Physics–Informed Neural Networks for the Factored Eikonal Equation

Resumo

The Eikonal equation often appears in problems including, but not limited to, geometric optics, shortest
path problems, image segmentation, seismic and medical imaging. While there are efficient and stable techniques
for solving the Eikonal equation for regular or arbitrary geometries in several dimensions, it remains a big challenge
to solve inverse problems governed by this equation, especially when it comes to uncertainty quantification. As an
alternative to classical methods for solving forward and inverse problems governed by PDEs, the Physics-Informed
Neural Networks (PINNs) have shown accuracy and versatility for solving problems in fluid dynamics, inference of
hydraulic conductivity, velocity inversion, phase separation, and others, being also able to quantify uncertainty in
these problems. Here we study PINNs for solving the forward and inverse probabilistic Factored Eikonal Equation.
We solve the probabilistic inverse problem using variational inference and the Flipout technique. Our results for a
benchmark problem show excellent accuracy.