Solver-in-the-loop approach to closure of shell models of turbulence

TL;DR

This paper introduces a solver-in-the-loop, data-driven method for turbulence modeling that leverages differentiable physics, resulting in more accurate and stable closures capable of capturing high-order statistics in shell models.

Contribution

It presents a novel a posteriori training approach for turbulence closure models that integrates neural networks with differential equation solvers over time.

Findings

The learned closure reproduces high-order statistical moments.

Unrolling in time improves model performance.

Potential extension to Navier-Stokes equations.

Abstract

This work studies an a posteriori data-driven approach (known as solver-in-the-loop) for sub-grid modeling of a shell model for turbulence. This approach takes advantage of the differentiable physics paradigm of deep learning, allowing a neural network model to interact with the differential equation solver over time during the training process. The closure model is, then, naturally exposed to equations-informed input distributions by accounting for prior corrections over the temporal evolution in training. Such a characteristic makes this approach depart from the conventional a priori instantaneous training paradigm and often leads to a more accurate and stable closure model. Our study demonstrates that the closure learned via this a posteriori approach is able to reproduce high-order statistical moments of interest also in closures of high Reynolds number turbulence. Moreover, we investigate the performance of the learned model by experimenting with the effect of unrolling in time, which has remained for the most part unexplored in the literature. Finally, we discuss potential extensions of this approach to Navier-Stokes equations.