Least-Squares Neural Network (LSNN) Method for Scalar Hyperbolic Partial Differential Equations

TL;DR

This paper introduces the LSNN method, a neural network-based approach for solving scalar hyperbolic PDEs that captures shocks accurately without oscillations, using a least-squares formulation and physics-preserving differentiation.

Contribution

It presents a novel LSNN framework that employs ReLU neural networks and avoids traditional penalization techniques for hyperbolic PDEs, effectively capturing discontinuities.

Findings

Captures shock features without oscillations

Uses least-squares formulation for hyperbolic PDEs

Employs physics-preserved numerical differentiation

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.