The finite expression method for turbulent dynamics with high-order moment recovery

TL;DR

This paper introduces a two-stage data-driven framework combining symbolic regression and generative models to accurately identify and predict the statistical properties of turbulent dynamical systems.

Contribution

It presents the Finite Expression Method for discovering closed-form deterministic dynamics and integrates generative models to capture stochastic residuals, improving higher-order moment prediction.

Findings

Successfully recovers nonlinear interaction terms and external forcing.

Accurately predicts statistical moments up to order five.

Demonstrates effectiveness on stochastic triad models across regimes.

Abstract

Turbulent dynamical systems are characterized by nonlinear interactions and stochastic effects that generate coupled statistical quantities, such as non-zero higher-order moments, which are difficult to capture from data with accuracy. We propose a two-stage data-driven modeling framework that combines symbolic regression with generative models to jointly identify the governing dynamics and predict their key statistical quantities. In Stage I of the framework, the Finite Expression Method (FEX) is adopted to discover closed-form expressions of the deterministic dynamics, recovering nonlinear interaction terms and external forcing without predefined libraries. In Stage II, generative models are introduced to learn the residual stochastic components as a refined correction to the model error from the Stage I approximation, enabling accurate characterization of higher-order statistics. Theoretical analysis establishes the consistency of the symbolic estimator and quantifies the estimation error in terms of data size and numerical discretization. The model performance is verified through detailed numerical experiments on the stochastic triad models across multiple regimes, demonstrating that the framework successfully recovers interaction terms and forcing expressions, and accurately predicts statistical moments up to order five. These results highlight the potential of integrating interpretable symbolic discovery with data-driven stochastic modeling for complex turbulent systems.