Eikonal Solution Using Physics-Informed Neural Networks

The eikonal equation is utilized across a wide spectrum of science and engineering disciplines. In seismology, it regulates seismic wave traveltimes needed for applications like source localization, imaging, and inversion. Several numerical algorithms have been developed over the years to solve the eikonal equation. However, they suffer from computational bottleneck when repeated computations are needed for perturbations in the velocity model and/or the source location, particularly in large 3D models. Here, we employ the emerging paradigm of physics-informed neural networks (PINNs) to solve the eikonal equation. By minimizing a loss function formed by imposing the validity of the eikonal equation, we train a neural network to produce traveltimes that are consistent with the underlying partial differential equation. More specifically, to tackle point-source singularity, we use the factored eikonal equation. We observe sufficiently high traveltime accuracy for most applications of interest. We also demonstrate how machine learning techniques like transfer learning and surrogate modeling can be used to massively speed up traveltime computations for updated velocity models and source locations. These properties of the PINN eikonal solver are highly desirable in obtaining an efficient forward modeling engine for seismic inversion applications.