# Deep Eikonal Solvers

arXiv: https://arxiv.org/abs/1903.07973 · HTML: https://mlsanity.com/paper/1903.07973/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Mar 2019.

## Built on by (most cited first, 9 of 9)

- [Lie Point Symmetry Data Augmentation for Neural PDE Solvers](https://mlsanity.com/paper/2202.07643/) (2022)
- [Interpreting and Improving Diffusion Models from an Optimization Perspective](https://mlsanity.com/paper/2306.04848/) (2023)
- [Wormhole Loss for Partial Shape Matching](https://mlsanity.com/paper/2410.22899/) (2024)
- [Follow the Energy, Find the Path: Riemannian Metrics from Energy-Based Models](https://mlsanity.com/paper/2505.18230/) (2025)
- [Physics-guided Data Augmentation for Learning the Solution Operator of Linear Differential Equations](https://mlsanity.com/paper/2212.04100/) (2022)
- [Parallelised Differentiable Straightest Geodesics for 3D Meshes](https://mlsanity.com/paper/2603.15780/) (2026)
- [Deep Signatures -- Learning Invariants of Planar Curves](https://mlsanity.com/paper/2202.05922/) (2022)
- [Learning Differential Invariants of Planar Curves](https://mlsanity.com/paper/2303.03458/) (2023)
- [Deep Accurate Solver for the Geodesic Problem](https://mlsanity.com/paper/2602.22275/) (2026)

## Abstract (arXiv)

A deep learning approach to numerically approximate the solution to the Eikonal equation is introduced. The proposed method is built on the fast marching scheme which comprises of two components: a local numerical solver and an update scheme. We replace the formulaic local numerical solver with a trained neural network to provide highly accurate estimates of local distances for a variety of different geometries and sampling conditions. Our learning approach generalizes not only to flat Euclidean domains but also to curved surfaces enabled by the incorporation of certain invariant features in the neural network architecture. We show a considerable gain in performance, validated by smaller errors and higher orders of accuracy for the numerical solutions of the Eikonal equation computed on different surfaces The proposed approach leverages the approximation power of neural networks to enhance the performance of numerical algorithms, thereby, connecting the somewhat disparate themes of numerical geometry and learning.