ANN-Flux Method for Nonconvex Fluxes in Riemann Problems

Abstract

Artificial neural networks (ANNs) [3] have been successfully applied to solve partial differential equations, particularly since the emergence of physics-informed neural networks (PINNs) [7]. Applications to nonlinear conservation laws have also shown promising results, including PINNs for high-speed flows [6], conservative PINNs [4], and weak PINNs [8]. In this work, we propose a novel method, called ANN-Flux, to solve the Riemann problem for the nonlinear scalar conservation law u_t + (F(u))_x = 0, x, t ∈ ℝ × (0, t_f], (1) u(x, 0) = u_L, x ∈ ℝ−, u(x, 0) = u_R, x ∈ ℝ+, (2) with prescribed left and right states u_L, u_R ∈ ℝ, and a flux function F : ℝ → ℝ. When the flux function is nonconvex, the entropy solution of problem (1) may consist of a combination of shock and rarefaction waves, depending on the initial states. [...] The ANN-Flux (ANN-F) method is designed for intricate flux functions that are computationally expensive to evaluate, differentiate, or invert. Given that neural networks are universal approximators [3], ANN-F approximates the flux F using a multilayer perceptron (MLP) ỹ = N_F(u) trained to minimize the mean squared error (MSE) loss ε. In the rarefaction case, however, a second set of MLPs, denoted by N_G^(k), is introduced to approximate the inverse of the derivative of F, i.e., [N'_F]^-1 ≈ [F']^-1. By combining the ANN-based shock and rarefaction approximations, the ANN-Flux method is capable of solving Riemann problems involving nonconvex flux functions. The Buckley–Leverett equation models two-phase immiscible and incompressible fluid flow through porous media [5]. Its flux function is nonconvex, exhibiting a single change in concavity within the admissible domain of the variable. As a preliminary result, we compared the ANN-F solution with a reference numerical solution of the Buckley–Leverett equation. The MLP architectures were selected based on numerical experiments, from which we concluded that a 1–50 × 3–1 network for N_F and a 1–50 × 4–1 network for N_G were suitable for approximating F and G, respectively, achieving an L₂ error of 10^-6. Training was performed using hyperbolic tangent activation functions in the hidden layers and the identity function in the output layer, with the mean squared error as the loss function. The ANN-F method yielded accurate results for the Buckley–Leverett equation, with a relative L₂ error of 10^-4. Future work will focus on extending the method to more intricate conservation laws, such as those arising in traffic flow modeling [1]. [...]