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

TL;DR

The paper develops a data-driven discretization for hyperbolic conservation laws within a flux-limited finite-volume framework, learning spatial derivatives via a CNN-based operator to achieve high-fidelity, shock-capturing solutions on coarse grids. It extends Bar-Sinai's approach to include boundary-condition-aware padding, entropy/TVD regularizations, and a transfer-learning training regime, validated on 1D Burgers and Euler equations and 2D Euler cases. Von Neumann stability analysis supports the method's robustness, and extensive 1D/2D tests demonstrate improved accuracy and shock resolution on coarse meshes, albeit with higher compute time. This work offers a promising path to accurate, data-driven, high-performance simulations of hyperbolic PDEs with flexible boundary handling and potential industrial applicability.

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.

Figures (20)

• Figure 1: Example of a 1D finite volume mesh.
• Figure 2: Ghost cell approach at boundaries, the ghost cell is represented in dashed line
• Figure 3: Architecture of our model. First, all primitives are normalized between 0 and 1. Subsequently, there are N blocks, comprising a padding layer and a convolutional layer. The padding layer serves to address the boundaries and to guarantee that the size of the output solution vector is identical to that of the input solution vector, as a convolution operation results in a reduction in vector size. Finally, the linear constraint guarantees the consistency of the model and reads $\sum_{j} \alpha_j=0$.
• Figure 4: Numerical dissipation (a) and dispersion (b) of the ML scheme and the centered scheme. The green line corresponds to the exact dispersion relation for (\ref{['eq:lin_advection']}).
• Figure 5: Sine-wave test case, 1024 cells for fine integration, 32 cells for ML and coarse solution, T = 0.39