ANN-Flux Method for Nonconvex Fluxes in Riemann Problems

Artificial neural networks (ANNs) have been successfully applied to solve partial differential equations, mainly after the emergence of the physics-informed neural networks (PINNs). Applications to nonlinear conservation laws are also notable, including PINNs for high-speed flows, conservative PINNs (cPINNs), and weak PINNs (wPINNs). In this work, we propose the new ANN-Flux method to solve the Riemann problem for the nonlinear conservation law with given left and right states and flux function. The entropic solution of the problem when the flux is nonconvex can be a mixture of shock and rarefaction waves, depending on the left and right states. The ANN-Flux method is designed for complex fluxes, which are costly to be evaluated, differentiated, or inverted. The method consists of approximating the flux by a multilayer perceptron (MLP) neural network classically trained to minimize the mean squared error (MSE) loss with a generated data set. For the simple shock wave case, the solution is approximated by substituting the flux by the neural network. For a rarefaction case, a second MLP neural network learns to approximate the inverse of the derivative of the flux. The evaluation of the derivative can be efficiently computed by automatic differentiation. The method has shown very good results for the Buckley-Leverett equation and will be applied to more complex conservation laws in future work.