Parametrization of subgrid scales in long-term simulations of the shallow-water equations using machine learning and convex limiting

TL;DR

This paper introduces a neural network-based parametrization method for sub-grid processes in the shallow-water equations, improving energy balance and solution accuracy in long-term turbulent simulations.

Contribution

It presents a local, four-point stencil neural network approach that enhances energy conservation and robustness in long-term fluid flow simulations.

Findings

Improved energy balance in long-term turbulent simulations.

Accurate reproduction of individual solutions.

Robustness in regimes not included in training data.

Abstract

We present a method for parametrizing sub-grid processes in the Shallow Water equations. We define coarse variables and local spatial averages and use a feed-forward neural network to learn sub-grid fluxes. Our method results in a local parametrization that uses a four-point computational stencil, which has several advantages over globally coupled parametrizations. We demonstrate numerically that our method improves energy balance in long-term turbulent simulations and also accurately reproduces individual solutions. The long-term simulations refer to numerical studies where a fluid flow is simulated over a duration long enough to reach a statistical steady state. The neural network parametrization can be easily combined with flux limiting to reduce oscillations near shocks. More importantly, our method provides reliable parametrizations, even in dynamical regimes that are not included in the training data.