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 use this solve the eikonal equation that describes first-arrival traveltimes waves propagating in smooth heterogeneous velocity models. However, challenge is exacerbated by models producing caustics, resulting instabilities and deterioration accuracy non-smooth solution behavior. In paper, we revisit problem using tackle caustic pathologies. We introduce novel Neural Eikonal Solver (NES) isotropic two formulations: one-point fixed source location; two-point an arbitrary source-receiver pair. present several techniques which provide stability case caustics: improved factorization; non-symmetric loss function based on Hamiltonian; gaussian activation; symmetrization. our tests, NES showed relative mean-absolute error 0.2-0.4% from second-order factored Fast Marching Method with similar inference time, outperformed existing neural-network solvers giving 10-60 times lower errors 2-30 faster training. provides most accurate solution, while gives slightly but extremely compact representation all spatial derivatives. It can be many seismic problems: massive computations millions pairs Kirchhoff migration; modeling ray amplitudes derivatives; traveltime tomography; earthquake localization; multipathing analysis.