Score-based physics-informed learning framework for stochastic dynamics

The Fokker-Planck equation is crucial for characterizing the dynamics of stochastic systems and predicting their evolutionary behavior. In practice, the forward problem of the Fokker-Planck equation becomes ill-posed when the system dynamics and initial conditions are incompletely known. Meanwhile, the inverse problem for inferring dynamical coefficients is fundamentally constrained by the inherent unobservability of probability densities, which restricts available data to discrete-time particle observations. To address these challenges, we propose a novel Score-based Physics-informed Learning Framework that leverages score matching to connect particle observations with the forward and inverse problems of the Fokker-Planck equation, without requiring density reference solution or complete system dynamics. Experimental results demonstrate that our method achieves superior accuracy, computational efficiency, scalability to high dimensions, and robustness to data sparsity and noise.