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

TL;DR

The paper tackles the challenge of representing subgrid processes in the Shallow Water Equations for long-term simulations by learning nonlinear subgrid fluxes on a coarse grid using a feedforward neural network with a four-point stencil. It couples this NN parametrization with Monolithic Convex Limiting to preserve physical admissibility, including positivity of the water height, and demonstrates improved energy transfer across scales and accurate reproduction of DNS-like solutions. The approach generalizes beyond training regimes, remaining robust to larger forcing and to changes in geometry such as topography and Manning friction, while maintaining the energy spectra and reducing spurious oscillations near shocks. Overall, the work provides a local, parallelizable, physics-informed ML surrogate for subgrid dynamics in conservation-law-based geophysical models, with clear potential for extension to multi-layer SWE and primitive equations.

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.

Figures (6)

• Figure 1: Visual representation of the neural network architecture used to approximate the subgrid flux. The input layer consists of four resolved modes, each with two values (height and discharge). The hidden layers process this input, and the output layer provides the estimated subgrid flux components.
• Figure 2: Top: a typical profile of the coarse solution $H_I(t)$; Bottom: the corresponding smoothness indicator $\beta_I$ in \ref{['eq:beta']}.
• Figure 3: Energy spectra for the water height $h(x,t)$ in long stationary simulations of the full model and reduced models with different resolutions. Note that the reduced NN model is trained in the $N_c=128$ regime and is used for other resolutions without retraining. Blue line - DNS-1024, Red Line - Reduced NN model with resolution $N_c$, Green Line - LLF solver with resolution $N_c$. Simulations with $N_c = 64$, $128$, $256$, $512$ - top left, top right, bottom left, bottom right, respectively.
• Figure 4: Energy spectra for $h$ (left) and $q$ (right) in simulations with $N_c=128$ and forcing amplitudes $A=0.1$ (top row), $0.12$ (middle row), and $0.14$ (bottom row). Note that the NN parametrization is originally trained on $A=0.1$ and $N_c=128$; it is used without retraining in other regimes.
• Figure 5: Comparison between snapshots $h(x,t)$ in simulations of NN-256 (red) and DNS-1024 (blue) (left column) and NN-MCL-256 (red) and DNS-1024 (blue) (right column) at times $t=198$, $209$, $334$. The NN-256 model shows very good agreement with the DNS reference, successfully capturing the hydraulic jump and the nonlinear wave propagation. MCL strategy is essential in reducing oscillations near shocks in simulations of the NN reduced model.