The finite expression method for turbulent dynamics with high-order moment recovery
For researchers studying turbulent dynamical systems, this work provides a data-driven method to jointly identify interpretable dynamics and capture non-Gaussian statistics, though it is an incremental extension of existing symbolic regression and generative modeling techniques.
The paper proposes a two-stage framework combining symbolic regression (FEX) with generative models to discover governing dynamics and predict higher-order statistical moments in turbulent systems, achieving accurate recovery of interaction terms and moments up to order five in stochastic triad models.
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.