Weak collocation networks: A deep learning approach to reconstruct stochastic dynamics from aggregate data

Stochastic differential equations (SDEs) play a key role in modeling the dynamics of systems significantly influenced by random perturbations. In this work, we propose a novel Weak Collocation Networks(WCN) method to determine the hidden dynamics-namely, the drift and diffusion functions of an SDE-from aggregate data. This method employs efficient neural networks, particularly the Kolmogorov-Arnold network (KAN), to parameterize the unknown functions. Instead of using the conventional metric that compares the data distribution with the predicted distribution, we introduce an efficient physics-informed loss derived from the weak form of the Fokker-Planck equation. By leveraging Monte Carlo spatial integration and a linear multi-step method for temporal differentiation, our approach facilitates a rapid and accurate estimation of the Fokker-Planck equation residual directly from observational data. Moreover, through an adaptive selection method for sampling the test function, we can enhance the robustness of our method. Numerical experiments demonstrate the efficiency and accuracy of our method.