The Application of Physics Informed Networks to Solve Hyperbolic Partial Differential Equations with Nonconvex Flux Function and Diffusion Term

In this paper, we study the possibility of using neural networks to solve rare derivatives, in particular transport problems in a porous medium. Neural networks can approximate the solution of differential equations, particularly multivariate partial differential equations (PDEs). We use physics-informed neural networks (PINN) for the classic hyperbolic model problem, namely the Buckley–Leverett. The experiment shows fairly accurate results; the error is RSME \(=\) 3.7356e-01. However, there is the open question of whether a solution to the Buckley–Leverett problem with a nonconvex flow function can be learnt by deep neural networks without the aid of artificial physical constraints.