Learning high-accuracy numerical schemes for hyperbolic equations on coarse meshes

TL;DR

This work addresses the challenge of achieving high-accuracy solutions of hyperbolic PDEs on coarse meshes by learning interface-flux reconstruction weights with a weight-learning neural network (WLNN) on a six-point stencil. The WLNN enforces hard consistency constraints so that the learned weights satisfy $w_1+\cdots+w_6=1$ and $-5w_1-3w_2-w_3+w_4+3w_5+5w_6=0$, yielding a data-driven finite-difference scheme that outperforms classical 6th-order central and 5th-order upwind schemes for smooth problems. Extensive numerical tests on 1D and 3D scalar equations and 2D/3D Euler flows show improved accuracy, favorable spectral properties via approximate dispersion relation, and robustness to unseen parameters, with demonstrated efficiency advantages in certain regimes. The approach offers a practical path to high-fidelity simulations on coarse grids for multi-scale hyperbolic problems, including turbulence-related benchmarks, while highlighting the potential for extensions to non-smooth solutions.

Abstract

When solving partial differential equations using classical schemes such as finite difference or finite volume methods, sufficiently fine meshes and carefully designed schemes are required to achieve high-order accuracy of numerical solutions, leading to a significant increase in computational costs, especially for three-dimensional (3D) time-dependent problems. Recently, machine learning-assisted numerical methods have been proposed to enhance accuracy or efficiency. In this paper, we propose a data-driven finite difference numerical method to solve the hyperbolic equations with smooth solutions on coarse grids, which can achieve higher accuracy than classical numerical schemes based on the same mesh size. In addition, the data-driven schemes have better spectrum properties than the classical schemes, although the spectrum properties are not explicitly optimized during the training process. Numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method, as well as its good performance on dispersion and dissipation.

Figures (13)

• Figure 1: The discretization of the spatial domain.
• Figure 2: Schematic of the neural network architecture.
• Figure 3: Learning framework and forward solver workflows.
• Figure 4: Approximate dispersion relation for the selected schemes. Real parts of $\Phi(\varphi)$ are shown in (a) and imaginary parts of $\Phi(\varphi)$ are shown in (b).
• Figure 5: Test 1-1: The $L_2$ errors of WLNN, CE6 and UP5 over time.