Conservative approximation-based feedforward neural network for WENO schemes
This work addresses the challenge of improving numerical accuracy and robustness in computational fluid dynamics simulations, representing an incremental advancement by integrating neural networks into existing WENO frameworks.
The authors tackled the problem of designing weighted essentially non-oscillatory (WENO) schemes for hyperbolic conservation laws by replacing the classical weighting procedure with a feedforward neural network based on conservative approximation, resulting in WENO3-CADNNs that outperform WENO3-Z and achieve accuracy comparable to WENO5-JS across various benchmarks and resolutions.
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.