A data-driven learned discretization approach in finite volume schemes for hyperbolic conservation laws and varying boundary conditions

TL;DR

This paper introduces a data-driven finite volume method that leverages neural networks to accurately solve hyperbolic conservation laws with varying boundary conditions on coarse meshes, improving efficiency and stability.

Contribution

The work extends previous data-driven discretization methods to flux-limited finite volume schemes, enabling accurate capture of discontinuities and boundary conditions with neural network-based coefficients.

Findings

Accurately reproduces fine-grid solutions on coarse meshes

Enhances stability and accuracy of finite volume schemes

Effective in 1D and 2D hyperbolic PDEs

Abstract

This paper presents a data-driven finite volume method for solving 1D and 2D hyperbolic partial differential equations. This work builds upon the prior research incorporating a data-driven finite-difference approximation of smooth solutions of scalar conservation laws, where optimal coefficients of neural networks approximating space derivatives are learned based on accurate, but cumbersome solutions to these equations. We extend this approach to flux-limited finite volume schemes for hyperbolic scalar and systems of conservation laws. We also train the discretization to efficiently capture discontinuous solutions with shock and contact waves, as well as to the application of boundary conditions. The learning procedure of the data-driven model is extended through the definition of a new loss, paddings and adequate database. These new ingredients guarantee computational stability, preserve the accuracy of fine-grid solutions, and enhance overall performance. Numerical experiments using test cases from the literature in both one- and two-dimensional spaces demonstrate that the learned model accurately reproduces fine-grid results on very coarse meshes.