swMATH ID: 
44191

Software Authors: 
Grubas, Serafim; Duchkov, Anton; Loginov, Georgy

Description: 
Neural eikonal solver: improving accuracy of physicsinformed neural networks for solving eikonal equation in case of caustics. The concept of physicsinformed neural networks has become a useful tool for solving differential equations due to its flexibility. A few approaches use this concept to solve the eikonal equation that describes the firstarrival traveltimes of waves propagating in smooth heterogeneous velocity models. However, the challenge of the eikonal is exacerbated by the velocity models producing caustics, resulting in instabilities and deterioration of accuracy due to the nonsmooth solution behavior. In this paper, we revisit the problem of solving the eikonal equation using neural networks to tackle caustic pathologies. We introduce the novel Neural Eikonal Solver (NES) for solving the isotropic eikonal equation in two formulations: the onepoint problem is for a fixed source location; the twopoint problem is for an arbitrary sourcereceiver pair. We present several techniques which provide stability in the case of caustics: improved factorization; nonsymmetric loss function based on Hamiltonian; gaussian activation; symmetrization. In our tests, NES showed the relative meanabsolute error of 0.2–0.4% from the secondorder factored Fast Marching Method with a similar inference time, and outperformed existing neuralnetwork solvers giving 10–60 times lower errors and 2–30 times faster training. The onepoint NES provides the most accurate solution, while the twopoint NES gives slightly lower accuracy but an extremely compact representation with all spatial derivatives. It can be useful in many seismic problems: massive computations of traveltimes for millions of sourcereceiver pairs in Kirchhoff migration; modeling of ray amplitudes using spatial derivatives; traveltime tomography; earthquake localization; ray multipathing analysis. 
Homepage: 
https://sgrubas.github.io/NES/

Source Code: 
https://github.com/sgrubas/NES

Dependencies: 
Python 
Keywords: 
physicsinformed neural network;
eikonal equation;
seismic;
traveltimes;
caustics

Related Software: 
PINNtomo;
HypoSVI;
Keras;
TensorFlow;
DiffSharp;
DeepXDE;
Adam;
EikoNet;
PINNeik;
DGM

Cited in: 
1 Document
