A third-order finite difference weighted essentially non-oscillatory scheme with shallow neural network

TL;DR

The paper introduces WENO3-SNN, a shallow neural network that learns nonlinear WENO3 weights via a Delta layer to map three-point stencils to weights, enabling less dissipative, ENO-like performance near discontinuities while maintaining high accuracy in smooth regions. The method uses a two-stage training process and two loss functions (MSE and MSLE) to bias the network toward linear weights in smooth regions and WENO3-JS behavior near shocks, without a post-processing ENO layer during testing. Numerical experiments across 1D/2D scalar and system problems show that WENO3-SNN schemes outperform standard WENO3-JS and WENO3-Z in accuracy and shock-capturing capability, with WENO3-SNN2 often delivering the best resolution of discontinuities. The results demonstrate a promising integration of shallow neural networks into finite-difference WENO schemes for hyperbolic conservation laws, with potential extension to higher-order (e.g., fifth-order) schemes in the future.

Abstract

In this paper, we introduce the finite difference weighted essentially non-oscillatory (WENO) scheme based on the neural network for hyperbolic conservation laws. We employ the supervised learning and design two loss functions, one with the mean squared error and the other with the mean squared logarithmic error, where the WENO3-JS weights are computed as the labels. Each loss function consists of two components where the first component compares the difference between the weights from the neural network and WENO3-JS weights, while the second component matches the output weights of the neural network and the linear weights. The former of the loss function enforces the neural network to follow the WENO properties, implying that there is no need for the post-processing layer. Additionally the latter leads to better performance around discontinuities. As a neural network structure, we choose the shallow neural network (SNN) for computational efficiency with the Delta layer consisting of the normalized undivided differences. These constructed WENO3-SNN schemes show the outperformed results in one-dimensional examples and improved behavior in two-dimensional examples, compared with the simulations from WENO3-JS and WENO3-Z.

Figures (26)

• Figure 1: The construction of the numerical flux $\hat{f}^-_{i+1/2}$ depends on the stencil $\left( f^+_{i-1}, f^+_i, f^+_{i+1} \right)$.
• Figure 2: The construction of the numerical flux $\hat{f}^+_{i+1/2}$ depends on the stencil $\left( f^-_{i+2}, f^-_{i+1}, f^-_i \right)$.
• Figure 3: Schematic of the SNN architecture.
• Figure 4: Solution profiles with $C=0.01, \, 35, \, 10^7$ for the Riemann problem of \ref{['eq:advection_1d']} at $T=0.5$ approximated by WENO3-SNN1. The dashed black line is the exact solution.
• Figure 5: Solution profiles with $D=0.01, \, 2.5, \, 100$ for the Riemann problem of \ref{['eq:advection_1d']} at $T=0.5$ approximated by WENO3-SNN2. The dashed black line is the exact solution.

Theorems & Definitions (18)

• Remark 2.1
• Remark 3.1
• Remark 3.2
• Example 4.1
• Example 4.2
• Example 4.3
• Example 4.4
• Example 4.5
• Example 4.6
• Example 4.7