Machine learning-based statistical closure models for turbulent dynamical systems.

Area of Science:

• * Physics
• * Applied Mathematics
• * Data Science

Background:

• * Statistical closure modeling is crucial for turbulent dynamical systems but often hindered by limited training data.
• * Traditional supervised learning methods struggle when data is scarce, particularly due to stationary statistics beyond decorrelation times.
• * The 40-dimensional Lorenz-96 model exemplifies systems where only short-time transient statistics provide informative training data.

Purpose of the Study:

• * To develop a machine learning (ML) non-Markovian closure modeling framework for accurate statistical predictions.
• * To address the challenge of insufficient training data in turbulent dynamical systems.
• * To create a unified, agnostic ML approach applicable across various truncation regimes.

Main Methods:

• * Employed a Long-Short-Term-Memory (LSTM) architecture within a unified closure framework.
• * Represented higher-order unresolved statistical feedbacks using the LSTM.
• * Incorporated an ansatz to ensure stability and accurate long-time predictions despite intrinsic system instabilities.

Main Results:

• * The ML closure model demonstrated strong performance across different truncation scenarios.
• * Accurately predicted long-time statistical responses to time-dependent external forces.
• * Successfully predicted responses to forces with larger amplitudes and unseen characteristics compared to the training dataset.

Conclusions:

• * The proposed ML non-Markovian closure framework effectively overcomes data scarcity issues.
• * The unified, agnostic approach offers robust predictions for turbulent dynamical systems.
• * This data-driven method enhances the predictability of complex systems under various forcing conditions.