ANN-Flux Method for Solving Nonlinear Scalar Conservation Laws: Primal Aspects

Artificial neural networks (ANNs) have been successfully applied to solve partial differential equations (PDEs). Conservation laws are PDEs that arise in a variety of fields such as fluid dynamics, traffic modeling, and fluid flow through porous media. We've recently proposed the original ANN-Flux method to solve the Riemann problem for nonlinear scalar conservation laws. The ANN-Flux method is designed for complex fluxes, which are costly to evaluate, differentiate, or invert. Based on the entropy solution form, the method applies two ANNs to approximate the flux $F$ and its derivative inverse $G = [F']^{-1}$ for the shock and the rarefaction cases. The $F$ is approximated by a multilayer perceptron (MLP) $\mathcal{N_F}$. Automatic differentiation is then applied to approximate the derivative $F' \approx \mathcal{N}'_F$ and train a new MLP $\mathcal{N}_G\approx G$. Once the ANNs are trained, they can be applied to solve any Riemann problem in the flux convexity region. In this work we review the primal aspects of the generalization of the method to non-Riemann initial conditions by combining the ANN-Flux with the Godunov Method. The results show that the ANN-Flux produced precise results when compared with analytical (when available) or numerical alternatives.