🤖 machine learning

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

This paper introduces Equivariant Neural Eikonal Solvers, a novel framework that combines Equivariant Neural Fields with Physics-Informed Neural Networks to efficiently predict travel times on arbitrary homogeneous spaces by leveraging a shared backbone conditioned on latent point clouds for enhanced geometric grounding, scalability, and solution steerability.

Original authors: Alejandro García-Castellanos, David R. Wessels, Nicky J. van den Berg, Remco Duits, Daniël M. Pelt, Erik J. Bekkers

Published 2026-07-30

📖 7 min read🧠 Deep dive

CC BY 4.0

Original authors: Alejandro García-Castellanos, David R. Wessels, Nicky J. van den Berg, Remco Duits, Daniël M. Pelt, Erik J. Bekkers

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a master cartographer trying to draw the fastest route for a traveler to get from point A to point B. But here's the twist: the ground beneath them isn't flat or uniform. Sometimes it's a smooth highway, other times it's a muddy swamp, and sometimes the traveler is moving on the curved surface of a giant ball or even a saddle-shaped world. In physics and engineering, figuring out these "fastest arrival times" is governed by a tricky math puzzle called the eikonal equation. Think of it as the ultimate traffic calculator that tells you how long a wave of sound, light, or a robot's movement will take to reach its destination through a complex landscape.

For decades, scientists have used rigid, grid-based tools to solve this puzzle, like trying to draw a smooth curve using only square Lego bricks. If the landscape gets too complex, you need millions of tiny bricks, which takes forever to compute and eats up all your computer's memory. Recently, smart computers (neural networks) started trying to solve this without the bricks, learning the shape of the world directly. But these new methods often had to be retrained from scratch every time the landscape changed, making them too slow for real-time use. They were like a chef who could make a perfect pizza, but if you asked for a burger, they had to go back to culinary school and learn everything again.

This is where a team of researchers from the Netherlands steps in with a clever new idea. They built a "shape-shifting" neural network that doesn't just learn one specific map; it learns the rules of how maps change when you rotate or shift the world. By treating the problem like a game of geometric symmetry, they created a system that can instantly adapt to new terrains without needing to relearn everything. It's like teaching a chef the secret ingredient that makes any dish taste right, so they can instantly switch from pizza to burgers just by changing the ingredients, not the recipe.

The Magic of the "Steerable" Map

The paper introduces a framework called Equivariant Neural Eikonal Solvers (E-NES). To understand how it works, imagine you have a magical, stretchy rubber sheet representing a landscape. If you slide a rock across this sheet, the ripples (the travel times) move with it perfectly. If you rotate the sheet, the ripples rotate too. This property is called equivariance.

Most old neural networks are like rigid statues; if you rotate the input, the output gets scrambled and the network has to guess what happened. The E-NES, however, is built with a "geometric brain." It understands that if you rotate the input (the velocity field or the terrain), the solution (the travel time) must rotate in a matching, predictable way. The researchers achieved this by representing the "conditions" of the problem not as a simple list of numbers, but as a point cloud floating in a special mathematical space called a Lie group. Think of this point cloud as a set of floating anchors that tell the network where the "wind" is blowing or where the "mud" is located.

The genius move here is that the network shares its "muscle memory" (weights) across all these transformations. Instead of learning a separate solution for every possible rotation or shift, it learns one master solution that can be "steered" to fit any orientation. If you tell the network, "Rotate the wind 90 degrees," the network doesn't need to retrain; it simply applies that rotation to its internal logic, and the travel-time map updates instantly and correctly.

What They Found and How They Tested It

The team put their new solver to the test in three very different worlds: flat 2D maps, 3D cubes, and the curved surface of a sphere (like the Earth). They compared their method against the current state-of-the-art, which includes "Neural Operators" (like FC-DeepONet) and other grid-free methods.

In the 2D tests, which used a benchmark dataset called OpenFWI with 10 different types of velocity fields (some flat, some curved, some with faults), the E-NES showed it could be incredibly accurate. When they let the network "autodecode" (fine-tune its internal settings) for about 100 rounds, it beat the competition on seven out of ten datasets. For the trickier, more complex maps, the improvement was substantial. Even when they rushed the process to save time (using a "meta-learning" approach that only took a few steps), the E-NES was still competitive, but with a massive speed boost.

Here is where the numbers get exciting: The traditional methods took hundreds of seconds to fit a single new map. The E-NES, using its meta-learning trick, could fit 100 different velocity fields in less than 6 seconds total. That is a two-orders-of-magnitude improvement. While the "full convergence" mode (where it runs for longer to get the absolute best accuracy) took about 222 seconds for 100 fields, it still produced lower errors than the competitors in most cases.

They also tested the system in 3D. As they increased the grid size from a small 2x2x2 block up to a massive 6x6x6 block, the E-NES kept its error rates stable. Unlike the old grid-based methods (like the Fast Marching Method), which get slower and slower as you add more detail, the E-NES stayed fast because it doesn't rely on a grid at all. It treats the world as a continuous flow, so adding more resolution doesn't break it.

Finally, they took the system to a sphere (the 2-sphere, $S^2$ ). This is a non-Euclidean world, meaning the geometry is curved like a ball, not flat like a table. They tested it with constant speeds and with "Gaussian obstacle" fields (where a slow zone acts like a bump on the sphere). The E-NES handled this beautifully, achieving a relative error of just 0.013 for constant speeds and 0.015 for the obstacle fields. It even successfully planned geodesic paths (the shortest routes on a sphere), bending the trajectory around obstacles just as physics dictates.

Why This Matters

The paper explicitly rules out the idea that you need to retrain a network for every new scenario or that you are stuck with the limitations of a grid. They argue against the "per-instance training" of older methods, showing that it creates a bottleneck for real-time applications. Instead, they demonstrate that by embedding geometric symmetry directly into the network's architecture, you get a system that is not only more accurate but also "steerable."

This steerability is the key. It means that if you change the input (like rotating a velocity field), the output changes in a mathematically guaranteed way. This allows the system to generalize to arbitrary Riemannian manifolds—whether they are flat, spherical, or hyperbolic—without needing special adjustments. The authors suggest that this approach could revolutionize fields like seismic imaging (mapping underground structures), robotics (planning paths for robots in complex environments), and computer vision (understanding 3D shapes).

In short, the researchers didn't just build a faster calculator; they built a calculator that understands the geometry of the world it's calculating for. They showed that by respecting the symmetries of the universe, you can solve complex travel-time puzzles with a fraction of the effort, opening the door to real-time, high-precision modeling in 2D, 3D, and even on the surface of a sphere.

Technical Summary: Equivariant Eikonal Neural Networks

Problem Statement

The Eikonal equation, a first-order nonlinear partial differential equation (PDE), is fundamental to computing shortest arrival times (travel times) in scalar velocity fields across diverse domains such as computer vision (Signed Distance Functions, geodesic segmentation), robotics (motion planning), and geophysics (seismic wave propagation).

Conventional numerical solvers, including the Fast Marching Method (FMM) and Fast Sweeping Method (FSM), rely heavily on spatial discretization. This creates a trade-off where complex velocity models require high-resolution grids, leading to prohibitive computational and memory costs. While Physics-Informed Neural Networks (PINNs) offer a grid-free alternative by integrating PDE constraints into the loss function, they typically require training a separate network for each distinct velocity field, limiting their applicability in real-time or large-scale scenarios.

Neural Operators (e.g., DeepONet, Fourier Neural Operators) attempt to learn mappings between function spaces (velocity fields to travel-time solutions) using shared backbones. However, existing approaches often rely on partial discretization, lack resolution invariance, or depend on supervision from numerical solvers. Furthermore, they often fail to explicitly leverage the geometric symmetries inherent in the problem, missing opportunities for improved representation efficiency and generalization.

Methodology: Equivariant Neural Eikonal Solver (E-NES)

The authors propose Equivariant Neural Eikonal Solver (E-NES), a framework that integrates Equivariant Neural Fields (ENFs) with Neural Eikonal Solvers. The core innovation lies in modeling the solution as a conditional neural field where the conditioning variables are geometric point clouds within a Lie group, ensuring equivariance under group actions.

1. Theoretical Framework

The method defines a conditional neural field $T_\theta(s, r; z)$ that approximates the travel-time solution $T(s, r)$ for a given velocity field. The conditioning variable $z$ is represented as a set of pose-context pairs $z = \{(g_i, c_i)\}_{i=1}^N$ , where $g_i$ are group elements (poses) and $c_i$ are context vectors.

The framework enforces a steerability constraint:
$T_\theta(s, r; g \cdot z) = T_\theta(g^{-1} \cdot s, g^{-1} \cdot r; z)$
This property ensures that applying a group transformation $g$ to the latent conditioning variable induces a corresponding, predictable transformation on the travel-time solution. Consequently, the network generalizes across the entire orbit of group transformations without explicit data augmentation.

The solution is factorized as $T(s, r) = \tilde{d}(s, r) \tau(s, r)$ , where $\tilde{d}$ is a group-invariant semimetric (e.g., geodesic or Euclidean chordal distance) and $\tau$ is the learnable scalar field.

2. Model Architecture

The E-NES architecture consists of two main components:

Invariant Cross-Attention Encoder: This module computes invariant representations of the input coordinates $(s, r)$ and the conditioning point cloud $z$ . It utilizes the moving frame method to derive a complete set of functionally independent invariants. Specifically, for each pose $g_i$ in the point cloud, it computes invariants via $Inv(s, r, g_i) = (g_i^{-1} \cdot s, g_i^{-1} \cdot r)$ . These invariants are processed through Random Fourier Features (RFF) and fed into a cross-attention mechanism where the query is derived from the invariant features and keys/values from the context vectors.
Bounded Velocity Projection: The encoder output is passed through a Multi-Layer Perceptron (MLP) with AdaptiveGauss activations to model sharp wavefronts. A final projection layer maps the output to the physically valid range $[1/v_{max}, 1/v_{min}]$ .

3. Invariant Computation

A key theoretical contribution is the extension of the moving frame method to product manifolds where the group action may not be free. The authors prove that by augmenting the space with a learnable latent group element, a complete set of functionally independent invariants can be systematically derived. This guarantees zero information loss during canonicalization and ensures the model's expressivity.

4. Training Strategy

The model is trained using Physics-Informed Neural Network (PINN) principles. The loss function penalizes deviations from the Hamiltonian form of the Eikonal equation ( $H=0$ ) at sampled source-receiver pairs. The authors employ two training modes:

Autodecoding: Optimizing network weights $\theta$ and latent variables $z$ simultaneously.
Meta-learning: An outer loop optimizes $\theta$ , while an inner loop optimizes $z$ for specific velocity fields in a few steps, enabling rapid adaptation to new fields.

Key Contributions

Generalization of Equivariant Neural Fields: The authors extend ENFs to functions defined over products of Riemannian manifolds with regular group actions, covering homogeneous spaces associated with linear Lie groups.
E-NES Framework: They implement a solver that leverages geometric symmetries to solve Eikonal equations, enabling generalization across group transformations without data augmentation.
Theoretical Guarantees: They provide a formal expressivity analysis establishing a complete set of independent invariants, addressing gaps in prior work where invariants were selected heuristically.
Grid-Free Scalability: The approach offers a grid-free, resolution-independent method for solving Eikonal equations on arbitrary Riemannian manifolds, including Euclidean, spherical, and hyperbolic spaces.

Experimental Results

The authors validated E-NES on 2D, 3D, and spherical benchmarks:

2D OpenFWI Benchmarks: On ten velocity field categories (e.g., FlatVel, CurveFault, Style), E-NES outperformed FC-DeepONet in 7 out of 10 datasets when using full autodecoding convergence. It achieved significantly lower relative errors (RE) and demonstrated superior adaptability to complex geometries.
Comparison with Functa: E-NES consistently outperformed Functa (a standard Conditional Neural Field baseline) across all datasets, attributed to E-NES's geometric point-cloud conditioning versus Functa's global latent vectors.
Scalability (3D): When extended to 3D, E-NES maintained stable error metrics as grid dimensions increased, whereas traditional methods like FMM suffer from exponential computational growth. E-NES demonstrated computational advantages over FMM even at minimal grid sizes, with efficiency gains amplifying at larger scales.
Non-Euclidean Domains: The method successfully generalized to the 2-sphere ( $S^2$ ) with $SO(2)$ steerability, achieving low relative errors on constant and complex velocity fields. It enabled geodesic path planning via gradient integration, correctly navigating around obstacles.
Efficiency: While autodecoding requires optimization time, the meta-learning approach reduced fitting time for 100 velocity fields from ~1000 seconds (DeepONet) to under 6 seconds, representing a two-order-of-magnitude improvement in inference speed for new fields.

Significance and Claims

The paper claims that E-NES represents a significant advancement in scientific machine learning for PDE solving by unifying geometric deep learning with physics-informed constraints.

Steerability and Control: The framework allows users to induce geometrically meaningful modifications in the solution by transforming the latent point cloud, offering a level of controllability absent in standard Neural Operators.
Representation Efficiency: By sharing weights across the group orbit, the model achieves higher representation efficiency and robustness compared to methods requiring separate training for each instance.
Flexibility: The grid-free nature allows for continuous evaluation at arbitrary resolutions, making it suitable for applications requiring high precision or continuous path planning (e.g., robotics, ray tracing).
Test-Time Adaptability: Unlike fixed-weight Neural Operators, E-NES allows practitioners to dynamically adjust the trade-off between accuracy and computational cost at test time by varying the number of optimization iterations.

The authors position this work as a step toward more generalizable, geometrically grounded solvers for complex physical systems, with future work planned for position-orientation spaces and hyperbolic geometries.