A note on the error analysis of data-driven closure models for large eddy simulations of turbulence

TL;DR

The paper addresses error propagation in trajectory predictions for large eddy simulations (LES) that employ data-driven closures. It develops an a-posteriori upper bound on the prediction error, decomposed into a modeling term $\beta_t$ and an optimization term $\gamma_t$, showing that the total error grows like $e^{R T}$ with rollout horizon $T$ and Jacobian bound $R$, and derives explicit bounds for each component. The main contributions are the bounds for $\beta_t$ and $\gamma_t$, their combination into a total error bound, and the insight that the time-step size $Δt$ and the Jacobian critically influence error amplification, pointing toward regularization strategies. This framework links LES discretization, data-driven closures, and autoregressive deployment, providing a foundation for designing more robust, regularized turbulence closures in practice.

Abstract

In this work, we provide a mathematical formulation for error propagation in flow trajectory prediction using data-driven turbulence closure modeling. Under the assumption that the predicted state of a large eddy simulation prediction must be close to that of a subsampled direct numerical simulation, we retrieve an upper bound for the prediction error when utilizing a data-driven closure model. We also demonstrate that this error is significantly affected by the time step size and the Jacobian which play a role in amplifying the initial one-step error made by using the closure. Our analysis also shows that the error propagates exponentially with rollout time and the upper bound of the system Jacobian which is itself influenced by the Jacobian of the closure formulation. These findings could enable the development of new regularization techniques for ML models based on the identified error-bound terms, improving their robustness and reducing error propagation.