The concept of physics-informed neural networks has become a useful tool for
solving differential equations due to its flexibility. There are a few
approaches 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 which provide stability in velocity models producing caustics:
improved factorization; non-symmetric loss function based on Hamiltonian;
gaussian activation; symmetrization. In our tests, NES showed the
relative-mean-absolute error of about 0.2-0.4% from the second-order factored
Fast Marching Method, and outperformed existing neural-network solvers giving
10-60 times lower errors and 2-30 times faster training. The inference time of
NES 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 are required (millions of
source-receiver pairs): ray modeling, traveltime tomography, hypocenter
localization, and Kirchhoff migration.Comment: The paper has 14 pages and 6 figures. Source code is available at
https://github.com/sgrubas/NE