Least-Squares Neural Network (LSNN) Method for Scalar Hyperbolic Partial Differential Equations
Min Liu, Zhiqiang Cai
TL;DR
The LSNN framework addresses scalar first-order hyperbolic PDEs, including linear advection-reaction and nonlinear hyperbolic conservation laws, by recasting the problem as an equivalent least-squares minimization on an admissible, possibly discontinuous solution set. It combines ReLU neural networks with a physics-preserved, weak formulation via a directional derivative $D_{\boldsymbol{\beta}}$ and a divergence-based approach to capture shocks without artificial viscosity or entropy penalties, while adaptively refining numerical integration and using a structure-guided Gauss-Newton solver to mitigate non-convexity. Key innovations include the use of physics-preserved differentiation, a space-time divergence formulation, and an SgGN-based solver that handles separable linear and nonlinear NN parameters; the method is demonstrated on 2D linear advection with variable velocity, a 1D Riemann problem with a convex flux, and a 2D Burgers equation, showing accurate shock representation with manageable degrees of freedom. The results indicate LSNN’s potential for efficient, physics-faithful hyperbolic PDE solvers, while highlighting ongoing challenges in solver robustness and high-dimensional performance.
Abstract
This chapter offers a comprehensive introduction to the least-squares neural network (LSNN) method introduced in [14,16], for solving scalar first-order hyperbolic partial differential equations, specifically linear advection-reaction equations and nonlinear hyperbolic conservation laws. The LSNN method is built on an equivalent least-squares formulation of the underlying problem on an admissible solution set that accommodates discontinuous solutions. It employs ReLU neural networks (in place of finite elements) as the approximating functions, uses a carefully designed physics-preserved numerical differentiation, and avoids penalization techniques such as artificial viscosity, entropy condition, and/or total variation. This approach captures shock features in the solution without oscillations or overshooting. Efficiently and reliably solving the resulting non-convex optimization problem posed by the LSNN method remains an open challenge. This chapter concludes with a brief discussion on application of the structure-guided Gauss-Newton (SgGN) method developed recently in [21] for solving shallow NN approximation.