Machine Learning-Based Statistical Closure Models for Turbulent Dynamical Systems

TL;DR

This paper introduces a machine learning-based non-Markovian closure model for turbulent systems, effectively predicting long-term statistics despite limited training data and various model truncations.

Contribution

It develops a unified ML closure framework using LSTM architectures that performs well across different truncation regimes and external forcings.

Findings

Accurately predicts long-term statistical responses.

Performs well with limited short-term training data.

Handles various external forcing scenarios.

Abstract

We propose a Machine Learning (ML) non-Markovian closure modeling framework for accurate predictions of statistical responses of turbulent dynamical systems subjected to external forcings. One of the difficulties in this statistical closure problem is the lack of training data, which is a configuration that is not desirable in supervised learning with neural network models. In this study with the 40-dimensional Lorenz-96 model, the shortage of data (in temporal) is due to the stationarity of the statistics beyond the decorrelation time, thus, the only informative content in the training data is on short-time transient statistics. We adopted a unified closure framework on various truncation regimes, including and excluding the detailed dynamical equations for the variances. The closure frameworks employ a Long-Short-Term-Memory architecture to represent the higher-order unresolved statistical feedbacks with careful consideration to account for the intrinsic instability yet producing stable long-time predictions. We found that this unified agnostic ML approach performs well under various truncation scenarios. Numerically, the ML closure model can accurately predict the long-time statistical responses subjected to various time-dependent external forces that are not (and maximum forcing amplitudes that are relatively larger than those) in the training dataset.