Deep Eikonal Solvers
This work improves numerical geometry algorithms for applications in fields like computer graphics and physics, though it is incremental as it builds on the existing fast marching method.
The authors tackled the problem of numerically solving the Eikonal equation by replacing the local solver in the fast marching scheme with a trained neural network, resulting in smaller errors and higher accuracy orders for various geometries and surfaces.
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.