Evolving Neural Network (ENN) Method for One-Dimensional Scalar Hyperbolic Conservation Laws: I Linear and Quadratic Fluxes

TL;DR

The paper introduces an evolving neural network method for solving one-dimensional scalar hyperbolic conservation laws, demonstrating improved accuracy and efficiency over traditional methods for linear and quadratic fluxes.

Contribution

It develops a neural network-based approach that evolves initial data representations over time, providing theoretical error bounds and numerical validation for linear and quadratic flux cases.

Findings

The ENN method is not limited by time step size for linear flux.

The method achieves higher accuracy than traditional mesh-based methods.

Numerical results confirm improved efficiency and precision.

Abstract

We propose and study the evolving neural network (ENN) method for solving one-dimensional scalar hyperbolic conservation laws with linear and quadratic spatial fluxes. The ENN method first represents the initial data and the inflow boundary data by neural networks. Then, it evolves the neural network representation of the initial data along the temporal direction. The evolution is computed using a combination of characteristic and finite volume methods. For the linear spatial flux, the method is not subject to any time step size, and it is shown theoretically that the error at any time step is bounded by the representation errors of the initial and boundary condition. For the quadratic flux, an error estimate is studied in a companion paper. Finally, numerical results for the linear advection equation and the inviscid Burgers equation are presented to show that the ENN method is more accurate and cost efficient than traditional mesh-based methods.