# Deep Eikonal Solvers

## TL;DR

This paper introduces a deep learning-based method to solve the Eikonal equation more accurately and efficiently by replacing traditional local solvers with neural networks, applicable to both flat and curved geometries.

## Contribution

It presents a novel neural network approach integrated into the fast marching scheme, improving accuracy and generalization for solving the Eikonal equation on various surfaces.

## Key findings

- Smaller errors in numerical solutions
- Higher orders of accuracy achieved
- Effective on both flat and curved geometries

## Abstract

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.