Rational-WENO: A lightweight, physically-consistent three-point weighted essentially non-oscillatory scheme

Abstract

Conventional WENO3 methods are known to be highly dissipative at lower resolutions, introducing significant errors in the pre-asymptotic regime. In this paper, we employ a rational neural network to accurately estimate the local smoothness of the solution, dynamically adapting the stencil weights based on local solution features. As rational neural networks can represent fast transitions between smooth and sharp regimes, this approach achieves a granular reconstruction with significantly reduced dissipation, improving the accuracy of the simulation. The network is trained offline on a carefully chosen dataset of analytical functions, bypassing the need for differentiable solvers. We also propose a robust model selection criterion based on estimates of the interpolation's convergence order on a set of test functions, which correlates better with the model performance in downstream tasks. We demonstrate the effectiveness of our approach on several one-, two-, and three-dimensional fluid flow problems: our scheme generalizes across grid resolutions while handling smooth and discontinuous solutions. In most cases, our rational network-based scheme achieves higher accuracy than conventional WENO3 with the same stencil size, and in a few of them, it achieves accuracy comparable to WENO5, which uses a larger stencil.

Figures (22)

• Figure 1: Location of cell centers, faces and stencils. Here $\bar{u}_i$ is the cell average of the solution at the $i$-th cell, $u^{-}_{i+1/2}$ is the left-value of the solution at the interface between the $i$-th and $i+1$-th cells. Similarly, $u^{+}_{i+1/2}$ is the right-value of the solution at the same interface. WENO schemes seek to interpolate the value of the solution at the interfaces using neighboring cell averages. WENO3 uses three adjacent cells to compute the interpolation and WENO5 uses five cells.
• Figure 2: Sketch of the architecture, the network takes cell averages $[\bar{u}_{i+1}, \bar{u}_i, \bar{u}_{i-1}]$ as input and applies rational featurization (\ref{['eq:rational_feats']}). The resulting features are passed to a rational MLP (\ref{['eq:rational_layers']}). During inference, an ENO layer (\ref{['eq:eno_layer']}) generates weights used in a convex combination (\ref{['eq:convex_combination']}) to approximate boundary values $u_{i\pm 1/2}$. These values are inputs to the numerical flux calculation (\ref{['eq:num_flux']}).
• Figure 3: Convergence of rational neural models on the evaluation functions where the order is estimated as the coefficient of linear regression. The models with convergence order closest to the theoretical optimal 3 are selected. These are better than the conventional WENO3 schemes.
• Figure 4: Advection of cosine wave (shaded region depicts the confidence interval of WENO3-NN-Rational-1 to WENO3-NN-Rational-6 in \ref{['tab:net_hyper_params']}).
• Figure 5: Advection of sigmoid wave (shaded region depicts the confidence interval of WENO3-NN-Rational-1 to WENO3-NN-Rational-6 in \ref{['tab:net_hyper_params']}).