have dimension r [Olver 1995]. A group acts regularly on M if each point has arbitrarily small neighborhoods whose intersections with each orbit are connected. In practical applications, the groups of interest typically act regularly.
As discussed in Section 2, it is crucial for building Equivariant Neural Fields to obtain a complete set of functionally independent invariants. As demonstrated by Olver [2001], when Lie group acts freely and regularly, such a set can be systematically derived locally using the moving frame method - detailed in the Appendix (Section B
3.2 Eikonal Equation Formulation
On a Riemannian manifold (M,9), the two-point Riemannian Eikonal equation with respect to velocity field e M -> [Umin; ] (where < Umin S Umax < 00) is:
grad, T(s, r)]ls = u(s)_1
grad, T(s, u(r)-
(I)
T(s,r) = s),, T(s,s) =

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where grad, and grad, denote the Riemannian gradients with respect to the source E MM and the receiver E M, respectively. The solution T: M x M M R corresponds to the travel-time function, and the interval [Umin; Umax] specifies the minimum and maximum velocity values in the training set.
To prevent irregular behavior as r + 8, it is standard to factorize the travel-time function as r(s, d(ss T(s,r),
where d is semimetric- a function satisfying non non-ncgativity (d(s,r) > 0). identity of indiscernibles (d(s, r) iff = r), and symmetry (d(s,r) = d(r, s)) that approximates the ground -ruth Riemannian distance dg [Smith et al. [202]] Waheed et al., 2021 Grubas et al. 2023 Li et al., 2024b Kelshaw and Magri 2024]. The scalar field ((8, r) then represents the unknown travel-time factor.
In this work, further require that d(ss r) be invariant under the group action of G; that is, d(s,r) d(g 8,g r)) for all 9 G. This invariance condition is essential ensure that the travel-time function preserves its steerability property, as will be demonstrated in Section 4 When the group acts by isometries, d can, as previously discussed, be taken as the geodesic distance on the manifold. In particular, for manifolds embedded in Euclidean space where the group action extends to isometries of the ambient Euclidean space, the Euclidean (chordal) distance offers computationally efficient approximation [Kelshaw and Magni 2024). For more general group actions, one may instead adopt the discrete semimetric d(s,, r) 1s which equals when s A and 0 otherwise. To maintain gradient compatibility during training, this discrete indicator can be incorporated using straight-through estimator [Bengio et al. 2013].
4 Method
We introduce Equivariant Neural Eikonal Solver (E-NES), which extends Equivariant Neural Fields to efficiently solve eikonal equations by leveraging geometric symmetries. Our approach incorporates steerability constraints that enable generalization across group transformations without explicit data augmentation. We present the theoretical framework (Section 4.1), detail our equivariant architecture (Section 4.2), introduce a technique for computing fundamental cinttiivvariants (Section 4.3) and describe our physics-informed training methodology (Section
4.1 Theoretical Framework
We extend the Equivariant Neural Field architecture introduced in Wessels et al. (024) to represent solutions of the eikonal equation. Let (M,,9) g) denote the input Riemannian manifold on which the eikonal equations are defined, and let G be a Lie group acting regularly on M. We introduce a conditioning variable, represented as geometric point cloud z = ((g,, c;) which consists of N so-called pose-context pairs. Here, each gi E G is referred to as pose, and each corresponding
C, E Rd is the associated context vector. We will denote the space of pose-context pairs as the product manifold Z G x Rd, so that is an element of the power set (2). This representation naturally supports a G-group action defined by g 2 {(g 9:: c.))
In the setting of the factored eikonal equation, consider a solution Tj satisfying Equation ① for the velocity field et- We associate this solution with the conditioning variable 21, such that our conditional neural field (s r; 21) = d(s, r) (a r; 2)) is trained to approximate To r; 2)) v Ti(s, r). for all 8, r E M. Here, 0 denotes the network weights.
The steerability constraint, i.e.,
To(s, g: 2) = -t.
2) for all (s,7, E X (Z),
(2)
incorporates equivariance, enabling the network to generalize across all transformations y E G with- out requiring explicit data augmentation, thus significantly enhancing data efficiency. Consequently. solving the eikonal equation for one velocity field automatically extends to its entire family under group actions, as illustrated in Figure [ This property is formally stated in the following proposition:
Definition 4.1 (g-steered metrie). For all g e G. define the g g-sterred metric G TM x TM R as:
G3 (ú, t) = Gup ((dgg-1 (g [u], (dLy-> (g (b)) for M, and ú,ú úe T.M.
5

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where : - M the diffeomorphism defined by Lp- = g
Proposition 4.1 (Steered Eikonal Solution). Let To M M (2) + R be a conditional
neural field satisfying the steerability property 2). and let 2 be the conditioning variable representing
the solution of the eikonal equation for v : -M- + Riz i.e, To(s, (18,,;,1 Ti f. r) for Ti satisfying
Equation ① for the velocity field Let G9 be g-steered metric (Definition 4.1 Then:
1. The map p. G. (M R2, (M R1) defined by
u(g. (s) = ggrad Ti 5, 9
rr)
go go
where is arbitrary point in M, is well-defined group action.
2. For any E G, To(s, 2)) solves the eikonal equation with velocity field p(g, m)).
For the common cases where the group action is either isometric conformal, the expression for the associated velocity fields admits simpler form:
Corollary 4.1. Assume the hypotheses of Proposition 4.1 then the group action μ : G x (M
R + (M - 1R.) is given by:
1. p(g. (s) = vilg- 1. s) i acts isometrically on M.
2. p(g, (s) = Mgg. s) 11 s) if G acts conformally on M with conformal factor ((g,s) > ie. Ggs (1)]1,, (L)((){(2))) = ()(g,s)² Go (51, 52), V V 82 E T.M.
Since μ G (M (M- (M R2) constitutes a group action on the space of velocity fields, its orbit space induces partition. Therefore, by obtaining the conditioning variable associated with one representative of orbit, we effectively learn solve the eikonal equation for all velocities within that equivalence class.
Finally, steerability also relates grad, To 2) to grad, (s =) (see Lemma A.1 Hence, any geodesic extracted by backtracking the gradient of To for one field generalizes to its transformed counterpart. This property essential for applications such geodesic segmentation (Chen and Cohen 2019], motion planning [Ni and Qureshil 2023 and ray tracing (Abgrall and Benamou [999].
Further details regarding the steerability property for eikonal equations, as well as proofs for Proposi- tion 4.1 and Corollary 4.1 can be found in the Appendix (Section A).
4.2 Model Architecture
We define the Equivariant Neural Eikonal Solver (E-NES) as Ta ===== Po where Wessels al. 22024 and P R + R4 is the bounded projection head from Grubas et al. [2023].
1. Invariant Cross-Attention Encoder. To enforce the steerability, our encoder builds invariant representations under G-symmetries of (s, g.).. For each Ji E 2, we compute:
(e.r) = RFF (Inv (s,
(n,=) = RFF (Tnv 9,))),
where Inv(-) yields a complete set of functionally independent invariants via the moving frame method (as we will explain in Section 4.3), and RFF is a random Fourier feature mapping [Taneik et al. 2020 To enforce (2( =
(e 2), we use a; (*,))) + a. (()=))) //2, the Reynolds operator over S2 Dym et al. 2024). Then the invariant cross-attention encoder is computed as:
N
E(s, ry = FFN
Σ
a va,
with a, = Σ exp(((a))
where the attention maps and values are parameterized as:
(()) W,ā, k(c) = W& LN(W.c),
v v(ā c) = (W, LN(W.c) @ (1 (F), + (a))
with FFN E, FFN,, FFN,,, FFN, being small multilayer perceptron (MLP) using GELU activation functions, and WG, W., W., W.,, W W, learnable linear maps.

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2. Bounded Velocity Projection. The encoder output h Ess, 2) passes through a second MLP network FFNp with AdaptiveGauss activations to model sharp wavefronts and caustics [Grubas et al.) 2023 The final output is projected into [1 /Umaax 1/ //Umnn] by:
1
P(h)
o FFNp(h))
Umin
Umax
Umax
where a is the sigmoid function and cog E IR., is a learnable temperature parameter.
4.3 Computation of Fundamental Joint-Invariants
=1 Mi x.... MM denote a product of Riemannian manifolds, each equipped with a smooth, regular action ò; G x M; + M, by Lie group G. These individual actions induce natural diagonal action on the product II given by 5(g.
Pm.) = (61 (1).P....... ...)........ (..)))).
As observed in Olver [2001) when the group action on II is not free, the standard moving frame method is not directly applicable. In such cases, alternative techniques- such as those discussed in Section 2 are typically employed to compute invariants.
We show that the moving frame method can be restored in this setting by augmenting the space II with an auxiliary (learnable) group element, yielding extended space IT II x G. On this augmented space, the group action admits a canonicalization procedure with explicitly computable invariants:
Theorem 4.1 (Canonicalization via latent-pose extension). Let II and G be as above. Define a new group action 8 : G x II + II by 8(h, (PI....,Pry. 9)) (8(h, .oss...,Pmm))... h- g). Then, the set forms complete collection of functionally independent invariants of the action ş.. {ö:(g (p)) 1=
Sketch of proof (full at Appendix, Section By To verify that the action şI is free, we show that the isotropy group of any point (P1,-.. Pm. g) E IT is trivial. Specifically, this subgroup satisfies G Gipp.... Pmp.....P... Gp, 7..-7 n Gpm. n G. where Gp, denotes the isotropy subgroup of pt under d, and Gg is the isotropy subgroup of g E G under left multiplication. Since f 9 = g implies h = e in group, we have G2 {e). Thus, the intersection is trivial, and 5 defines free action. The moving frame method then guarantees a complete set of invariants, which are exactly
p.)) O
This result formally justifies the construction proposed in Wessels et al. [024], showing that the method yields complete set of functionally independent invariants and thus guarantees full expres- sivity. Moreover, extends the applicability of Equivariant Neural Fields to settings where G acts regularly-but not necessarily freely transitive-on on product manifolds. In particular, the invariant computation used in our E-NES architecture, as presented in Equation (4), takes the form
Inv(s, r. =(g s, r) E M x M.
4.4 Training details
Let V = {ur : M + [imin: mmax]] KKK be our training set of velocity fields over the domain M.
At each iteration, we sample batch B with velocity fields {w C V and N., source-receiver pairs
C. .2 for each Un Let { be the conditioning variables associated with {vx)_ To enforce the Eikonal equation, we express it in Hamiltonian form H(s r. T) [ grad, T(s, r)l - 1, where the Eikonal equation is satisfied when H 0 |Grubas al. 2023). We then minimize physics-informed loss that penalizes deviations from this zero-level set at both source and receiver locations:
Nar
L(0, ()) N
1
ΣΣ [stg)) grad,, (s Figs 2 1|
(5)
(r:,) gragd, To Tig: 2))1 -1)
Fitting performed in the two modes presented in Wessels et al. [2024). The first one is Autodecoding [Park et al. 2019 where 2 and 0 are optimised simultaneously over dataset. The second one is Meta-learning [Tancik al. [02 Cheng and Alkhalifah 2024) where optimization split into an outer and inner loop, with being optimized in the outer loop and z being re-initialized every outer step to solve the eikonal equation of the velocity fields the batch in a limited number of SGD steps in the inner loop. We refer the readers to the Appendix (Section C for further details.
7

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Table Performance comparison on OpenFWI datasets against FC-DeepONet. Colours denote Best, Second best, and Third best performing setups for each dataset. Fitting time represents the total computational time required to fit the latent conditioning variables for all 100 testing velocity fields.
E-NES
FC-DeepONet
Autodccoding (100 epochs)
Autodecoding (convergence)
Meta-leaming
Dataset
RE ())
Fitting (8)
RE (U)
Fitting (8)
RE (1)
Fitting (8)
RE (4)
Fitring (6)
FlarVel-A
0.00277
0.615
0.00952
223.31
0.00506
1120.25
0.01065
5.92
CurveVel-A
0.01878
0.615
0,01348
222.72
0.00955
1009.67
0.02196
5,91
FlutFault-A
0.00514
0.615
0.00857
222.61
0.00568
1014.45
0.01372
5.92
Curvefrault-A
0.00963
0.615
0,01108
222.89
0.00820
1123,90
0,02086
5.92
Style-A
0,03461
0.615
0.01034
222.00
0.00833
1117.99
0.01317
5.92
FlatVel-B
0.00711
0.615
0.01581
222.74
0.00860
1010.32
0.02274
5.91
CurveVel-B
0.03410
0.615
0,03203
222.97
0.02250
1127.87
0,03583
5.90
FlutFaultBB
0.04459
0.01989
222.70
0.01568
1133.98
0.03058
5.93
CurveFault-B
0.07863
0.615
0,02183
222.89
0.01885
893.84
0.03812
5.89
Style-B
0.03463
0.615
0.01171
221.90
0.01069
896.06
0,01541
Experiments
We evaluate Equivariant Neural Eikonal Solvers (E-NES) on the 2D OpenFWI benchmark (Deng al., 2022] and extend our analysis to 3D settings to assess scalability and spherical geometry to
show its generalization capabilities. Implementation details are provided in the Appendix (Section D). public repository.
The code, including the experiments, is provided in the
5.1
Benchmark on 2D-OpenFWI
Following Mei et al. 2024 we utilize ten velocity field categories from OpenFWI: FlatVel-A/B, CurveVel-A/BB PlatFault-A// CurveFault-A/B, and Style-A/B, each defined on 70 x 70 grid. We train E-NES 500 velocity fields per category and evaluate on 100 test fields, positioning four cquidistant source points at the top boundary and computing travel times to all receiver coordinates. Additional evaluations using denser 14 x 14 source grid are presented in the Appendix (Section E.2
Performance is quantified using relative error (RE) and relative mean absolute error (RMAE):
N,
Mp レ T -
1 N. RMAE = N 1
RE:
N.
Σ Σ
TH
where N₂ represents the total number of samples, Mp denotes the total number of evaluated source- receiver pairs, Т indicates the j-th point of the i-th ground truth travel time, and 7 represents the model's 's predicted travel times. The ground truth values are generated using the second-order factored Fast Marching Method (Triister and Haber 2016).
5.1.1 Impact of Steerable Geometric Conditioning
To empirically validate the theoretical benefits of equivariance in our formulation, con- ducted a controlled ablation study comparing
0.10
10-1
No Equivariant Equivariant
MSE (log snale)
Elhocal Lees
E-NES with equivariance (Z SF((2 Re)
against variant without equivariance con- straints (Z e IR9) the Style-B dataset. Fig- ure Tre@inn illlustaats consistent performance advan- tages with equivariance, demonstrated by lower values in both Eikonal loss and mean squared er- ror (MSE) throughout the training process. This empirical validation substantiates our theoreti- cal motivation for incorporating explicit equiv- ariance constraints into the model architecture.
10-3
0.04
500
1000
1500
500
1000
1500
Step (x700))
Step (x1000))
Figure 2: Comparative analysis of cquivariant con- ditioning variables on the Style-B dataset. For non-equivariant models R while equivari- ant models use Z = (2) x R

---

To further assess the role of geometric condi- tioning, we compare our method against Functa Dupont et al. 2022), common baseline in the literature on conditional neural fields. Functa employs SIRENs (Sitzmann et al. 2020] with sample-specific scale and shift modulation but relies global latent variables without geomet- ric constraints, providing a clear contrast to our geometric point-cloud formulation of the con- ditioning variables. As shown in Table 2 our method consistently outperforms Functa across all datasets. We attribute this to Functa's global conditioning, which lacks localized representa- tion and explicit geometric constraints.
Table 2: Performance comparison on OpenFWI B-type datasets against Functa. Fitting time repre- sents the total computational time required fit the latent conditioning variables for all 100 testing velocity fields. Here both methods perform 100 epochs of autodecoding to fit the latents.
Funeta
E-XES
Dataset
RE (L)
Fitting (6)
RE (L)
Fitting (6)
FlatVel-B
0,11854
12.55
0.01581
222.74
CurveVel-B
0.11210
12.49
0.0320.
222.97
FlatFault-B
0.06428
12.50
0.01989
222.70
CurveFault-B
0.06146
12.72
0,02183
222.89
Style-B
0.03106
12.33
0.01171
221.90)
5.1.2 Performance Comparison
AD RMAE
Table presents a systematic comparison between E-NES and FC-DeepONet [Mei et al. 2024) across all OpenFWT benchmark datasets. We evaluate three configurations of E- NES: autodecoding with 100 epochs (tradcoff between computational effi- ciency and performance), autodecoding until conver- gence (optimizing for accuracy), and meta-learning (prioritizing computational efficiency).
META RMAE
Error
2x2x2
3x3xx3
4xp4x4
5xcoc5
6x6x6
Source points grid size
Our results demonstrate that E-NES with full au- todecoding convergence outperforms FC-DeepONet in seven out of ten datasets, with particularly sub-
FMM
Computation (x.00)
AD Inference
stantial improvements on the more challenging variants-Supe Striant-SStyleA/BB FlatFault-B, and CurveFault-B Even with the reduced computational budget of 100 epochs, E-NES maintains competitive performance across most datasets. The meta-learning approach, while exhibiting moderately higher error rates, deliv- remarkable computational efficiency efficiency-reduccng reducing fitting time from approximately 1000 seconds to un- der seconds for the total 100 velocity fields, rep- resenting a two orders of magnitude improvement. Additional analyses provided in Appendix F33
META Inference
2x2x2
3x3x3
4x4xtt
5x5x5
6x6x6
Source
(b)
Figure 3: Scaling analysis of E-NES versus FMM on 3D OpenFWI data. (a) Both au- todecoding and meta-learning maintain con- sistent error metrics (RE and RMAE, x10 2) across increasing grid dimensions. (b) E-NES demonstrates computational advantages (sec- onds x x10²) over FMM sizes, with efficiency gains amplifying as di- mensions increase. Note that meta-learning fitting times (approximately 3 seconds) are barely visible in (b) due their minimal mag- nitude relative to other displayed times.
The quantitative results are supplemented by qualita- tive evaluations in the Appendix (Section(..6) includ- ing visualizations of travel-time predictions and spa- tial error distributions across all datasets. For more detailed analysis the trade-off between computa- tional efficiency and prediction accuracy, including performance with varying numbers of autodecoding epochs, refer to the ablation studies in the Ap- pendix (Section
minimal grid
5.2 Extending to 3D: Scalability Analysis
To evaluate scalability to higher dimensions, we extended the Style-B B dataset 3D by extruding 2D velocity fields along the z-axis. Figure Ba shows that both autodecoding and meta-learning approaches maintain stable error metrics as grid dimensions increase, demonstrating -ESS's ability to model continuous fields independent of discretization resolution. Figure (3b/e//b/hwww E-NES maintains efficiency advantages over Fast Marching Method (FMM) across all evaluated grid dimensions, with this advantage becoming more pronounced at larger scales (total computation time for all 100 test velocity fields). This stability stems from E-NES's continuous representation, which adapts the underlying physics without requiring increasingly fine discretization.

---

Polar Coords: Constant Velocity Field
Vobood Food
Coords: Obstacle
2-
2
2 9
Constant Velocity Field
Gaussian Obstacle Velocity Field
Figure 4: Geodesic path planning on the sphere using gradient integration of the travel-time function under velocity fields. Left panels show the trajectories in local polar coordinates, while right panels visualize the corresponding paths on the spherical surface. The constant velocity field (top) yields great-cirele path, whereas the Gaussian obstacle velocity field (bottom) causes the trajectory to bend around the low-speed region. The diamond denotes the start and the circle the goal.
Generalizability to Non-Euclidean Domains
We validate the generality of our framework on the 2-sphere, i.e., on 82 C IR$, with SO(2) steerability (rotations about the z-axis). This setting demonstrates two key capabilities: (i) handling non-trann -antraanittvve Lie group actions, and (ii) extending to non- non-Euclidnn Euclidean geometries.
Table 3: Performance of our method Eikonal
solvers over the 2-sphere.
Dataset
RE (L)
RMAE (L)
Fitting Time (s)
Constant Speed
0.013
0.012
209.2
Sphericul Style-B
0.015
0.012
207.1
We test on two velocity field types: constant speed fields with velocities uniformly sampled, and Spherical Style- fields, obtained by projecting OpenFWI's 2D Style-B fields onto the sphere via spherical coordinates. As shown in Table our method achieves strong performance on both benchmarks, effectively learning the sphere's 's intrinsic geometry and correctly modeling wavefront propagation despite using Euclidean chordal distance d(s,r) r) in the factorized representation (as described in Section 3.2)
Moreover, Figure 4 demonstrates that E-NES enables geodesic path planning via gradient integration, yielding optimal trajectories under configurations with and without obstacles. Additional details on how to perform this path- aathffnding task are provided the Appendix C.2
Discussion and Future Work
In this work, we proposed a systematic approach to incorporate equivariance into neural fields and demonstrated effectiveness through our Equivariant Neural Eikonal Solver (E-NES), Our experiments show that E-NES outperforms both Neural Operator methods (e.g., FC-DeepONet) and Conditional Neural Field approaches (e.g., Functa) across most benchmark datasets. The grid-free formulation is particularly advantageous for gradient integration tasks and naturally extends to Riemannian manifolds.
While our method requires explicit optimization test time. FC-DeepONet's encoder forward pass performs implicit latent fitting (0.615 seconds for 100 velocity fields, indicated Table [T). Criti- cally, test-time optimization enables practitioners to dynamically adjust the accuracy-efficcees trade-off by varying the number of iterations (Appendix E. This adaptability parallels recent test- time optimization advances in large language models (Zhang al.) 2025), whereas FC-DeepONet's performance fixed post-training. Additional comparative analyses are provided in Appendix [..T
For future work, we plan to extend our analysis to homogeneous spaces beyond Euclidean and spherical domains, including position- coatiionorieetationn spaces for systems with nonholonomic constraints (e.g., vehicle path planning) and hyperbolic spaces for hierarchical interpolation tasks.
10

---

Acknowledgements
We wish to thank Maksim Zhdanov for his help on the JAX implementation. Alejandro Garefa Castellanos is funded by the Hybrid Intelligence Center, 10-year programme funded through the research programme Gravitation which is (partly) financed by the Dutch Research Council (NWO). This publication is part of the project SIGN with file number VI. Vidi. fidi.233.220 of the research programme Vidi which is (partly) financed by the Dutch Research Council (NWO) under the grant httpss:/doi. org/11 /0. 61686//6680611 61686/PKQG277 /P/G//KK2Z20000 1165 David Wessels is partially funded Ellogon.Al Al and public grant of the Dutch Cancer Society (KWF) under subsidy (15059/2022-PPS)
Remco Duits and Nicky van den Berg gratefully acknowledge NWO for financial support via VIC- C 202.031.
References
Rémi Abgrall and Jean-David Benamou. Big ray-tracing and eikonal solver on unstructured grids: Application to the computation of multivalued traveltime field in the Marmousi model. GEO- PHYSICS, 64(1):230-239, January 1999. ISSN 001 0016-8,33, 1942-2156. doi: 10. 1..1190/1.144555 URL https:///ibraat seg. org httg//oi/110..o R00.0...190/// 1190/1. 1444519
P.-A. Absil, R. Mahony, and R. Sepulchre. Optimization algorithms matrix manifolds. Prince- ton University Press, Princeton, N.J. ; Woodstock, 2008. ISBN 978-0-691-11298-3. OCLC: 0c1974129993.
Fredrik Andreassen. Joint Invariants of Symplectic and Contact Lie Algebra Actions. 2020.
Matthias Bauer, Emilien Dupont, Andy Brock, Dan Rosenbaum, Jonathan Richard Schwarz, and Hyunjik Kim. Spatial Functa: Scaling Functa to ImageNet Classification and Generation, February 2023. URL httpp:///aaa org/ 03130 arXiv:2302.03 [cs].
Erik Bekkers, Remco Duits, Alexey Mashtakov, and Gonzalo R. Sanguinetti. A PDE Approach Data-driven Sub-Riemannian Geodesics in SE(2), April 2015, URL http://arxiv org/ /bs//
1503. 01433
arXiv:1503.0143: [math].
Yoshua Bengio, Nicholas Léonard, and Aaron Courville. Estimating propagating gradients through stochastic neurons for conditional computation. arXiv preprint arXiv: /308.33333..33 2013.
Ben Blum-Smith, Ningyuan Huang, Marco Cuturi, and Soledad Villar. Learning functions on symmetric matrices and point clouds via lightweight invariant features, May 2024. URL http:
//axxiv. petriivorrg/a /tap////00.0.0 08097 arXiv:2405.080% [cs, math].
Evangelos Chatzipantaziss Stefanos Pertigkiozoglou. Edgar Dobriban, and Kostas Daniilidis. SE(3)- Equivariant Attention Networks for Shape Reconstruction in Function Space, February 2023. URL
httt::/aaxiv.. org/ www.:orgg/bbbbbaeeeeo www. 220pp//2222002444444 arXiv:2204.02394 [cs].
Da Chen and Laurent D. Cohen. From Active Contours to Minimal Geodesic Paths: New Solutions to Active Contours Problems by Eikonal Equations, September 2019. URL http::ttaxxii.ooo/// orrg abs/ 1907 09828, arXiv:1907.0999 009828 [cs].
Yunlu Chen, Basura Fernando, Hakan Bilen, Matthias Nießner, and Efstratios Gavves. 3D Equiv- ariant Graph Implicit Functions, March 2022. URL httt:://txxxi org. abs 2203 httgg/aa//....11///// 17178 arXiv:2203.1 17177 [cs].
Shijun Cheng and Tariq Alkhalifah. Meta-pinn: Meta learning for improved neural network wavefield solutions. arXiv preprint 1oXiv:2401.11002. 2024.
Jose Luis Blanco Claraco. A tutorial on SE(3) transformation parameterizations and on-manifold optimization. Technical report, 2022.
Chengyuan Deng, Shihang Feng, Hanchen Wang, Xitong Zhang, Peng Jin, Yinan Feng, Qili Zeng, Yinpeng Chen, and Youzuo Lin. Openfwi: Large-scale multi-structural benchmark datasets for full waveform inversion. Advances in Neural Information Processing Systems, 35:6007-6020. 2022.
11

---

Weitao Du, He Zhang, Yuanqi Du, Qi Meng, Wei Chen, Nanning Zheng, Bin Shao, and Tie-Yan Liu. SE(3) Equivariant Graph Neural Networks with Complete Local Frames. In Proceedings of the 39th International Conference on Machine Learning, pages 5583-5608. PMLR, June 2022. URL ISSN: 2640-3498.
mlr wwwpsreessvveeee v12 du22e. html
Emilien Dupont, Hyunjik Kim, S. M. Ali Eslami, Danilo Rezende, and Dan Rosenbaum. From data to functa: Your data point is a function and you can treat like one, November 2022. URL http://arxiv.cm org/ abs/ 2201 12204 arXiv:2201.1220 [cs].
Nadav Dym and Steven J. Gortler. Low Dimensional Invariant Embeddings for Universal Geometric Learning, November 2023. URL Bttp:/tp///axii wwt.:/og///........o aaep/bbbs//..... 02956 arXiv:2205.0221 02956..
Nadav Dym, Hannah Lawrence, and Jonathan W. Siegel. Equivariant Frames and the Impossibil- ity Continuous Canonicalization, June 2024. URL arXiv:2402.16077.
16077
Serafim Grubas, Anton Duchkov, and Georgy Loginov. Neural Eikonal solver: Improving accuracy of physics-informed neural networks for solving eikonal equation in of caustics. Journal of Com- putational Physics, 474: 11179, February 2023. ISSN 0021-9991. doi: 10. 10.10161i.ccp222221117788
URL buttps:/wwat http://www..ceeecccirr...... com ott.a//w.......a.........//////a
Xin Hu, Xiaole Tang, Ruixuan Yu, and Jian Sun. Learning 3D Equivariant Implicit Function with Patch-Level Pose-Invariant Representation. November 2024. URI. htpppre:///eeerreiiee
Aryaman Jeendgar, Aditya Pola, Soma S. Dhavala, and Snehanshu Saha. LogGENE: smooth alternative check loss for Deep Healthcare Inference Tasks, June 2022. URL http::/arp//
htg/abs/2206 09333 arXiv:2206.0923 099333 [cs] version: 1.
M.W. Jones, J.A. Baerentzen, and M. Sramek. distance fields: a survey of techniques and applications. IEEE Transactions Visualization and Computer Graphics, 12(4):581 July 2006. ISSN 1941-0506, doi: 10.1100/TVCG.2006. URL/https: /tte:://eeeeeep//eeeee ieee. ieee.org/ Conference Name: IEEE Transactions on Visualiza-
org/
tion and Computer Graphics.
Amirhossein Kazerouni, Soroush Mehraban, Michael Brudno, and Babak Taati. LIFT: Latent Implicit Functions for Task- and Data-Agmostic Encoding, March 2025. URL http:/ttp///xii org/ /bs/ 2503. 15420 arXiv:2503. 15420 [cs].
Daniel Kelshaw and Luca Magri. Computing distances and means on manifolds with metric- constrained Eikonal approach, April 2024. URL http://arxiv.. wttp://..orraaa//////0/ 08754
arXiv:2404.08754 [cs, math).
David M. Knigge, David R. Wessels, Riccardo Valperga, Samuele Papa, Jan-Jakob Sonke, Efstratios Gavves, and Erik Bekkers. Space-Time Continuous PDE Forecasting using Equivariant Neural Fields, June 2024. URLphttp:// /ttp///axvvvvv. www.xiiv.rrgaeoo org/ abs/ 2406. 06660 arXiv:2406.06660 [cs].
John M. Lee. Introduction to Riemannian Manifolds, volume 176 of Graduate Texts in Mathe- matics. Springer International Publishing, Cham, 2018. ISBN 978-3-319-9175422 978-3 91755-9. doi:
URL http::/link: springer hwwppiiger.co/11/// 1007
116-3-319--111 319-5-5---7
Mario Lezcano-Casadee Trivializations for Gradient- Based Optimization on Manifolds, October 2019. URL/attp://arxiv org/ httgg//b//19...... 09501 arXiv:1909.09501.
Yikang Li, Yeqing Qiu, Yuxuan Chen, Lingshen He, and Zhouchen Lin. Affine Equivariant Networks Based on Differential Invariants. 2024a.
Yiming Li, Jiacheng Qiu, and Sylvain Calinon. A Riemannian Take on Distance Fields and Geodesic Flows in Roboties, December 2024b. http:///://arxiv org/ abs/aaa// 05197 arXiv:2412.05197 [es].
12

---

Lu Lu, Pengzhan Jin, and George Em Karniadakis. DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators. Nature Machine Intelligence, 3(3):218-22) March 2021. ISSN 2522-5839. doi: 10. 1038/
042256-021-00030255. wtt.tttpp///xxxi..ooo//////////a
1910. 03193 11X1v:1910003193 [es].
Joost Luijmes, Alexander Gielisse, Roman Knyazhitskiy, and Jan van Gemert. ARC: Anchored Representation Clouds for High-Resolution INR Classification, March 2025. http:
org/ 15156, 15156 [cs].
Yifan Mei, Yijie Zhang, Xueyu Zhu, Rongxi Gou, and Jinghuai Gao. Fully Convolutional Network- Enhanced DeepONet- -ased Surrogate of Predicting the Travel-Time Fields. IEEE Transactions
doi: 10.1109/TGRS. on Geoscience and Remote Sensing, 62:1-12, 2024. ISSN wtt.eeeenettt1n99//ee / leee. org/ dww.m/ot
2024.340196. URL htttps://eeexxple.
10530929 Conference Name: IEEE Transactions on Geoscience and Remote Sensing.
Jean-Marie Mirebeau and Jorg Portegies. Hamiltonian fast marching: A numerical solver for anisotropic and non-holonomic eikonal pdes. Image processing on line, 9:47-93, 2019.
Ruiqi Ni and Ahmed H. Qureshi. NTFields: Neural Time Fields for Physics-Informed Robot Motion Planning, March 2023. URL http://///aaxiii org. 22110 00120 arXiv:2210.00120 [es].
Marcus M. Noack and Stuart Clark. Acoustic wave and eikonal equations in a transformed metric space for various types of anisotropy. Heliyon, 3(3):000200 March 2017. ISSN 2405-8440. doi: 10. http://heeiyooo217.00022222 p.1/$2406844006320933.
URL https: www. wwwp://ccceneediret.cccomm www http://rrrrrreecoccccccc///raaaa
Peter J. Olver. Equivalence, Invariants and Symmetry. Cambridge University Press, June 1995. ISBN
978-0-521-4781115 Google-Books-ID: HUIZZ61HILAC.
Peter J. Olver. Joint Invariant Signatures. Foundations of Computational Mathematics, 1(1): --.. January 2001. ISSN 1615-3375.
doi: 10. 10.10077/112008010001 URL http:tttt///:k..ttt .priiinersr
com/10 12. 1001000011228001000111
Peter J. Olver. Lectures Moving Frames. In Decio Levi, Peter Olver, Zora Thomova, and Pavel Winternitz, editors, Symmetries and Integrability of Difference Equations, pages 207-246.
Cambridge University Press, 1 edition, June 2011. ISBN 978-0-521-13658-1 978-0-511-99771144
doi: 10.1017/CB097780511 199719339111 http:ttss:///wet wtt.://awwbbrege..ooooooooo/eopa wowp/pprreepe////o// (lle::/4eee020077771111111116660
Jeong Joon Park, Peter Florence, Julian Straub, Richard Newcombe, and Steven Lovegrove. Deepsdf: Learning continuous signed distance functions for shape representation. In Proceedings of the IEEF/CVF conference on computer vision and pattern recognition, pages 165-174, 2019.
N. Rawlinson, M. Sambridge, and J. Hauser. Multipathing, reciprocal traveltime fields and raylets. Geophysical Journal International, 1812))10077-1092 2010. doi: 136111111136--266..222 04
Mateus Sangalli, Samy Blusseau, Santiago Velasco-Forera and Jesus Angúlo. Differential Invariants for (35(2)-Eqiivart) Networks. In 2022 IEEE International Conference on Image Processing (ICIP), pages 2216 2220, October 2022. doi: 10. 10.11199/17946 211...2222.9997300 URL https / //ieeexplore www.eeeeeeeeeee...eeeeo.o.
org/ abtpraattttattteeee http://dommott////9000/ ISSN: 2381-8549.
Mateus Sangalli, Samy Blusseau, Santiago Velasco-Forero, and Jesús Angulo. Moving frame net: SE(3)-cqiivarian network for volumes. In Proceedings of the 1st NewrIPS Workshop Symmetry and Geometry in Neural Representatiions pages 91-97. PMLR, February 2023. URL ISSN: 2640-3498.
https://prooceddnns. mlr. www.ppresssvvie wwwr.press/v19///aaaas wttp:////v11///aaall1aaa
html
Gerard T. Schuster and Aksel Quintus Quistus-Bosz. Wavepath eikonal traveltime inversion: Theory. GEO-
PHYSICS, 11493333 1993. doi: 10.1199/1.1443514. URL httthtttt//ooiii ort.1/...rr11..000 1190/ 1 1443514
J A Sethian. A fast marching level set method for monotonically advancing fronts. Proceedings of the National Academy of Sciences the United States of Americu, 15145195 February 1996. ISSN 0027-8424. URLltttps //www//wwww//.oo ncbi www.mmmm...ii.mmmm.iimi..aa..iww
13

---

Zakhar Shumaylov, Peter Zaika, James Rowbottom, Ferdia Sherry, Melanie Weber, and Carola- Bibiane Schönlieb. Lie Algebra Canonicalization: Equivariant Neural Operators under arbitrary Lie Groups, October 2024. URL math].
02698 arXiv:2410.02698 [cs,
Vincent Sitzmann, Julien Martel, Alexander Bergman, David Lindell, and Gordon Wetzstein. Im- plicit neural representations with periodic activation functions. Advances in neural information processing systems, 33:7462-7473, 2020.
Jonathan D. Smith, Kamyar Azizzadenesheli, and Zachary E. Ross. EikoNet: Solving the Eikonal equation with Deep Neural Networks. IEEE Transactions on Geoscience and Remote Sensing, 59(12):10685-1000 10696, December 2021. ISSN 0196-2892, 1558-0644. doi: 10.1109/TGRS.000 3039165, URL
arXiv:2004.0036 [physics, stat].
Joan Solà, Jeremie Deray, and Dinesh Atchuthan. A micro Lie theory for state estimation in robotics, December 2021. URL http:://axxi..rrr/g///// http://///abbb/aagoeee 01537 arXiv: [cs].
Chao Song, Tianshuo Zhao, Umair Bin Waheed, Cai Liu, and You Tian. Seismic Traveltime Simulation for Variable Velocity Models Using Physics-Informn Informed Fourier Neural Operator. IEEE Trunsactions on Geoscience and Remote Sensing, 62:1-9, 2024. ISSN 1558-0644. doi: 10. N0.1109/7GRRS..04444377777 Arnumber=106788 Conference Name: TEEE Transactions on Geoscience and Remote Sensing.
URL htttps://iieexpporre ieee. www..e..e....o.eeeeeeeeeeeeeeeee wwwp/rourgg..//////ooooooo/o (ttccummnn/1167889777 106788977
Matthew Tancik, Pratul Srinivasan, Ben Mildenhall, Sara Fridovich-Keill Nithin Raghavan, Utkarsh Singhal. Ravi Ramamoorthi, Jonathan Barron, and Ren Ng. Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains. In Advances in Neural Information Processing Systems, volume 33, pages 7537-7547. Curran Associates. Inc., 2020. URL 550538226667777779999662222333 Abstratttttttttttt...
nne neurips..... httpps..cpppaee wwtpcc/pppppp.....ee files/le/// wwles/paper/ 2220/22000hhhha/
Matthew Tancik, Ben Mildenhall, Terrance Wang, Divi Schmidt, Pratul P Srinivasan, Jonathan T Barron, and Ren Ng. Learned initializations for optimizing coordinate-baset neural representations. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 2846-2855, 2021.
Eran Treister and Eldad Haber. A fast marching algorithm for the factored eikonal equation. Journal of Computational physics, 324:210-222 225, 2016.
Soledad Villar, David W Hogg, Kate Storey-Fisher, Weichi Yao. and Ben Blum-Smith. Scalars are universal: Equivariant machine learning. structured like classical physics. In Advances in Neural Information Processing Systems, volume 34, pages 28848 228463 Curran Associates, Inc., 2021. URL htttps: http://prooeedinggg..ueeusssspps cc/paper www.ffillepaaaeeee httpp/papp2////aa///e/ I (1007794bbc333333322266066333353 www.:e/sssttbbttttttttoooo
Umair bin Waheed, Ehsan Haghighat, Tariq Alkhalifah, Chao Song, and Qi Hao. PINNeik: Fikonal
solution using physics-infrrmed neural networks. Computers Geosciences, 155: 105:33 October 2021. ISSN 00983004. doi: 10.101 10.1016/j.cagen
1044833.. URL http:://atxii.....///// wrw. abs/ arXiv:2007.0833 08330 [math-ph: physics:physics].
Dian Wang, Robin Walters, and Robert Platt. So(()-equivariant reinforcement learning. arXiv preprint arXiv:2203.044 04439, 2022.
Ling Wang, Runfa Chen, Yikai Wang, Fuchun Sun, Xinzhou Wang, Sun Kai, Guangyuan Fu, Jianwei Zhang, and Wenbing Huang. Equivariant Local Reference Frames for Unsupervised Non-rigid Point Cloud Shape Correspondence, April 2024a. URL http://a//a/ org/aba/////eoooooooooogooabaebt 2404. 00959 MxXiv:2404.00950 [cs].
Sifan Wang, Jacob H. Seidman, Shyam Sankaran, Hanwen Wang, George J. Pappas, and Paris Perdikaris. Bridging Operator Learning and Conditioned Neural Fields: A Unifying Perspective, May 2024b. URt.
13998 Stti::2405.19999 [cs, stat].
David R. Wessels, David M. Knigge, Samuele Papa, Riccardo Valperga, Sharvaree Vadgama, Efstra- tios Gavves, and Erik J. Bekkers. Grounding Continuous Representations Geometry: Equivariant Neural Fields, June 2024. URL h
sattp://arx:x //tp:v//axivvaa/ ort//ogg//////eo/oo/ooooeeeeoe/t abs/ 05753 arXiv:2406.05753 [cs].
14

---

Hermann Weyl. The classical groups: their invariants and representations. Princeton landmarks in mathematics and physics Mathematics. Princeton University Press, Princeton, N.J. Chichester, 2nd ed., with suppi edition, 1946. ISBN 978-0-691-00923-3 978-0-99977-------775---
Tianhe Yu, Deirdre Quillen, Zhanpeng He, Ryan Julian, Karol Hausman, Chelsea Finn, and Sergey Levine. Meta-world: A benchmark and evaluation for multi-task and meta reinforcement learning. In Conference robot learning. pages 1094-1100. PMLR, 2020.
Qiyuan Zhang, Fuyuan Lyu, Zexu Sun, Lei Wang, Weixu Zhang, Wenyue Hua, Haolun Wu, Zhihan Guo, Yufei Wang, Niklas Muennighoff, et al. A survey on test-time scaling in large language models: What, how, where, and how well? arXiv preprint urXiv:2503.24235 2025.
Zhiyuan Zhang, Binh-Son Hua, David W. Rosen, and Sai-Kit Yeung. Rotation Invariant Convolutions for 3D Point Clouds Deep Learning. In 2019 International Conference on 3D Vision (3DV), pages 204 213, September 2019. doi: 10.1109/3DV.2019 ,000311 URL URL htttpss://eeexploreeeeeee
http://cmmat//88866////aaaaaaaap 88860 8886052
ISSN: 2475-7888.
Hongkai Zhao. A fast sweeping method for Eikonal equations. Mathematics of Computation, 74 (250))6036627, May 2004. ISSN 0025-5718, 1088-6842. doi: 10. 000110//0000005571--000000066660
URL https: / wwww//wwww..om ams
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Appendix
Table of Contents
A Steerability and Gradient Equivariance
17
Computing Invariants via Moving Frame
20
Preliminaries.
20
Canonicalization via patent-pose extension
20
C Training and Inference Details
21
C.I Training
21
Inference
22
D Experimental Details
23
D.I 2D OpenFWI Experiments
23
D.2 3D OpenF Experiments
24
D.3 Spherical Experiments)
24
E
Additional results
25
E.1
Impact of Autodecoding Pretraining on Meta-Learning Performance
25
E.2
Comprehensive Grid Evaluation of Open! WT Performance
26
E.3
Ablation Study: Choice Embedding for Computing Invariants
26
Ablation Study: Latent Conditioning Design]
27
Ablation Study: Impact Autodecoding Fitting Epochs
27
E.6
Qualitative Analysis of Travel- Time Predictions
29
E.7
Empirical Stecrability Test
32
F
Extended Discussion
32
F1
Improved Memory Scalability
32
F2
Applicability Beyond Seismic Travel-Time Modeling
33
Additional Discussion on Performance Comparison against FC-DeepONel
34
16

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A Steerability and Gradient Equivariance
Let G be Lie group acting smoothly on the Riemannian manifold (M,9) by left-translations
Lg: M + M,
Lg(p) 9-P
and write (s): T.M - Te for its differential. Let (dLg(s))) Tg.M T.M denote the adjoint of ( with respect to the metric g.
Lemma A.1 (Gradient Equivariance). Let To: M M (2) - + be steerable conditional neural field, i.e. for all g G and all 8,r M. To(s, r: 2) TT(g 8, g r;22). Then, for each fixed a E 2, fixed receiver M, and every g E G.
grad, To(s,r:g- 2) (dLg is)) [grad Tolg s, 9 - .r;,)]] T.M.
Proof. By steerability, one has
Ta(s, gi9. =
= (L (r);
Fix E M and differentiate with respect For any i E T,M, the chain rule yields
(:g d)] = dg ( s, g r; 2) [dL, [i]] By the defining property of the Riemannian gradient, have (grad f, df, that:
g grad, To(s,r:g.22 2), i) 11-1a( (grad, t.. To 8.9 2) (8))))
ET.M
ET,-I,M
This exactly characterizes adjoint (s)) and the result follows.
Definition 4.1 Restated), For all G, define the g-steered metric G :TM TM R as:
9g (ú. ú) gup ((d [ú) (g (g)))) for M, and ú, i E T.M.
Proposition 4.1 (Restated). Let MMxMxXPP
- E. be conditional neural field satisfying the steerability property (2) and let 2 be the conditioning variable representing the solution of the eikonal equation for un MM- RR,,, iee, To(s, velocity field vn Let G be a
((s,);2) 2)) = (i(s,) for Ti satisfying Equation (1) for the
a g-steered metric (Definition) 4.1. Then:
1. The map p. G (M RR,) (M Ri) defined by
u(g, (s) Igrad Ti 5, 9
(6)
where is arbitrary point in M, is well-defined group action.
2. For any g E G. To 21) solves the eikonal equation with velocity field p(g, vt).
Proof. By the steerability of To, for every g E G and 8, E E we have
Te(s,T: 2) Tolg s,g m 21). Since Te(s, (((s r). follows that
To(s, 2)) Ti(g s. .r)
Define the steered arrival time
(s, Ti ...."... s.g-1.r)
We aim to show that 2 satisfies the eikonal equation with velocity field p(g. v)).
17

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Gradient transformation.
By Lemma
A.1
the gradient of Tÿ is related to that of Tj via
grad, (s, (dly-1 (8))" gradd- Ti 8, g gr.r
Fix g E and write - grad Ti 5, r) E T 1g Taking the squared G-norm we get:
grad TP (s. r = G. (ldL (s)) (s))
Then, for G defined by Definition 4.1
gradg 1, TT(g 8, 1
gー (tt, 1b)
Agg t ((dLg- (gg [vô], (dgg-119g (wo))
Gs ( ( (ub]
grad, T (s.
By Eq.
B). we now get
|| grad, (s,
plg, ms)(s)
i.., TV solves the eikonal equation with velocity u(g, v1).
Group action properties of p. (1) Identity: Let G denote the identity. Then e and (s Id. hence:
g Ga:
ule, (s) = || Ti(s, rll = v7(s),
using the eikonal equation for Ti.
(2) Compatibility: For all h E G, we show that:
p(g, u(h, ut)) = p(gh, ut).
We note that left and right hand side are respectively given by
p(g. p(h, vi) ggaad,-1--1, Ti h h l
and
p(gh, u1) = Igradd ( Ti(i,)ii((h_LL1 (gh) r)llgun"
(8)
Both equations 7 and 8 are the same iff goh = (g*)², By Definition 4.1 it suffices to show
(dL(gh)-1)) =dL (dLn which follows readily:
(dL(g))=))* = (dIn-1 o dLg-1)" = (dLg-1)* (dLn-1)"
Hence, the group action μ is compatible.
Corollary (Restated). Assume the hypotheses of Proposition 4.1 then the group action G (M RT) is given by:
1. u(g. (s) vi(9 if acts isometrically on M.
2. u(g, (s) Agg 8) vnlg
G acts conformally M with conformal factor Mgg,8) 0, iee. 9ns (dL, (8)), ())(6))())) = 2(g, 3)² GN (81, 62), V 82 E T.M.
Proof. (1) Isometric case: an isometry, then (dI. (+))) (dl. 1 clog
so that GP is equal to the pull-back metric and preserves inner products. Hence.
|| ( 8) (d s)[j),
)){{)s G. 8) ||
18

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Therefore,
grad, To 2)]l = grad, 1, To 8, "..g..re r; m220 = ullg-1 s)² so plg, (s) vr s).
(2) Conformal case: Lg acts conformally with factor Sig, s), then for 81. 82 T.M,
Ggs (+) (=)(_)) = Sgg, 8)² g, (81, 82).
Hence , (s) scales lengths by (2 ,, () and its adjoint satisfies (dLg-1 ((g, 8) (dLg-1(s)) because (dLg(s))² Stgg, 8)² (dLy(s))
Then
grad, (s,,r: 2) () (dL.- (dl) (s)) Igrad To -1 s, 9 1; 21).
and thus
Igrad To(s, 2 وا]هة ((g 1r
Therefore plg. (s) = Gg
As an example of an isometric action, consider 2D rotations with G S0(2) acting on M given a velocity field v(s) and its cor- responding conditioning variable 2, the encodes Simi-
R²a In Figure |
2.0
g
1.5
transformed variable RT/6
the velocity field u( u(Ros/65). larly, as an example of a conformal ac- tion, consider the positive scaling group G = Ri acting on M = =R. In the graph shown in Figure 5 given veloc- ity field v(s) encoded by z, the scaled conditioning variable 2 encodes the velocity field 2s(s/2). Additional ex- amples the stecrability property well as the emperical validation on our implementation pendix E.7 Remark A.1 (Implementation in Euclidean Coordinates). In local coordinates, the metric tensor Op at any point p M is represented by symmetric positive-defnite matrix, which induces an inner product on the tangent space T M.M. Specifically, for any tangent vectors PL.22 E T,M, the inner product is given by 9p(Pr. P2) = (pr.22)g = pl G,dr, and the corresponding norm is defined as lir (pn p1)g.
1.0
0.5
0.0
-5
-4
-3
3
4
Figure 5: Example of conformal group action on u(s) =
be found in Ap-
Moreover, in local coordinates, the Riemannian grudient is related to that of the Euclidean gradient f via the inverse metric tensor: grad f(p) V AAbsil et al. 2008). If M is embedded in Euclidean space, then the norm of the Riemannian gradient can be computed as ligrad f'lg
| fll--1, and the adjoint can be computed as ( (,,)
(d)
Therefore, the group action ;.: G (M + R=) + (M + I2.)) in local coordinates is given by:
μ(g, (()s)) = -,, s.g l
where r is an arbitrary point in M, and 3a T.M x TpM + R the local coordinated expression of the g-steered metric given by
0 dLy igg p) (dL, (g p)) for any p M.
Remark A.2 (Relation buck-back metric). G is isometric conformal, then the g-steered metric the pull-back metric G = (Lg)*g " and one directly has
g (g = u(g192, vi) = u(gi, p(gz, v1)).
19

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B Computing Invariants via Moving Frame
B.1 Preliminaries
In this section, we present the essential mathematical foundations of the Moving Frame method, following Olver [2001]. Sangalli al. [2023]. For a comprehensive treatment, we refer readers to Olver 201 1995].
Let M be m-dimensional smooth manifold and G be an r-dimensional Lie group acting M. A (right) moving frame is a smooth equivariant map .M - G satisfying p(g p) = p(p) gg for all
g E G and p M. Every moving frame p induces canonicalization function k M + M defined by
k(p) p(p) P-
which is G-invariant:
E 9 EG kgg p) plg p) - p(p) -p k(p).
p==
The existence of moving frames characterized by the following fundamental result:
Theorem B.1 (Existence of moving frame; Olver 2001)). A moving frame exists in neighbor- hood of a point E M if and only if G acts freely and regularly near
Moving frames be constructed via cross secssssections group orbits. A cross-section to the group orbits is submanifold K S .M of complementary dimension to the group (i.e., dim = m r) that intersects each orbit transversally at exactly point.
Theorem B.2 (Moving frame from cross-section, see Olver 2001 D. IfG acts freely and regularly on M, then given a cross-section K to the group orbits, for each p M there exists a unique element Op G such that Op E K. The function p : + G mapping p to Op is a moving frame.
While any regular Lie group action admits multiple local cross-sections, coordinate cross-sections (obtained by fixing of the coordinates) are particularly useful for determining fundamental invariants of M with regards G.
Theorem B.3 (Fundamental invariants via coordinate cross-sections. see Olver 2001 Given a free, regular Lie group action and a coordinate cross-section K, let be the associated moving frame. Then the non-constant coordinates of the canonicalization function image
k(p) = p(p) p K
form complete system of finctionally independent invariants.
Moreover, this theorem aligns with the classical result on the structure and separation properties of invariants:
Theorem B.4 (Existence of Fundamental Invariants; see OOler 1995)). Let G be a Lie group acting freely and regularly on an m-dimensional manifold M with orbits of dimension s. Then, for each point p M, there exist m functionally independent invariants I,...., defined on neighborhood U of p such that any other invariant I on can be expressed as I = H (II...... Im-e) for some function H. Moreover, these invariants separate orbits: two points y, E U lie in the same orbit if and only if T, Tz(2) (2))fo for all u =, 1,..., m
These foundational
soullts-xistennee construction via cross- cross-sections, and the properties of invari- ants ants-underlie all moving-frame computations and guarantee that one obtains a complete, orbit- separating set of invariants.
B.2 Canonicalization via latent-pose extension
Theorem 4.1 (Restated). Let G be Lie group acting smoothly and regularly (but not necessarily freety) on each Riemannian manifold Mi via 8, : G M; + Mi for =1, 1,..., m, and hence diagonally on
=1 x ... x Mms 8(g. (m....,,.. =
6m pm))
On the augmented space II I1 x G. define oh,, (pi.... (pt...., Pm. 9)) (8(h, (PI...., ,m)) hg).
20

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Then:
1. 6 is free.
2. A moving frame is given by 1I + G, such that p(p 9) =g
3. The set (ā:(g-) .p)) forms complete collection of functionally independent invariants of the action p.
Proof.
(1) Freeness:
Fix (P1,..., 9) E II. Its isotropy subgroup is
G(pi...pm..))
{ E 8(h, (P1,...,Pm) -p.....-Pa Pmn: hg -9}-
Hence
Gpp Gp, n n Gpm.. Gg.
where Gy, denotes the isotropy subgroup of p. under 8,, and G2 the isotropy subgroup of g E G under left multiplication. Since hg = g implies h in group, we have Gy {e}. Thus, the intersection is trivial, and z defines a free action.
(2) Moving frame existence: Because is free and regular, Theorem B.1 equivariant map
g guarantees unique smooth
pIII-C
associated to the cross-section K ((p,, . 9) E [I == e). A direct check shows
.(PI....,Pr.. 9) =
and one verifies equivariance
p(8(h, (p), (p Pm, g))) = (hgg) 11 = = - h-- = p(p)..... .mm. 9) h-!
(3) Functional independence and completeness: The canonicalization map k(p1,... . Pmn g)
.P......Pms p(plp..... ...... Pmn g), (Pi...., Pm: 9)) E gives
k(p Pm g) = "6g (PI...,,,P... Pm.g]) g))
= (g .I)..... 6m Pm), e).
By Theorem B.3 the nonconstant coordinates 6, 1 p4) are functionally independent and generate all invariants of 6.
C
Training and Inference Details
C.1 Training
Autodecoding. autodecoding, we jointly optimize both the network parameters and the latent conditioning variables across the dataset. As detailed in Algorithm [1 this approach yields tighter fit the eikonal equation, at the expense of longer training times. Empirically, we observe that convergence on the validation set requires between 250 and 500 fitting epochs.
Meta-learning. Our meta-learning framework separates training into two loops: an inner loop for optimizing latents and an outer loop for updating network parameters. Algorithm 2 summarizes this bi-level optimization procedure.
By leveraging meta-learning, we achieve significantly faster fitting and impose additional structure on latent space [nigge et al. 2024). However, as noted by Dupont et al.) [2022), the small number of inner-loop updates typically used meta-learning can restrict expressivity. To mitigate this, we initialize the network with weights pretrained via autodecoding, which accelerates convergence and often leads to better local minima (see Section E.I
21

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Algorithm Autodecoding Training
Require:
Velocity fields {w)) epochs num_epochss, batch size B, pairs per field N learning rate 1)
Randomly initialize shared base network Ta
2: Initialize latents + {(9,. c.) for all velocity fields
3: for epochs to num cerm_ppochssceee do
4: while dataloader not empty do
5:
Sample batch B = {[Sij. Ti.j. us (8ig)) (e(ri,) )) R.N
1=1j==
6:
Compute loss L(0, [ B3) (see Equation 5)
7:
Update t 0 - V
Update each 2; 2, 7 V L 9: end while
8:
end for
1 Ensure: Trained @ and latents ( 1=1
Algorithm 2 Meta-learning Training
Require: Velocity fields V = [w)]== outer epochs num_epochs, inner steps S, batch size B, pairs
per field Nar. learning rates r7, 7/SGD
1: Initialize shared base network Ta (optionally pretrained), and learnable learning rate T/s-
2: for epochs to maum_epocts do
3:
while dataloader not empty do
4: Sample batch of velocity fields {u)) R S
Initialize latents (0) for each for t do
5:
6:
Inner loop: Update latents
7:
Sample Nor source-receiver pairs {( ,ii
(t-1) ((-1)) Ner c M², for each
Construct batch Btt = [(5 (1-1) (t-1)
sia [u((s (1-1)), (1-1) B.N.r
8:
Compute İ(0, 12 (t-1) 1==1 Bet,, ([1-1))
9:
Update each 2
11 V
11:
end for Sample Ner rececerreceiver pairs ((5 (5) Nor C M2, for each in
Construct batch (( {(5 (ij U/ (5 19) U/ (r i. -=
(S)
Compute T.mata (f) T.(00, {2 1==1 B(5)
(8)
15:
Update + A - n7 VoLmete
16:
Update 7) Tl2 T|GGD V n, Lieea end while
17:
18:
end for
Ensure: Trained
In our implementation, we utilize an alternative loss function L. in place of the original loss defined in Equation 5 Specifically, employs the log-hyperbolic cosine loss log cosh(r)) [Jeendgar et al. 2022]. as a differentiable substitute for the absolute value term in Equation [ This substitution is critical for effective meta-learning, as the log-cosh function provides smooth approximation to the absolute value while maintaining convexity and ensuring well-behaved gradients throughout its domain. The differentiability properties of this function enable us to obtain high-quality higher-order derivatives, which are essential for the backpropagation process through all inner optimization steps, as outlined in lines 15 and 16 of Algorithm
C.2 Inference
Solving for new velocity fields. Given new set of velocity fields, we will obtain their correspond- ing latent representation as follows:
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usiin Autodecoding: we perform Algorithm Jusing the frozen weights 6*, ii.., we do not perform
steps 1 and 7.
Meta-lcarning: we perform the inner loop of the Algorithm 2 using frozen weights 0*, i.e., we do steps 5 to 11.
Execution of bidirectional backward integration. As stated in Section given the solution of the eikonal equation, you can obtain the shortest path between two points under the given velocity via backward integration. Indeed, we perform SGD over the normalized gradients grad, [ss, r)ll grad, T(s. r) [Bekkers et al. 2015). Furthermore, as shown in Ni and Qureshi [023], our model's symmetry behavior allows us to perform gradient steps bidirectionally from source to receiver and from receiver to source. Hence, we compute the final path solution bidirec- tionally using iterative Riemannian Gradient Descent [Absil et al. 2008) by updating the source and receiver points follows, where R. is step size hyperparameter.
(t9) R3 (-a grad, To- To-(s((-1) r(tt-);; 21]lg grad, Te- (s(t-1), x((111;21)) (()) +- 1, (-a grad, To- (s(t-1), p (t-11); 21) grad, To. (1(1-1), ( r(((-1); :1))
(9)
where Ry: TT.M- M is retraction at p M. The retraction mapping will provide a notion of moving in the direction of tangent vector, while staying on the manifold:
Definition C.1 (Retraction; see Absil et al. [2008]). A retraction on manifold M is a smooth mapping R from the tangent bundle TM onto M with the following properties. Let R2 denote the restriction of R to M.
(i) Rp()p) p. where Op denotes the zero element of T.MM
(ii) With the canonical identification To, Th T. = M, Rp satisfies
d.P((0p)) idT,.M:
where idr,M denotes the identity mapping on T.
D Experimental Details
This section presents the comprehensive training and validation hyperparameters employed in the experiments described in Scction All experiments were conducted using single NVIDIA H100 GPU.
D.1 2D OpenFWI Experiments
Model Architecture. Our invariant cross- cross-attention implementation utilizes hidden dimension of 128 with 2 attention heads. The conditioning variables are defined as 2 e P(Z) with cardinality |x| = 9, where Z = S2(() x 32 for each velocity field. We initialize the pose component of the latents derived from S = R x 5 at equidistant positions in TR?,, with orientations randomly sampled from uniform distribution over [- n,, r). The context component of the latents is initialized as constant unit vectors.
For embedding the invariants, employ RFF-Net, variant of Random Fourier Features with trainable frequency parameters. This approach enhances training robustness with respect to hyperpa- rameter selection. Following the methodology of Wessels et al. [2024), we implement two distinct RFF embeddings: one for the value function and another for the query function of the cross-attention mechanism. The frequency parameters initialized to 0.05 for the query function and 0.2 for the value function.
Detailed ablations on these architectural choices can be found in Section H. and Section
Dataset Configuration. For each OpenFWWI dataset, 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. For each velocity field, we uniformly sample 20,480 coordinates, producing 10,240 pairs per velocity field. Each batch comprises two velocity fields with 5,120 source-receiver pairs per field.
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Training Protocol. The autodecoding phase consists of 3,000 epochs, while the meta-learning phase comprises 500 epochs. To mitigate overfitting. we report results based on the model that performs optimally on the validation set. Under this criterion, the effective training duration for autodecoding averages approximately 920 epochs (3.6 hours per dataset), while meta-learning averages 440 epochs (17.8 hours per dataset).
SE(2) Optimization with Srundd-xxpnential Map. For optimizing parameters in SE(() RR² x St, employ standard simplification of Riemannian optimization for affine groups, known as parameterization via the "pseudo-exponential map." This approach substitutes the exponential map of SE(n) R" x SO(n) with the exponential map of g SO(n) [Solà et al. 2021 Claraco 2022] This technique is applied in both autodecoding and meta-learning phases, though with different optimizers as detailed below.
Autodecoding Optimization Strategy. For autodecoding, we optimize all parameters using the Adam optimizer with different learning rates for each component. The model parameters are trained with learning rate of 10 4. For the latent variables, context vectors use learning rate of 10-2, while pose components in S2) are optimized with a learning rate of 10-3 Both latent variable components (context and pose) optimized using Adam.
Meta-learning Configuration. For meta-learning, we jointly optimize the model parameters 0 and inner loop learning rates 7)= using Adam with cosine scheduler. This scheduler implements single cycle with an initial learning rate of 10 and a minimum learning rate of 10-6. For the SGD inner loop optimization, we initialize the learning rates at 30 for context vectors and 2 for pose components, executing 5 optimization steps in the inner loop.
For the experiment presented in Table 2 adapt the model presented in Dupont et al. [022] to have characteristics similar to E-NES model. Specifically, we use global conditioning variable of 2 E R315 to match the total number of parameters in our geometric point cloud. Moreover, will use hidden dimension of 128 and latent modulation size of 128 match the dimensionality of our architecture.
Functa Model Architecture.
D.2 3D OpenFWI Experiments
Model Architecture. For the 3D experiments, we adapt the architecture described in the 2D case with several key modifications. Most notably, we reduce the cross-attention mechanism to single head rather than the two employed in the 2D experiments. Additionally, we utilize a set eight elements in 2 SE(3) x R as conditioning variables instead of the nine used in the 2D case.
Pose Representation. While we maintain the same general approach for pose optimization as in the 2D experiments, the 3D case requires parameterization of (3) rather than (2).). We employ the pseudo-exponential map as described previously, but with an important distinction in the rotation component. Specifically, we parameterize the SO(3) component using Euler angles.
Training and Optimization. All other aspects of the training procedure-includee dataset con- figuration, optimization strategies, learning rates, and epoch nemnan-remain consistent with those detailed in the 2D experiments. We maintain the same distinction between optimization approaches in autodecoding (Adam for all parameters) and meta-learning (Adam for model parameters in the outer loop, SGD for latent variables in the inner loop). This consistency allows for direct compari- son between 2D and 3D experimental results while accounting for the specific requirements of 3D modeling.
D.3 Spherical Experiments
Model Architecture. For the spherical experiments, we adapt the architecture from the 2D Eu- clidean case with several key modifications. First, we reduce the cross-altention mechanism to single head for the constant-velocity dataset, while using two heads for the non-constant cases as in the 2D experiments. Additionally, we employ conditioning variables from Z = 50(() x 32 with nine elements for the non-constant case, compared to four elements from SO(2) R 16 for the constant
24

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case. These differences reflect the reduced representational requirements of the constant-velocity scenario, which exhibits simpler dynamics than its non-constant counterparts.
Dataset Configuration. sphere:
We evaluate our method on three types of velocity fields defined over the
Constant Speed Fields: For each sample, draw a scalar velocity (((011,200 and define constant velocity field over the entire sphere.
Spherical Style-B Fields: We construct contially-vayyingg velocity fields by projecting OpenFWI's 2D Style-B velocity models onto the sphere. We query the velocity field at continuous coordinates via RBF interpolation using cosine distance kernels.
Gaussian Obstacle Fields: For each sample, we generate random von Mises-Fisher distri- butions on the sphere with concentration parameters sampled uniformly from U(1 The distributions are normalized to the interval [0.1, 10.0) to enforce distinctions between low-velocity regions (obstacles) and high-velocity regions.
Ground truth travel times are computed using the Hamiltonian Fast Marching method of Mirebeau and Portegies
2019 with the canonical spherical metric tensor.
E
Additional results
E.1 Impact of Autodecoding Pretraining on Meta-Learning Performance
This section examines the effectiveness of initializing meta-learning with parameters derived from standard autodecoding pretraining. We present a comparative analysis using the Style-A and Curve 2D OpenFWI datasets, evaluating performance through both eikonal loss and mean squared error (MSE) metrics throughout the training process.
Figure 6 illustrates the training dynamics across both initialization strategies. Our results demonstrate that utilizing pretrained model parameters from the autodecoding phase substantially enhances con- vergence characteristics in two critical dimensions. First, pretrained initialization enables significantly faster convergence, reducing the number of required epochs to reach performance plateaus, Second, and more importantly, this approach allows the optimization process achieve superior local minima compared to random initialization.
StylesA Vanilla
Eikonal Loss (log scale)
Style-A Preirain
-
CurseVall-A
10-7
MSE (log scale)
CurseVel-A Pretrain
10-3
20
40
100
120
20
40
100
120
Step (x1000)
Step (x10000)
Figure 6: Comparative analysis of meta- neta-legrning convergence with pretrained versus random initial- ization on Style-A and VurveVel-A OpenFWT datasets.
These findings highlight fundamental efficiency in our methodology: by leveraging pretrained autodecoding parameters, the meta-learning phase is effectively transformed from learning from scratch to a targeted adaptation task. Specifically, the pretrained model has already established robust conditional neural field representation of the underlying physics. The subsequent meta-learning process then primarily needs to adapt this existing representation to interpret conditioning variables obtained through just steps SGD, rather than those refined over 500 epochs of autodecoding. This represents significant computational advantage, as the meta-learning algorithm can focus exclusively on learning the mapping between rapidly-obtaines SGD variables and the already-establisher neural field, rather than simultaneously learning both the field representation and the optimal conditioning.
25

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E.2 Comprehensive Grid Evaluation of OpenFWI Performance
In Section 5 we evaluated E-NES performance on the OpenFW1 benchmark following the method-
ology established by Mei et al. [024)), which utilizes four equidistant source points the top of the velocity fields and measures predicted travel times from these sources to all 70x70 receiver coordinates. This approach allows direct comparison with both the Fast Marching Method (FMM)
and the FC-DeepOnet model [Mei et al. 2024].
To validate the robustness and generalizability of our approach, we conducted additional experiments using a substantially denser sampling protocol specifically, a uniform 14x source point grid similar to that employed by Grubas et al. [2023] As demonstrated in Table 4 E-NES maintains performance metrics comparable to those presented in Table [ despite the significant increase in source points and resulting source-receiver pairs.
Table 4: Performance on OpenFWI datasets on a 14 x 14 grid of source points. Fitting time represents the total computational time required to fit the latent conditioning variables for all 100 testing velocity fields.
Autodecoding (100 epochs)
Autodecoding (convergence)
Meta-leaming
Dataset
RE (U)
RMAE (1)
Fitting (s)
RE (L)
(J)
Fitting (s)
RE (1)
RMAE (L)
Fitting (s)
FlarVel-A
0.01023
0.00827
223.31
0.00624
0.00509
1010.90
0.01304
0.01003
5.92
CurveVel-A
0.01438
0.01139
222.72
0.01069
0.00841
1009.67
0.02460
0.01878
5.91
FlatFault-A
0,01050
0.00751
222.61
0.00744
0.00510
1014.45
0.01749
0.01255
5.92
CurveFault-A
0.01380
0.00976
222.89
0.01088
0.00745
1007.97
0.02471
0.01 807
5.92
Style-A
0.00962
0.00785
222.00
0.00795
0.00646
783.13
0.01326
0.01036
5.92
FlatVel-B
0.01988
0,01586
222.74
0.01178
0.00906
786.48
0.03077
0.02474
5.91
CurveVel-B
0.04291
0.03349
222.97
0.03297
0.02528
1010.70
0.04977
0.03930
FlatFuult-B
0.01889
0,01413
222.70
0.01557
0.01147
898.28
0.02998
0.02214
5,93
CurveFault-B
0.02244
0.01728
222.89
0.01991
0.01537
161.22
0.03824
0.02945
Style-B
0.01061
0.00860
221.90
0.00984
0.00798
1120.09
0.01566
0.01227
5.90
This consistency across sampling densities provides strong evidence that E-NES effectively captures the underlying travel-time function for arbitrary point pairs throughout the domain. We attribute this capability to two key architectural decisions in our approach. First, the grid-free architecture allows the model to operate on continuous spatial coordinates rather than discretized grid positions. Second, training methodology leverages physics-informen neural network (PINN) principles rather than relying on numerical solver supervision as implemented in Mei et al. [2024]. This physics-based learning approach enables E-NES internalize the governing eikonal equation, resulting in more generalizable representation of travel-time fields that remains accurate across varying evaluation protocols.
E.3
Ablation Study: Choice of Embedding for Computing Invariants
To better understand the impact of architectural choices model performance, we conducted an ablation study examining different embedding functions for computing invariants in our cross- attention module. Table 5 compares three embedding approaches: standard MLP, a polynomial embedding (degree 3), and Random Fourier Features (RFF, our chosen approach).
The results demonstrate that RFF embeddings provide the best balance of accuracy and computational efficiency. While the polynomial embedding achieves comparable relative error (RE) and relative mean absolute error (RMAE) to RFF, it incurs prohibitively high computational cost, with fit and inference time approximately 6 times slower. The standard MLP embedding, though computationally efficient, shows degraded performance across both metrics.
These findings justify our choice of RFF with learnable frequencies (as discussed in Section D.2), which combines superior accuracy with reasonable computational requirements.
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Table 5: Ablation on the choice of embedding for computing invariants (CurveFault-B validation set).
Embedding
RE (4)
RMAE (4)
Fit + Inf. Time (s)
MLP
0,025
0.020
237.2
Polynomial (deg. 3)
0.024
0.019
1460.7
RFF (Ours)
0.021
0.016
255.8
E.4 Ablation Study: Latent Conditioning Design
The design of the latent conditioning mechanism significantly impacts both model performance and computational efficiency. We investigated different configurations by varying the number of latent points (NV) and their dimensionality (d), while maintaining fixed total parameter budget 288 (i.e., N x d 288). This ensures fair comparison across different architectural choices.
Table 6 presents results the CurveFault-B validation The configuration with latent points of dimension 32 provides the optimal tradcoff between accuracy and computational cost, achieving the lowest eтror metrics (RE 0.021, RMAE 0,016) with reasonable inference time (255.8 seconds).
Table 6: Ablation on latent conditioning design (CurveFault-B validation set). Total latent budget fixed at 288 parameters.
Latents (N)
Latent Dim. (d)
RE (L)
RMAE (L)
Fit + Inf. Time (s)
288
0.078
0.066
58.5
72
0.047
0.038
139.3
0.021
0.016
255.8
16
18
0.039
0.031
501.7
Notably, using single high-dimensional latent vector (1 x 288) results in significantly degraded performance, suggesting that spatial distribution of latent conditioning is crucial for capturing the complexity of the velocity field. Conversely, using too many low-dimensional latents (16 x 18) increases computational cost without improving accuracy, likely due insufficient expressiveness of the context vector per latent point. The selected configuration (9 x 32) thus represents an effective compromise that provides both spatial coverage and sufficient representational capacity per latent point.
These results highlight the importance of carefully balancing the number and dimensionality of latent conditioning variables, and demonstrate that our chosen architecture is well-tuned for the problem at hand.
E.5
Ablation Study: Impact of Autodecoding Fitting Epochs
This section presents a systematic analysis of how the number of autodecoding fitting epochs affects model performance and computational efficiency. We evaluate the E-NES model on the 2D-OpenFWF datasets across both grid configurations described Section E.2 measuring performance in terms of Relative Error while tracking computational costs. Additionally, we provide comparative analysis between the standard autodecoding approach and our meta-learning methodology.
Figures and illustrate the relationship between fitting time, number of epochs, and model performance across all datasets. Our analysis reveals that most datasets reach optimal solution convergence at approximately 400 autodecoding fitting epochs. Performance improvements beyond this threshold exhibit diminishing returns relative to the additional computational investment required. Notably, approximately 100 autodecoding epochs represents effective compromise between computational efficiency and performance quality.
The meta-learning approach demonstrates remarkable efficiency advantages. With negligible fitting times compared to standard autodecoding, meta-learning achieves performance levels comparable to 50- 100 epochs of autodecoding for the FlatVel-A/B and CurveVel-A/B datasets. This represents
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Meta-learming
Autocccocinn steps:
50
100
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250
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FlatVel-B
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0.040
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Style-B
Relative error
Relative error
0.012
0.014
0.010
0.012
10
Time (10"
Time (10'
Figure 7: Comparative analysis of Relative Error versus total fitting time for all 100 velocity fields in the 4-source configuration across all datasets. Circular markers represent autodecoding performance at varying epoch counts (color-coded from 50 to 500 epochs), while the star marker indicates meta- learning performance.
substantial reduction in computational requirements while maintaining acceptable performance characteristics.
These findings suggest that practitioners optimize computational resource allocation by selecting the appropriate training approach based on their specific performance requirements and computational constraints. For applications where rapid deployment is critical, meta-learning offers significant ad- vantages, while applications demanding maximum accuracy may benefit from extended autodecoding training, with the optimal epoch count determined by performance saturation points identified in analysis.
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Meta-learming
Autocccocing steps:
50
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FlatVei-B
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s)
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Rolative
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Relative error
0.012
0.014
0.010
0.012
0.008
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Time (10*
Time (10*
Figure 8: Comparative analysis of Relative Error versus total fitting time for all 100 velocity fields in the dense 14x14 source grid configuration. Circular markers represent autodecoding performance at varying epoch counts (color-coded from 50 to 500 epochs), while the star marker indicates meta- learning performance.
E.6 Qualitative Analysis of Travel-Time Predictions
This section provides visual assessment of the E-NES model's performance as quantitatively reported in T For each dataset in the 2D-OpenFWI benchmark, we present representative velocity field alongside the corresponding ground-truth and predicted travel-time surfaces. Additionally, we visualize the spatial distribution of relative error facilitate the identification of regions where prediction accuracy varies.
Figures 9 and 10 demonstrate the E-NES model's capacity accurately reconstruct travel-time functions across diverse geological scenarios. Particularly noteworthy is the model's performance on the challenging CurveFault-A/B and Style-A/B datasets, where the travel-time functions exhibit complex wavefront behaviors including caustic singulariies--rgiionn. At these caustics, seismic-ray trajectories intersect each other, forming singularity zones where gradients are discontinuous.
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Velocity Field
Ground Truth Time
Predicted Time
Absolute Error
45
0.0025
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0.0020
0.0035
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02
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Results for FlatVel-A
Velocity
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0.00 75
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+0,0125
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(b) Results for Vel-A
Velocity Field
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066
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-04
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Results for FlatFault-A
Velocity
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(d) Results for CurveFault-A
Velocity Field
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Absolute Error
0.01.0
35
0.6
0.0
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0.0000
330075
2.5
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5.0025
(e) Results for Style-A
Figure 9: Comparative visualization of E-NES predicted travel-times against reference solutions for OpInFWI type datasets. Each panel displays the velocity field (left), ground-truth travel-time surface (center-left), E-NES predicted travel-time surface (center-right). and relative error distribution (right). The red denotes the source point location from which wavefronts propagate.
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Velocity Field
Ground Truth Time
Predicted Time
Absolute Error
43
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3.5
0.000
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62
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Results for FlatVel--B
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Ground Truth Time
Predicted Time
Absolute Error
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06
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(b) Results for CurveVel-B
Velocity Field
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Predicted Time
Time
Absolute Error
0.15
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(c) Results for FlatFault-B
Velocity Field
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Predicted
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077
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(d) Results for CurveFault-B
Velocity Field
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Absolute Error
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04
04
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(e) Results for Style-B
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 -trutthtuhhh travel-time time surface (center-left), E-NES predicted travel-time surface (center-right), and relative error distribution (right). The red star denotes the source point location from which the wavefronts propagate.
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E.7 Empirical Steerability Test
In this section, we empirically validate the geometric inductive bias incorporated into our Eikonal Solver through a two-part test. First, we verify that [a(s, r; 21) correctly models the solution of the Eikonal equation for the velocity field vt. Second, we test the equivariance property: for any g E (2). we examine whether [s, rig =)) solves the Eikonal equation with the transformed velocity field p(g, vt), satisfying grad, (s, rig =)]lş== p(gs (i)(
For the Special Euclidean group, Corollary 4.1 establishes that p(g. vi)(s) 01(g-1. s). which enables direct geometric interpretation and visual verification of the steerability property. Figure IT demonstrates this behavior empirically: when we rotate the latent conditioning point cloud, the resulting travel-time field exhibits a corresponding rotation, and the gradient norm yields rotated version of the original velocity field. The same predictable equivariant behavior observed under y-axis translation, confirming that our model successfully captures the desired geometric structure.
ReGermton Time
Predicted Time
Poud. Time Buo 45
Pood: Time Rot. 90"
Pred. Time Dat. 180"
Pood. Time Trunslaie ad
45
ax
65
93
-0A
463
05
Origical Velocity
Recos. Vidocity
Becon, Vel. Bot. 45*
Rocon. Vel. Rui. 90"
Rocos Vel. Bot. 180
Bacon. Vol. Translated
10
-35
30
05
10
+10 -05 00 05
61 05
13
-55
30
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16
42 66
05
-35
30 10
Figure 11: Steerability test demonstrating SE) equivariance on the FlatVel-A dataset. When the latent conditioning point cloud transformed by elements of SE()) (rotations and translations), the predicted travel-time field and recovered velocity field undergo corresponding geometric transforma- tions, confirming the model's equivariant behavior. The red star denotes the source point from which wavefronts propagate.
F
Extended Discussion
F1 Improved Memory Scalability
In this section, expand our analysis to consider the relative merits of conditional versus uncondi- tional neural field approaches for solving the eikonal equation. We intentionally excluded comparisons against unconditional neural field-based eikonal equation solvers from main experimental results, as these comparisons require additional context to interpret appropriately.
Unconditional neural fields typically outperform conditional neural fields in both solution accuracy and inference speed for individual problem instances. This performance gap stems from the funda- overfitting
mental nature of the task: training unconditional neural field essentially constitutes problem, where even modest MLP architecture can achieve high accuracy on a single velocity field. In contrast, conditional neural fields must generalize across multiple instances, effectively learning mapping from conditioning variables to solution fields rather than memorizing a single solution.
However, this performance advantage diminishes significantly when considering parameter scalability across multiple velocity fields. The unconditional approach (e.g.. standard Neural Eikonal Solver) requires approximately 17,558 parameters per velocity field [Grubas et al.) 2023]. as it necessitates training an entirely new network for each travel-time solution. In contrast, our E-NES approach maintains a fixed baseline of 648,965 parameters for the core network Te, with each additional velocity field requiring only 315 parameters for the conditioning variables (in the A simple calculation reveals that E-NES becomes more parameter-efficient than the unconditional approach when handling more than velocity fields threshold easily surpassed in practical applications.
SE(2) x RR case).
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The memory scaling properties of E-NES offer additional advantages beyond parameter efficiency. The conditioning variables effectively serve as a quantization method for the travel-time function. significantly reducing memory requirements. For example, the 1xxxx000 grid configuration used in Table 4 would require storing 960,400 doating-point values per velocity field using traditional methods. With E-NES, we need only 315 parameters per velocity field regardless of the grid resolution. This resolution-invarian property becomes particularly valuable as problem dimensions increase, offering orders of magnitude improvements in memory efficiency for high-resolution applications.
These scalability advantages highlight
highhigh the complementary nature of conditional and unconditional approaches. While unconditional neural fields excel at individual problem instances where maximum accuracy required, conditional architectures like E-NES provide substantially better scalability for applications involving multiple velocity fields or varying resolution requirements.
F.2 Applicability Beyond Seismic Traeel-Time Modeling
While our manuscript focuses primarily on the Eikonal equation in the context of seismic raael-time modeling, the proposed generalization of Equivariant Neural Fields (ENFs) extends far beyond this specific application.
Eikonal Equation Beyond Seismic Imaging. Even within the realm of Eikonal solvers, there exist several diverse and impactful application domains. For example, in robotics, the Eikonal equation is used for optimal path planning, where travel-time functions are defined over the robot's configuration space [Ni and Qureshi 2023]. J. In image segmentation, the Eikonal equation can be employed to compute geodesic distances, with the velocity field derived from classical edge detectors, thereby transforming the segmentation task into minimal -ath extraction problem [Chen and Cohen 2019]. Although our experiments focus on the OpenFWI benchmark- currently the only publicly available dataset for learning-hai based Rikonal solvers-on our methods are directly applicable to these broader scenarios, discussed in Section 1
Equivariant Neural Fields in Reinforcement Learning. Beyond Eikonal solvers, the proposed framework is applicable continuous-conie reinforcement learning (RL). Specifically, the travel- time function [ss, r) obtained by solving the Eikonal equation can be interpreted as an optimal value
function:
In(00) v(t)
inf 0¹ (ssrr) T(s,r) (00)=,,(()====)
de.
where the state space is .M x M, the action space corresponds to T.M, and the optimal policy is the geodesic path y. The environment deterministic, with transitions r)()),r) for small e >> and terminal state (r,
In general, RL problems with continuous state and action spaces, the value function v and action-value function can naturally be modeled as neural fields. In multi-task RL (Yu al., 2020). these functions are conditioned on task embeddings. Our formulation of Conditional Neural Fields is well-suited to this, as it accommodates both continuous coordinate inputs and a conditioning signal. Moreover, when the state or action spaces admit Lie group symmetry [Wang et al.) 2022]. Equivariant Neural Fields offer inductive biases that promote better generalization and sample efficiency. Importantly, our proposed extension enables modeling such value functions in cases involving multiple input coordinates, which are common in practice.
Beyond Reinforcement Learning. The applicability of our method extends well beyond reinforce- ment learning. Consider, for instance, collection of signals f1, f1...., fn. where each fi M, -> Rd is defined over potentially distinct Riemannian manifold M, In many scientific and machine learning applications including computational biology, multimodal sensor fusion, and medical imaging is of interest to model a similarity metric of the form
which compares the values or features of these signals at specific points (P.....,Pp) E Mi x x M,. Traditional ENFs are limited to modeling functions defined over single manifold input. However, our proposed extension allows us to define and learn such similarity metries equivariantly with respect to group actions on the product manifold Mi ... X Mn.
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F.3 Additional Discussion on Performance Comparison against FC-DeepONet
As shown in Table I our method outperforms the FC-DeepONet baseline on out of 10 OpenFWI datasets, with performance on FlatFault-A comparable. For the two datasets where our method underperforms (FlatVel-A/B), this can be attributed two key factors:
Training signal difference:
FC-DeepONet is trained using the "ground-truth" travel time fields obtained from numerical solver (FMM), rather than learning from the Eikonal equation directly. As discussed in Section 1, the quality of FMM solutions improves with finer domain discretization. In the FlatVel-A/B datasets, the velocity profiles have low spatial frequency (see Figures 8 and 9), making the FMM-derived travel times particularly accurate and advantageous training targets. In contrast, our method learns from PDE constraints rather than supervised travel times. This makes our approach more broadly applicable, but also potentially disadvantaged in cases where numerical solvers already provide highly accurate approximations. Notably, this performance advantage for profiles more
FC-DeepONet diminishes in more complex scenarios involving intricate Riemannian manifolds, where numerical solvers like FMM may perform less reliably.
Inductive bias from domain discretization: As discussed in Section 2, FC-DeepONet requires discretized domain to produce conditional latents through its CNN encoder. While this introduces constraints generalizability (e.g., limited applicability on manifolds with multiple charts in tasks like geodesic backtracking), it can act as a strong inductive bias low frequecyyn settings. The FlatVel-A/B datasets exemplify such settings, where this discretization bias likely aids FC- DeepONet's performance.
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NeurIPS Paper Checklist
1. Claims
Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
Answer: [Yes]
The abstract and introduction accurately reflect the paper's dual contributions: (I) a theoretical advancement extending equivariant neural fields to functions over products of Riemannian manifolds with regular group actions via the moving frame method, and (2) an implementation contribution through Equivariant Neural Eikonal Solvers (E-NES), Both sections appropriately articulate how the theoretical framework provides mathematical foundations for equivariance across various manifolds while demonstrating practical benefits in computational efficiency, geometric grounding, and solution steerability. The introduc- tion adequately positions these contributions within the context of existing numerical and neural approaches to solving the eikonal equation, clearly identifying the limitations E-NES overcomes through its novel geometric formulation.
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Reviews:
[{"review_id": "IGKtud0Kv6", "reviewer": "Reviewer_uCF6", "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. \nThe authors validate their approach using 2D and 3D seismic travel-time modeling benchmarks. Ablation studies show that with equivariance, method achieves much lower losses.", "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?\n- would vanilla PINN be a reasonable baseline for table 1 or I am missing something?", "limitations": "Yes", "rating": 4, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 3, "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 \n\nIn 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.", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely appreciate the reviewer's recognition of both the theoretical and empirical contributions of our work. Moreover, we thank the reviewer for the insightful comments which strengthen the paper. Below we will go into all the specific questions of the reviewer.\n\n## On Test-Time Fitting Requirements\n\nWe acknowledge the reviewer's concern regarding the computational overhead of test-time fitting. However, we believe this represents a valuable trade-off that merits clarification.\n\n**Computational Comparison:** While our method requires optimization iterations at test time, FC-DeepONet's encoder forward pass can be viewed as an implicit latent fitting step. Specifically, FC-DeepONet takes approximately 0.615 seconds to process 100 velocity fields through its encoder (marked with a dashed line in Table 1 to indicate this implicit fitting). Our explicit optimization approach offers greater flexibility at the cost of additional computation.\n\n**Adaptive Performance Trade-offs:** Our test-time fitting introduces a crucial advantage: practitioners can dynamically balance computational cost against solution accuracy based on available resources and performance requirements. As detailed in Appendix E.3 and illustrated in Figures 6-7, users can achieve different accuracy levels by adjusting optimization iterations. This adaptability aligns with recent advances in test-time optimization for large language models [1], offering similar flexibility in the Neural Eikonal Solver domain. In contrast, FC-DeepONet's performance remains fixed after training.\n\nWe will expand this discussion in the revised manuscript to better highlight this strategic advantage.\n\n## Q1: Mixed performance in Table 1\n\nAs correctly noted, our method does not outperform FC-DeepONet on three of the ten OpenFWI datasets (though on the FlatFault-A case, the results are comparable). The performance differences on the FlatVel-A/B datasets can be attributed to two main factors:\n\n**Training signal difference:** FC-DeepONet is trained using the \"ground-truth\" travel time fields obtained from a numerical solver (FMM), rather than learning from the Eikonal equation directly. As discussed in Section 1, the quality of FMM solutions improves with finer domain discretization. In the FlatVel-A/B datasets, the velocity profiles have low spatial frequency (see Figures 8 and 9), making the FMM-derived travel times particularly accurate and advantageous as training targets. In contrast, our method learns from PDE constraints rather than supervised travel times. This makes our approach more broadly applicable, but also potentially disadvantaged in cases where numerical solvers already provide highly accurate approximations. Notably, this performance advantage for FC-DeepONet diminishes in more complex scenarios involving higher-frequency profiles or more intricate Riemannian manifolds, where numerical solvers like FMM may perform less reliably.\n\n**Inductive bias from domain discretization:** As discussed in Section 2, FC-DeepONet requires a discretized domain to produce conditional latents through its CNN encoder. While this introduces constraints on generalizability (e.g., limited applicability on manifolds with multiple charts or in tasks like geodesic backtracking), it can act as a strong inductive bias in low-frequency settings. The FlatVel-A/B datasets exemplify such settings, where this discretization bias likely aids FC-DeepONet's performance.\n\n## Q2: Vanilla PINNs as a baseline\n\nAs noted in Sections 1 and 2, vanilla PINNs can indeed serve as Neural Eikonal solvers and were among the earliest methods explored in this area. However, as discussed in Appendix E.5, unconditional neural fields (e.g., vanilla PINNs) typically outperform conditional neural fields (like FC-DeepONet or our method) in terms of per-instance accuracy and inference speed. That said, this performance advantage comes at the cost of parameter scalability: vanilla PINNs require a separate model to be trained per velocity field, making them impractical for tasks involving large numbers of queries. In contrast, conditional methods generalize across multiple velocity fields using shared parameters.\n\nFor this reason, we — as well as most works in the Conditional Neural Fields (CNF) literature — consider vanilla PINNs to belong to a different problem setting. While they are effective for solving individual PDE instances, they are not designed for the generalization and efficiency goals that conditional models target. Therefore, we believe it is not meaningful to include vanilla PINNs as a direct baseline in Table 1.\n\n### Enhanced baseline comparison\n\nOn the other hand, we agree that a more comprehensive baseline comparison significantly strengthens the evaluation of our proposed method. To address this, we have incorporated a well-established baseline from the literature on conditional neural fields: **Functa** [2]. Functa is particularly relevant for comparison as it leverages SIRENs with sample-specific scale and shift modulation, employing global latent variables without imposing any geometric constraints. This architecture provides a stark contrast to our geometry-grounded approach, allowing for a clearer understanding of the benefits conferred by our design choices.\n\nAs shown in the updated table below (we will add the remaining OpenFWI datasets in the camera-ready version), we present a comparative analysis against the Functa baseline on the Style-A and most -B datasets. The results indicate that Functa generally underperforms relative to FC-DeepONet, while E-NES outperforms it on most datasets. We believe this performance difference is primarily attributable to Functa's reliance on global conditioning, which inherently lacks the benefits of localized representation, and the absence of explicit geometric constraints within its formulation.\n\n**Table 1: Updated performance comparison including Functa baseline.**\n\n\n| | FC-DeepONet | FC-DeepONet | Functa (100 epochs) | Functa (100 epochs) | E-NES (100 epochs) | E-NES (100 epochs) | E-NES (Full AD) | E-NES (Full AD) | E-NES (META) | E-NES (META) |\n|---|---|---|---|---|---|---|---|---|---|---|\n| Dataset | RE ($\\downarrow$) | Fitting ($s$) | RE ($\\downarrow$) | Fitting ($s$) | RE ($\\downarrow$) | Fitting ($s$) | RE ($\\downarrow$) | Fitting ($s$) | RE ($\\downarrow$) | Fitting ($s$) |\n| Style-A | 0.03461 | 0.615 | 0.04377 | 19.80 | 0.01034 | 222.00 | **0.00833** | 1117.99 | 0.01317 | 5.92 |\n| FlatVel-B | **0.00711** | 0.615 | 0.13437 | 20.01 | 0.01581 | 222.74 | 0.00860 | 1010.32 | 0.02274 | 5.91 |\n| CurveVel-B | 0.03410 | 0.615 | 0.11602 | 19.89 | 0.03203 | 222.97 | **0.02250** | 1127.87 | 0.03583 | 5.90 |\n| CurveFault-B | 0.07863 | 0.615 | 0.06407 | 20.23 | 0.02183 | 222.89 | **0.01885** | 893.84 | 0.03812 | 5.89 |\n| Style-B | 0.03463 | 0.615 | 0.03145 | 20.05 | 0.01171 | 221.90 | **0.01069** | 896.06 | 0.03812 | 5.90 |\n\n\n### Closing Remarks\n\n\nWe hope these clarifications adequately address the reviewer’s concerns and highlight the theoretical and practical contributions of our work. We are committed to improving the manuscript based on this valuable feedback.\n\n**Refrences**\n\n[1] Zhang, Qiyuan, et al. \"A Survey on Test-Time Scaling in Large Language Models: What, How, Where, and How Well?\" arXiv preprint arXiv:2503.24235 (2025).\n\n[2] Dupont, Emilien, et al. \"From data to functa: Your data point is a function and you can treat it like one.\" arXiv preprint arXiv:2201.12204 (2022)."}, {"review_id": "MEHcNTo9Og", "reviewer": "Reviewer_uQ6k", "summary": "The 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.", "questions": "Can 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.", "limitations": "Yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 2, "strengths_and_weaknesses": "Paper 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.", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely thank the reviewer for their thoughtful and constructive assessment of our manuscript. We are grateful for their recognition of the clarity of our presentation and the strong empirical performance of our proposed method. We also appreciate their detailed comments, which have helped us further improve the manuscript. Below, we address the reviewer’s questions and concerns, particularly those regarding the novelty of our approach and its relation to prior work by Wessels et al.\n\n### Clarifying Novelty and Relation to Wessels et al.\n\nWe would like to clarify the precise nature of our contributions in the context of prior work. As noted by the reviewer, the concept of steerability and the general framework of Equivariant Neural Fields (ENFs) were originally introduced by Wessels et al. and others [1,2,3]. We do not claim the steerability property or its associated theoretical foundations as novel. This is explicitly stated in Section 2 of our manuscript.\n\nOur core theoretical contribution lies in significantly extending this framework. Specifically, we show that latent conditioning using a learnable point cloud on a Lie group can be generalized to the product of multiple input Riemannian manifolds. In contrast, Wessels et al. consider a conditional neural field with only a single input point, limiting its applicability to problems like solving the Eikonal equation.\n\nFurthermore, we provide a rigorous expressivity analysis of the invariants used in our model. We prove that it is always possible to construct a complete, maximally expressive set of independent invariants, ensuring no information bottleneck is introduced. This analysis is not present in Wessels et al. or Knigge et al., where invariants are selected heuristically without formal justification of their sufficiency.\n\nIn addition, our theoretical results apply to arbitrary Lie group actions on product manifolds, including those that are non-transitive. We demonstrate this empirically in Table 1 of the response to Reviewer Y7P6, where we solve the Eikonal equation on the 2-sphere with steerability over z-axis rotations. In this case, the pose component of our latent variables lies in SO(2), and the resulting invariants remain maximally expressive, even under this non-homogeneous group action.\n\nIn summary, while our work builds on ideas introduced by Wessels et al., it substantially generalizes and extends the theoretical foundations and practical utility of equivariant neural representations to a broader class of problems. We will further highlight this distinction in the revised manuscript.\n\n### Response to Reviewer’s Question: Applicability Beyond Eikonal Solvers\n\nWe thank the reviewer for raising this important question. While our manuscript focuses primarily on the Eikonal equation in the context of seismic travel-time modeling, the proposed generalization of Equivariant Neural Fields (ENFs) extends far beyond this specific application.\n\n#### Eikonal Equation Beyond Seismic Imaging.\nEven within the realm of Eikonal solvers, there exist several diverse and impactful application domains. For example, in robotics, the Eikonal equation is used for optimal path planning, where travel-time functions are defined over the robot's configuration space. In image segmentation, the Eikonal equation can be employed to compute geodesic distances, with the velocity field derived from classical edge detectors, thereby transforming the segmentation task into a minimal-path extraction problem. Although our experiments focus on the OpenFWI benchmark—currently the only publicly available dataset for learning-based Eikonal solvers—our methods are directly applicable to these broader scenarios, as discussed in the introduction.\n\n#### Equivariant Neural Fields in Reinforcement Learning.\nBeyond Eikonal solvers, the proposed framework is applicable in continuous-control reinforcement learning (RL). Specifically, the travel-time function $T(s, r)$ obtained by solving the Eikonal equation can be interpreted as an optimal value function:\n$$\nv^*(s, r) = T(s, r) = \\inf_{\\gamma(0) = s,\\, \\gamma(1) = r} \\int_0^1 \\frac{\\|\\dot{\\gamma}(t)\\|}{v(t)}\\, dt,\n$$\nwhere the state space is $\\mathcal{M} \\times \\mathcal{M}$, the action space corresponds to the tangent space of $\\mathcal{M}$, and the optimal policy is the geodesic path $\\gamma$. The environment is deterministic, with transitions $(\\gamma(\\varepsilon), r)$ for small $\\varepsilon > 0$, and terminal state $(r,r)$.\n\nIn general RL problems with continuous state and action spaces, the value function $v$ and action-value function $q$ can naturally be modeled as neural fields. In multi-task RL [4], these functions are conditioned on task embeddings. Our formulation of Conditional Neural Fields is well-suited to this, as it accommodates both continuous coordinate inputs and a conditioning signal. Moreover, when the state or action spaces admit a Lie group symmetry [5], Equivariant Neural Fields offer inductive biases that promote better generalization and sample efficiency. Importantly, our proposed extension enables modeling such value functions in cases involving multiple input coordinates, which are common in practice and are not solvable by standard ENFs.\n\n#### Beyond Reinforcement Learning.\nThe applicability of our method extends well beyond reinforcement learning. Consider, for instance, a collection of signals $ f_1, \\dots, f_n $, where each $f_i: \\mathcal{M}_i \\to \\mathbb{R}^d$ is defined over a potentially distinct Riemannian manifold $\\mathcal{M}_i$. In many scientific and machine learning applications—including computational biology, multimodal sensor fusion, and medical imaging—it is of interest to model a similarity metric of the form\n\n$$\n\\mathrm{sim}(f_1, \\dots, f_n)[p_1, \\dots, p_n],\n$$\nwhich compares the values or features of these signals at specific points $ (p_1, \\dots, p_n) \\in \\mathcal{M}_1 \\times \\cdots \\times \\mathcal{M}_n$. Traditional ENFs are limited to modeling functions defined over a single manifold input. However, our proposed extension allows us to define and learn such similarity metrics equivariantly with respect to group actions on the product manifold $\\mathcal{M}_1 \\times \\cdots \\times \\mathcal{M}_n$.\n\nWe thank the reviewer for highlighting the importance of clarifying these broader applications. We will incorporate a dedicated discussion of these potential extensions in the camera-ready version of the manuscript.\n\n### Paper Formatting Concerns\n\nWe agree with the reviewer’s observation that Figure 1 could more effectively communicate the notion of steerability. We will revise the figure to clearly illustrate the relationship between original and transformed fields, thereby improving its didactic value.\n\n### Clarification on Notation Consistency (Line 149)\n\nWe appreciate the reviewer’s attention to our notation. In Line 149, $\\dot{p_1}, \\dot{p_2} \\in T_{p} \\mathcal{M}$ are distinct tangent vectors at the same point $p \\in \\mathcal{M}$. Later, in Line 158, we introduce $\\dot{p} \\in T_p \\mathcal{M}$ and define $\\dot{q} \\in T_{\\phi(p)} \\mathcal{M}$, which lies in the tangent space at $\\phi(p) = q$. While this distinction is mathematically consistent, we acknowledge that it could be clearer and will revise the text at Line 158 to explicitly state the difference in tangent space domains.\n\n### Closing Remarks\n\nWe hope these clarifications adequately address the reviewer’s concerns and highlight the theoretical and practical contributions of our work. We are committed to improving the manuscript based on this valuable feedback.\n\n### References\n[1] Wessels, David R., et al. \"Grounding continuous representations in geometry: Equivariant neural fields.\" arXiv preprint arXiv:2406.05753 (2024).\n\n[2] Chatzipantazis, Evangelos, et al. \"SE (3)-equivariant attention networks for shape reconstruction in function space.\" arXiv preprint arXiv:2204.02394 (2022).\n\n[3] Chen, Yunlu, et al. \"3d equivariant graph implicit functions.\" European Conference on Computer Vision. Cham: Springer Nature Switzerland, 2022.\n\n[4] Yu, Tianhe, et al. \"Meta-world: A benchmark and evaluation for multi-task and meta reinforcement learning.\" Conference on robot learning. PMLR, 2020.\n\n[5] Wang, Dian, Robin Walters, and Robert Platt. \"$\\mathrm {SO}(2) $-Equivariant Reinforcement Learning.\" arXiv preprint arXiv:2203.04439 (2022)."}, {"review_id": "PRpL3EUFxo", "reviewer": "Reviewer_Bupx", "summary": "This 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.", "questions": "In 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).", "limitations": "Yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "The 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.", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely thank the reviewer for their thoughtful and constructive feedback. We are especially grateful for their recognition of the novelty and promise of our approach, including the benefits of enforcing equivariance, the potential of a grid-free framework, and the use of expressive latent embeddings for eikonal neural solvers. We also appreciate the reviewer’s careful reading and their suggestions to improve the clarity and accessibility of the manuscript. Below, we address each of the reviewer’s main points in detail.\n\n### Performance and Significance of Results.\n\nAs the reviewer notes, our method outperforms the FC-DeepONet baseline on 7 out of 10 OpenFWI datasets, with performance on FlatFault-A being comparable. For the two datasets where our method underperforms (FlatVel-A/B), this can be attributed to two key factors:\n\n1.  **Training Signal Source:** FC-DeepONet is trained on ground-truth travel times generated by the Fast Marching Method (FMM), a numerical solver that performs especially well on low-frequency velocity fields like those in FlatVel-A/B. Our method, by contrast, learns directly from the Eikonal PDE, which avoids reliance on solver-specific ground truth and enables broader applicability---but may underperform in simple regimes where solvers already yield high-quality approximations.\n\n2.  **Inductive Bias from Grid-Based Encoders:** FC-DeepONet uses a CNN-based encoder on a discretized domain, introducing a strong bias that is advantageous for smooth, low-frequency data. However, this also limits its generalization capabilities, especially in non-Euclidean or high-frequency settings where our grid-free approach excels.\n\nWe provide a more detailed discussion of these trade-offs, along with additional experimental comparisons, in our response to Reviewer uQ6k.\n\nWe would also like to highlight that, as shown in Table 1 of that response, our model outperforms other non-equivariant Conditional Neural Field models such as Functa \\[1]. While Functa was among the first to adopt auto-decoding for neural field conditioning (as we also do), it lacks our proposed equivariant inductive biases. Our results indicate that incorporating structured geometric priors and a grid-free formulation significantly improves both accuracy and generalization in neural Eikonal solvers.\n\nWe thank the reviewer for pointing out that these insights deserve more emphasis in the main paper, and we will revise the discussion accordingly.\n\n### Clarity and Motivation.\n\nWe acknowledge the reviewer’s concern regarding the accessibility of the manuscript. In the revised version, we will state the core research question more explicit upfront:\n*“What are the benefits of framing Neural Eikonal Solvers as Conditional Neural Fields (CNFs)?”*\n\nAdditionally, we will include a high-level overview of the proposed method before diving into the formalism. This will help readers unfamiliar with the technical details build a conceptual understanding early in the paper.\n\n### Steerability of the Latent Space.\n\nThe reviewer is correct in pointing out that, while we provide theoretical guarantees for steerability, we do not currently include an empirical demonstration. We agree this is an important omission and will add a qualitative visualization in the revised manuscript to illustrate this property. This will involve:\n\n1.  A reference velocity field and its corresponding travel-time solution.\n2.  The predicted travel-time field from our model.\n3.  The predicted field after applying a group transformation (e.g., rotation) to the latent representation.\n\nWe will present comparisons to a non-equivariant baseline to illustrate the improved consistency and generalization enabled by equivariant conditioning. This experiment will closely follow the style of Figure 10 in [2].\n\n### Applicability to Non-Euclidean Manifolds.\n\nAs the reviewer correctly noted, our framework is designed to generalize to arbitrary Riemannian manifolds that admit a well-defined Lie group action. In the current version, we focus on the Euclidean setting due to the availability of standard benchmarks: OpenFWI is the only existing dataset for neural-operator-based Eikonal solvers.\n\nThat said, we now include in Table 1 of our response to Reviewer Y7P6 a new experiment solving the Eikonal equation on the 2-sphere. The results show that our method generalizes well to non-Euclidean geometries. In the final version of the manuscript, we will also include a new experiment involving geodesic path planning on the sphere in the presence of obstacles, further showcasing the generality of our method. Importantly, such experiments are not feasible using FC-DeepONet, as it relies on discretizing the input domain in a regular grid to encode velocity fields—an approach that does not easily extend to general manifolds, as discussed in Section 2.\n\n### Clarification on Equation 5 and HJB Formulation.\n\nWe thank the reviewer for pointing out the need for clarification around Equation 5 and its connection to the Hamilton-Jacobi-Bellman (HJB) equation. To clarify: we reformulate the Eikonal equation in Hamiltonian form as\n\n$\\mathcal{H}(s, r, T) = v(s)^{2}||grad_{s}T(s, r) ||^2_{\\mathcal{G}} - 1 = 0,$\n\nwhich is equivalent to Equation 1 but written in terms of the Hamiltonian. This form emphasizes the interpretation of the Eikonal equation as the zero-level set of the Hamiltonian. In the final manuscript, we will make this equivalence explicit and provide additional verbal explanation to ensure this connection is clear to all readers.\n\n\n### Closing Remarks.\n\nWe are sincerely grateful for the reviewer’s insightful feedback and suggestions. We believe that the proposed revisions—clarifying the contributions, improving the accessibility of the text, empirically validating steerability, and expanding the experimental scope—will significantly strengthen the final manuscript. Thank again for your thoughtful review.\n\n\n### References \n[1] Dupont, Emilien, et al. \"From data to functa: Your data point is a function and you can treat it like one.\" arXiv preprint arXiv:2201.12204 (2022).\n\n[2] Wessels, David R., et al. \"Grounding continuous representations in geometry: Equivariant neural fields.\" arXiv preprint arXiv:2406.05753 (2024)."}, {"review_id": "R7befqkHOi", "reviewer": "Reviewer_Y7P6", "summary": "The 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.", "questions": "Can 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?", "limitations": "non-Euclidean and real-world datasets experiments are missing", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strengths \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", "was_revised": true, "final_justification": "The rebuttal resolves all my concerns. Thus, I raise my rating to boardline accept.", "rebuttal": "We sincerely thank the reviewer for their thoughtful and constructive feedback. We are especially grateful for their recognition of both the relevance of our work and its theoretical contributions.\n\n## Generalization Beyond the Euclidean Setting\n\nAs the reviewer correctly noted, our framework is designed to generalize to arbitrary Riemannian manifolds that admit a well-defined Lie group action. In the current version of the manuscript, we focused on the Euclidean setting due to the availability of standard benchmarks—specifically, OpenFWI, which remains the only publicly available dataset for neural operator-based Eikonal solvers.\n\nIn response to the reviewer's request, we now present two new experiments that evaluate our method on the 2-sphere, thereby extending our analysis beyond the Euclidean case. These experiments test the model's ability to solve the Eikonal equation on a curved manifold. It is worth emphasizing that our model operates using the Euclidean distance of the embedded manifold in the factorized travel-time function $T(s, r) = \\tilde{d}(s, r) \\tau(s, r)$ (as introduced in Section 3.2), following [1]. This design choice means the network must implicitly learn correct wavefront expansion behavior on the sphere to approximate the solution accurately.\n\nWe evaluate our method on two types of velocity fields defined over the sphere:\n\n- **Constant Speed**: To test generalization to geometrically nontrivial domains, we first use a dataset with constant velocity fields, each defined by sampling a scalar velocity from a uniform distribution over $[0.1, 2.0]$.\n- **Spherical Style-B**: To test generalization to more complex fields, we construct a dataset by projecting the 2D Style-B velocity fields from OpenFWI onto the sphere using spherical coordinates.\n\nTo empirically support our theoretical claim that the method applies to *arbitrary Lie group actions*, including *non-transitive* ones, we incorporate steerability with respect to $SO(2)$ (rotations about the $z$-axis). In this setting, the pose component of the latent representation lies in $SO(2)$, and the learned invariants remain maximally expressive, consistent with our theoretical predictions.\n\nAs shown in Table 1, our method generalizes effectively to non-Euclidean geometries. In the final version of the manuscript, we also plan to include an additional experiment involving geodesic path planning on the sphere with obstacles, further demonstrating the flexibility and scope of our framework.\n\nIt is worth noting that such experiments are not feasible for models like FC-DeepONet, which rely on discretizing the input domain into regular grids—an approach that does not naturally extend to general manifolds, as discussed in Section 2.\n\n**Table 1: Performance of our method on Eikonal solvers over the 2-sphere with steerability over $z$-axis rotations**\n\n| Dataset | RE (↓) | RMAE (↓) | Fitting Time (s) |\n|---------|--------|----------|------------------|\n| Constant Speed | 0.013 | 0.012 | 209.2 |\n| Spherical Style-B | 0.015 | 0.012 | 207.1 |\n\n## Ablation Studies and Architectural Design Justification\n\nWe appreciate the reviewer's concern regarding the need for further ablation studies. While the manuscript already includes a comparison of equivariant versus non-equivariant models, we would like to clarify that this is not the only ablation study provided.\n\n- **Section E.1** presents an ablation analyzing the role of pretraining in the meta-learning process. To our knowledge, this type of analysis is novel in the context of Equivariant Neural Fields [2, 3].\n\n- **Section E.3** provides an ablation examining the tradeoff between performance and computational efficiency as a function of the number of autodecoding steps. This highlights a key feature of our approach—its adaptability to resource constraints—unlike models such as FC-DeepONet, which have fixed cost-performance profiles post-training. A more detailed discussion of this tradeoff is included in our response to Reviewer uCF6.\n\nRegarding architectural choices, certain components—such as the bounded projection head and the invariant cross-attention module—were adopted based on strong empirical performance in prior work (Section 4.2) [2, 4]. However, we agree that a more focused discussion and further empirical analysis of these choices in the context of our specific task would strengthen the manuscript. In response, we have added a new ablation experiment focused on the choice of embedding function for computing invariants.\n\n**Table 2: Ablation on the choice of embedding for computing invariants (CurveFault-B validation set).**\n\n| Embedding | RE (↓) | RMAE (↓) | Fit + Inf. Time (s) |\n|-----------|--------|----------|---------------------|\n| MLP | 0.025 | 0.020 | 237.2 |\n| Polynomial (deg. 3) | 0.024 | 0.019 | 1460.7 |\n| RFF (Ours) | **0.021** | **0.016** | 255.8 |\n\nTo further validate the importance of our architectural design choices, we conducted an additional comparative experiment (detailed in Table 1 of our response to Reviewer uQ6k) against other conditional neural field models. Specifically, we demonstrate that our approach significantly outperforms non-equivariant methods such as Functa [5]. While Functa pioneered the use of auto-decoding for neural field conditioning—a technique we also leverage—it lacks the equivariant inductive biases that are central to our framework. These comparative results underscore that incorporating structured geometric priors within a grid-free formulation yields substantial improvements in both accuracy and generalization for neural Eikonal solvers.\n\n## Further Analysis: Resource Usage, Failure Modes, and Sensitivity\n\nWhile the reviewer raises a concern regarding the lack of analysis on resource usage and failure modes, we would like to clarify that the manuscript already provides relevant details on these aspects. Specifically, Section D includes hardware specifications, number of GPUs, parameter counts, and both training and inference times (see Figures 6 and 7).\n\nThat said, we agree that further discussion of potential failure modes and sensitivity to design choices adds value. In response, we expand our analysis accordingly.\n\nOne failure mode we identify occurs when sampling training data on general Riemannian manifolds. As discussed in [1], curvature-aware sampling is necessary to ensure geometric coverage and avoid degenerate training cases.\n\nRegarding hyperparameter sensitivity, the manuscript already highlights two mechanisms that reduce tuning requirements:\n\n1. The use of RFF with learnable frequencies (Section D.2).\n2. The adoption of a log-cosh loss function during meta-learning (Section C.1), which stabilizes gradients.\n\nWe also introduce a new ablation (Table 3) evaluating how different latent conditioning setups affect performance, under a fixed total latent budget of 288 parameters. The configuration with 9 latent points of dimension 32 provides the best tradeoff between accuracy and cost.\n\n**Table 3: Ablation on latent conditioning design (CurveFault-B validation set). Total latent budget fixed at 288 parameters.**\n\n| # Latents (N) | Latent Dim. (d) | RE (↓) | RMAE (↓) | Fit + Inf. Time (s) |\n|---------------|------------------|--------|----------|---------------------|\n| 1 | 288 | 0.078 | 0.066 | 58.5 |\n| 4 | 72 | 0.047 | 0.038 | 139.3 |\n| 9 | 32 | **0.021** | **0.016** | 255.8 |\n| 16 | 18 | 0.039 | 0.031 | 501.7 |\n\n## Closing Remarks\n\nWe believe the additional non-Euclidean experiments, detailed architectural ablations, and sensitivity analyses demonstrate the robustness, generality, and practical relevance of our approach. We hope this response addresses the reviewer's concerns and illustrates the merit of raising the overall evaluation score. Please don't hesitate to let us know if there are additional aspects we can clarify or elaborate further.\n\n## References\n[1] Kelshaw, Daniel, and Luca Magri. \"Computing distances and means on manifolds with a metric-constrained Eikonal approach.\" Proceedings A. Vol. 481. No. 2312. The Royal Society, 2025.\n\n[2] Wessels, David R., et al. \"Grounding continuous representations in geometry: Equivariant neural fields.\" arXiv preprint arXiv:2406.05753 (2024).\n\n[3] Knigge, David, et al. \"Space-time continuous pde forecasting using equivariant neural fields.\" Advances in Neural Information Processing Systems 37 (2024): 76553-76577.\n\n[4] Grubas, Serafim, Anton Duchkov, and Georgy Loginov. \"Neural Eikonal solver: Improving accuracy of physics-informed neural networks for solving eikonal equation in case of caustics.\" Journal of Computational Physics 474 (2023): 111789.\n\n[5] Dupont, Emilien, et al. \"From data to functa: Your data point is a function and you can treat it like one.\" arXiv preprint arXiv:2201.12204 (2022)."}]

Author rebuttal:

Meta-review: This manuscript proposes Equivariant Neural Eikonal Solvers, a framework that integrates Equivariant Neural Fields (ENFs) with Neural Eikonal Solvers. The claimed key advantages are: (1) enhanced representation efficiency through parameter sharing, (2) robust geometric grounding, and (3) solution steerability. The framework is evaluated on seismic travel-time modeling using 2D and 3D benchmark datasets, where it outperforms baseline methods.

All reviewers recognize the contributions of the theoretical framework, as well as the demonstrated improvements in both accuracy and efficiency over the baselines (e.g., FC-DeepONet). The AC concurs with the reviewers, which supports an initial assessment of accept (poster).

However, the AC shares the reviewers' concerns regarding the empirical demonstration of steerability (Bupx, Y7P6) and generalization to diverse manifolds (Bupx). The rebuttal provided additional experiments that help substantiate these claims, and the AC highly recommends including them in the final version of this manuscript.
Decision: Accept (poster)