Physics-constrained deep learninig for solving the Eikonal equation

The Eikonal equation is a non-linear PDE that is used for modeling seismic traveltimes. Here we test the idea of using neural networks for solving the 2D Eikonal equation. The concept of the physics-informed neural networks implies including the PDE and boundary conditions into the loss functions. Then no labeled data are required for training the network. While testing this approach we show that it is not sufficient to include only the equation and the boundary condition into the loss function as the training procedure may converge to solutions corresponding to various source terms. We propose supplementing the loss function with additional physics constraint promoting monotonic behavior (time increasing away from the source location). We were testing various neural-network architectures for several inhomogeneous velocity models: with linear vertical gradient, with a smooth high-velocity anomaly, the two-layered models. In the tests, the physics-informed neural network was able to reproduce the behavior of propagating fronts with the mean absolute relative error of about 5 % for all the considered tests. Further development of the training strategy is necessary for further accuracy improvement.