Neural Eikonal solver: Improving accuracy of physics-informed neural networks for solving eikonal equation in case of caustics

The concept of physics-informed neural networks has become a useful tool for solving differential equations due to its flexibility. A few approaches are using this concept to solve the eikonal equation which describes the first-arrival traveltimes of acoustic and elastic waves 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 non-smooth solution behaviour. In this paper, we revisit the problem of solving the eikonal equation using neural networks to tackle the caustic pathologies. We introduce the novel Neural Eikonal Solver (NES) for solving the isotropic eikonal equation in two formulations: the one-point problem is for a fixed source location; the two-point problem is for an arbitrary source-receiver pair. We present several techniques that provide relatively fast, stable, and accurate approximation of the eikonal in complex velocity models producing caustics: an improved factorization bounding the NES between the fastest and the slowest solutions to speed up the training; a non-symmetric loss function based on L 1 -norm and a Hamiltonian of the eikonal to account for the errors caused by caustics; gaussian activation for a more accurate approximation of solution in caustics; a symmetrization to account for the reciprocity principle in the two-point problem. The tests on the Marmousi model showed that NES provides the relative mean absolute error of about 0.2-0.4% from the second-order factored Fast Marching Method, and outperforms existing neural-network solvers giving 10-60 times lower errors and 2-30 times faster training. With using a GPU, the training takes 1-5 minutes, and the inference time is comparable with the Fast Marching. The one-point NES provides the most accurate solution, whereas the two-point NES provides slightly lower accuracy but gives an extremely compact representation. It can be useful in various seismic applications where massive computations of traveltimes are required (millions of source-receiver pairs): ray modeling, traveltime tomography, hypocenter localization, and Kirchhoff migration. Source code is available at https://github.com/sgrubas/NES two-point eikonal the PINN suggested several novelties to to to speed up training. the improved factorization of the to speed up by constraining the NES the fastest and the slowest solutions. To pathologies on