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

TL;DR

This paper analyzes how errors propagate in data-driven turbulence models for large eddy simulations, highlighting the influence of time step size and Jacobian on error growth and suggesting avenues for improving model robustness.

Contribution

It provides a mathematical formulation for error propagation in data-driven turbulence closure models, revealing key factors affecting error amplification.

Findings

Error propagates exponentially with rollout time.

Time step size and Jacobian significantly influence error growth.

Upper bounds for prediction error are derived under certain assumptions.

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.