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

TL;DR

This work presents a data-driven subgrid flux closure for the 1D shallow-water equations by approximating subgrid fluxes with a neural network and enforcing physicality through Monolithic Convex Limiting. The approach combines high-resolution DNS-derived training data with a four-point stencil and a convex-invariant limiting strategy to preserve positivity and entropy-like properties, enabling stable, accurate coarse-grid simulations. Results show the MCL-enhanced NN closures can reproduce DNS dynamics over long times, remain robust outside the training regime, and allow substantially larger time steps, offering practical acceleration for large-scale geophysical flows. The method demonstrates how ML closures can be safely integrated into traditional finite-volume schemes, providing a pathway for real-time forecasting and model verification via an a posteriori training-quality signal from the limiter.

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.

Figures (13)

• Figure 1: Schematic representation of the neural network used to approximate $G_{i+1/2}=[G_{i+1/2}^H,G_{i+1/2}^Q]$.
• Figure 2: Discharge $q(x,t)=hv$ at time $t=200$ in simulations of the full model (black solid line) with $\Delta x=0.05$ and reduced models with $k=20$ ($\Delta x=1$) and NN with $N_{hid}=128$ trained with $n=200$ and $T=80$; Blue Dash-Dot Line - NN-reduced model, Red Dashed Line - MCL-NN-reduced model. Initial conditions with parameters in Table \ref{['tab:ic']} (1 - top-left, 2 - top-right, 3 - bottom-left, 4 - bottom-right).
• Figure 3: Water height $h(x,t)$ at time $t=200$ in simulations of the full model (black solid line) with $\Delta x=0.05$ and reduced models with $k=20$ ($\Delta x=1$) and NN with $N_{hid}=128$ trained with $n=200$ and $T=80$; Blue Dash-Dot Line - NN-reduced model, Red Dashed Line - MCL-NN-reduced model. Initial conditions with parameters in Table \ref{['tab:ic']} (same location as in Figure \ref{['fig1']}).
• Figure 4:
• Figure 5: Discharge $q(x,t)=hv$ at time $t=200$ in simulations of the full model (black solid line) with $\Delta x=0.05$ and reduced models with $k=20$ ($\Delta x=1$) and NN with $N_{hid}=128$ trained with $n=100$ and $T^{train}=40$; Blue Dash-Dot Line - NN-reduced model, Red Dashed Line - MCL-NN-reduced model. Initial conditions with parameters in Table \ref{['tab:ic']} (same location as in Figure \ref{['fig1']}).