The finite expression method for turbulent dynamics with high-order moment recovery
Authors: Xingjian Xu, Di Qi, Chunmei Wang
Organizations: Department of Mathematics, University of Florida, Gainesvile, FL 32611 · Department of Mathematics, Purdue University, West Lafayette, IN 47907
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.
Discovering governing equations from observational data remains a fundamental challenge in scientific modeling, particularly when the underlying mathematical structure is unknown. Traditional sparse identification methods like SINDy excel at discovering parsimonious models but require researchers to specify candidate basis functions a priori, a limitation that often leads to model failure when critical terms are omitted or when systems exhibit unconventional dynamics. Purely symbolic regression approaches offer unlimited flexibility but struggle with noise sensitivity and frequently produce overly complex, unstable equations. We present AutoSINDy, a hybrid Discovery-then-Solve framework that combines the exploratory power of symbolic regression with the robust sparsity-promoting capabilities of SINDy. Our method operates in three stages: (1) PySR-based symbolic regression discovers candidate functional forms from bootstrapped data chunks; (2) a curation pipeline decomposes, expands, and filters these expressions using collinearity analysis to construct a minimal yet comprehensive library; and (3) SINDy identifies sparse governing equations from this custom-tailored library. Extensive experiments across canonical nonlinear systems demonstrate that AutoSINDy consistently recovers ground-truth equations even under high observational noise, achieving a ground-truth recovery rate of 92.8% across all trials. Compared with standard SINDy using enriched libraries and standalone symbolic regression, AutoSINDy achieves higher predictive accuracy, superior generalization to unseen trajectories, and substantially lower symbolic complexity.
A fundamental challenge in symbolic regression (SR) is efficiently recovering complex mathematical expressions from observational data. Although this problem is NP-hard, many expressions of practical interest decompose naturally into combinations of nonlinear feature modules, concentrating structural complexity into a small number of reusable components. Here, we introduce FePySR, a two-stage framework that reduces the SR search space by extracting valid features prior to equation search. FePySR first employs a heterogeneous neural network to constrain observational data to a set of candidate expressions, then performs structural optimization within this refined expression space using PySR. Across five standard benchmarks, FePySR outperforms state-of-the-art methods by achieving higher equation recovery rates. On a set of 75 highly complex synthesized equations, FePySR recovers 36 equations, while producing substantially smaller mean squared errors on the remaining unrecovered cases, with reduced computation time compared to PySR. FePySR's first stage also maintains consistent performance under varying numbers of selected top features and increasing levels of noise in the observational data. Applied to ordinary differential equations governing biological systems, FePySR successfully identifies governing equations in 24 out of 100 tests where PySR recovers none. Taken together, FePySR is a generalizable framework that can enhance the SR solvers, enabling the efficient and reliable recovery of symbolic expressions across scientific domains.
We introduce First-Order Trajectory Matching (FTM), a surrogate-modeling method that learns the first-order local transport of probability mass from trajectories of stochastic systems. By matching the symmetric first-order motion of trajectories, FTM learns the probability current velocity, whose flow preserves time marginals to match ensemble averages, while also capturing current-like trajectory quantities such as fluxes, circulations, and barrier-crossing currents. FTM learns the current velocity directly from trajectories, avoiding drift, diffusion, and score estimation. Our stability analysis separates discretization error from sampling variance and shows that the one-step simulation-free FTM loss is stable when temporal resolution and sample size are properly balanced. Across stochastic dynamical systems and PDE examples, we empirically demonstrate that FTM provides trajectory-aware ensemble predictions at low, deterministic-rollout cost.