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

2026-01-30

AI Summary

This work addresses the challenge of simultaneously preserving energy conservation, solution accuracy, and generalization in long-term simulations of the shallow water equations using conventional subgrid-scale parameterizations. The authors propose a local and scalable neural network–based parameterization framework that defines coarse-grained variables and local spatial averages on a four-point stencil, employs a feedforward neural network to learn subgrid fluxes, and incorporates a convex limiting technique to suppress non-physical oscillations near shocks. The approach avoids global coupling and demonstrates strong generalization even in dynamical regimes not covered by training data. It significantly improves energy balance in long-term turbulent simulations while accurately reproducing fine-scale solution structures.

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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 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.

Problem

• subgrid parametrization
• shallow-water equations
• long-term simulation
• turbulent dynamics
• machine learning

Innovation

• subgrid parametrization
• machine learning
• shallow water equations
• convex limiting
• neural network