Conservative approximation-based feedforward neural network for WENO schemes

2025-07-08

AI Summary

To address the complexity of nonlinear weighting in WENO schemes for hyperbolic conservation laws—and the resulting trade-off between accuracy and robustness—this paper proposes WENO3-CADNN, a conservative-approximation-driven feedforward neural network that replaces the classical nonlinear weighting procedure. Methodologically, fluxes are reconstructed from three-point cell-average values; training labels are generated via strict conservation constraints, and a symmetric balanced loss function is introduced to explicitly enforce high-order accuracy and numerical symmetry. Crucially, the network performs end-to-end learning of the shared weighting mechanism underlying both WENO3-JS and WENO3-Z. Experiments demonstrate that WENO3-CADNN exhibits superior robustness and generalization across diverse shock-smooth composite problems and grid resolutions. Its accuracy surpasses that of WENO3-Z and approaches that of WENO5-JS, while retaining the compact stencil and computational efficiency of third-order schemes.

Technology Category

• Machine Learning: Adversarial Learning & Robustness
• Computer Vision: Generative Adversarial Networks (GANs) for Vision
• Search and Optimization: Learning to Search

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Responsible Web: Measurement, analysis, and circumvention of Web censorship

Abstract

In this work, we present the feedforward neural network based on the conservative approximation to the derivative from point values, for the weighted essentially non-oscillatory (WENO) schemes in solving hyperbolic conservation laws. The feedforward neural network, whose inputs are point values from the three-point stencil and outputs are two nonlinear weights, takes the place of the classical WENO weighting procedure. For the training phase, we employ the supervised learning and create a new labeled dataset for one-dimensional conservative approximation, where we construct a numerical flux function from the given point values such that the flux difference approximates the derivative to high-order accuracy. The symmetric-balancing term is introduced for the loss function so that it propels the neural network to match the conservative approximation to the derivative and satisfy the symmetric property that WENO3-JS and WENO3-Z have in common. The consequent WENO schemes, WENO3-CADNNs, demonstrate robust generalization across various benchmark scenarios and resolutions, where they outperform WENO3-Z and achieve accuracy comparable to WENO5-JS.

Problem

• Replace classical WENO weighting with neural networks
• Train neural network for high-order derivative approximation
• Improve accuracy and generalization in WENO schemes

Innovation

• Feedforward neural network replaces WENO weighting
• Supervised learning creates new labeled dataset
• Symmetric-balancing term enhances loss function