A transfer learning method to solve Fokker-Planck equation based on the equivalent linearization

Efficient methods for solving the Fokker-Planck (FP) equation are crucial for studying stochastic systems. This paper proposes a transfer learning method to solve the FP equation, enabling the training process to proceed without starting from the beginning. The equivalent linearization is first applied to unify a class of stochastic differential equations into a single simplified form. Subsequently, a pre-trained neural network framework, inspired by transfer learning, is designed based on the FP equation of the simple system. By leveraging the pre-trained neural network, the solving process is accelerated by starting from a more advanced state. Finally, numerical experiments are conducted to verify the proposed approach, including one- and two-dimensional stochastic systems as well as a system driven by both Gaussian and Lévy noise. Results show that the contours of the FP equations can be learned by the network very expeditiously, greatly reducing training time while maintaining accuracy. The proposed method not only improves computational efficiency but also demonstrates strong generalization capabilities.