Application of Machine Learning and Convex Limiting to Subgrid Flux Modeling in the Shallow-Water Equations

Aug 3, 2026

A machine learning–flux limiting framework is proposed for property-preserving subgrid scale modeling within flux-limited finite volume methods for the one-dimensional shallow-water equations. Conservative target-scheme numerical fluxes are fitted to coarse-mesh averages using a neural network that parametrizes subgrid contributions derived from monotone fine-grid discretizations. A flux limiter constrains equivalent fluctuation intermediate states to remain in a convex admissible set, ensuring positivity and local maximum principle validity. Numerical studies demonstrate meaningful closures even when scenarios differ from training data.

LanguageEnglish

CategoryResearch & Report

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Pages14

Reading Time35 min

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Abstract

Introduction

Shallow-Water Equations

Discretization, Coarse-Scale Variables, and Subgrid Fluxes

arXiv :2407 . 17214v1 [physics .comp-ph] 24 Jul 2024 Application of Machine Learning and Convex Limiting to Subgrid Flux Modeling in the Shallow-Water Equations Ilya Timofeyev∗ Alexey Schwarzmann † Dmitri Kuzmin ‡ Abstract We propose a combination of machine learning and flux limiting for property-preserving subgrid scale modeling in the context of flux-limited finite volume methods for the one-dimensional shallow-water equations. The numerical fluxes of a conservative target scheme are fitted to the coarse-mesh averages of a monotone fine-grid discretization using a neural network to parametrize the subgrid scale components. To ensure positivity preservation and the validity of local maximum principles, we use a flux limiter that constrains the intermediate states of an equivalent fluctuation form to stay in a convex admissible set. The results of our numerical studies confirm that the proposed combination of machine learning with monolithic convex limiting produces meaningful closures even in scenarios for which the network was not trained. Key words: shallow water equations; large eddy simulation; subgrid scale modeling; parametrization; neural networks; positivity preservation; flux limiting 1 Introduction Many fluid flow models are based on nonlinear conservation laws that incorporate a vast range of spatial and temporal scales. Therefore, direct numerical simulations are often prohibitively expensive and, despite rapid progress in computer architecture, existing codes cannot be run at the discretization level required to resolve all relevant physical processes. Therefore, the derivation/calibration of reduced models has been an active area of research for many decades. The simplest Large-Eddy Simulation (LES) approaches use the Smagorinsky eddy viscosity model [40] for subgrid stresses that appear in the spatially filtered equations. More advanced alternatives include stochastic (e.g. [25, 32, 7, 21, 10, 41, 6, 35]) and deterministic (e.g.[1, 12, 27, 33]) closures, alpha models [20] etc. Recently, machine learning has also been used to represent the effects of unresolved degrees of freedom (e.g. [8, 3, 43, 2], review [28]) . The main focus of our work is on the Shallow-Water Equations (SWE) [36] . This nonlinear hyperbolic system is widely used to simulate flow in rivers and channels, as well as coastal and ocean dynamics. In the context of ocean dynamics, parametrization of mesoscale eddies requires developing appropriate subgrid scale models. Recent years have witnessed the advent of closures based on Machine Learning (ML) . The validity and effectiveness of such modern closures was demonstrated by many numerical examples. However, the use of Neural Network (NN) approximations makes it more difficult to prove the stability, consistency, and convergence of ML numerical methods for nonlinear PDEs. Moreover, for the SWE model, an NN-based closure may violate entropy inequalities or produce negative water heights. As a fail-safe remedy, we consider an NN model for representing subgrid fluxes (instead of estimating the whole right-hand side of the reduced model) and propose the application of a physics-aware flux limiter to ML-generated subgrid fluxes that may violate important inequality constraints. We use a relatively simple feed-forward network to extract subgrid fluxes from data generated by the fully resolved model. The monolithic convex limiting (MCL) strategy [22] ∗ Dept. of Mathematics, University of Houston, Houston, TX 77204, [itimofey@cougarnet.uh.edu](itimofey@cougarnet.uh.edu)[ ](itimofey@cougarnet.uh.edu)†Dept. of Mathematics, TU Dortmund, [alexey.schwarzmann@math.tu-dortmund.de](alexey.schwarzmann@math.tu-dortmund.de) ‡Dept. of Mathematics, TU Dortmund, [kuzmin@math.uni-dortmund.de](kuzmin@math.uni-dortmund.de) that we use in this work represents the coarse-mesh cell averages as convex combinations of intermediate states that belong to a convex admissible set. The proofs of desired pr

FAQs

What problem does the proposed method address in shallow-water simulations?

It targets property-preserving subgrid scale modeling for one-dimensional shallow-water equations using flux-limited finite volume methods, where unresolved effects must be represented reliably.

How does the method combine machine learning with numerical flux modeling?

It fits conservative target numerical fluxes to coarse-mesh averages of a monotone fine-grid discretization, using a neural network to parametrize the subgrid flux components.

Why is convex limiting used, and what properties does it enforce?

Convex limiting constrains intermediate states of an equivalent fluctuation form to stay within a convex admissible set, enabling positivity preservation and validity of local maximum principles.

Aug 3, 2026

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