Active-SINDy: Intelligent sampling for model discovery in the ultra-low-data limit

Recovering governing equations from limited data is a fundamental problem in modeling nonlinear partial differential equations, especially in fluid dynamics where simulations are expensive and experimental measurements are difficult to obtain. Traditional approaches, such as Reynolds-averaged Navier-Stokes models, rely on empirically tuned closure models and often lack generality. Data-driven frameworks, such as the Sparse Identification of Nonlinear Dynamics (SINDy), offer an alternative by learning interpretable models directly from data. However, they remain sensitive to noise and undersampling in space and time. This work presents an active learning approach that enhances Ensemble-SINDy by selecting spatial sampling points based on model uncertainty across an ensemble of individual SINDy models. This strategy improves robustness and reduces the number of samples required for accurate model identification. We apply this method to several benchmark PDEs, including the inviscid Burgers and the Kuramoto-Sivashinsky equation. In all cases, our method consistently recovers the correct dynamical structure with significantly less data than random sampling. Thus, active data selection enables accurate model identification in settings where dense sampling is impractical or very expensive.