Neural network-based analytical solver for Fokker–Planck equation

The Fokker–Planck equation has significant applications in dynamical systems. In recent years, some neural network methods have been used in combination with physical models to obtain its numerical solutions. However, it is also appealing if the analytical solution of the physical model can be obtained. This paper proposes a neural network-based method for the analytical solution of the FP equation. It relies on neural networks and uses their explicit model as the trial function for the FP equation. The trial function contains the weights and biases in the neural network. Therefore, the solving of the FP equation is converted into the calculation of the weights and biases. In the proposed method, the FP equations are first reduced to a set of easily solvable nonlinear algebraic equations using some trial functions, and then the corresponding weights and biases are determined using the method of pending coefficients. In this paper, linear and nonlinear numerical examples were used to verify the effectiveness of the proposed method. The results demonstrated that the proposed method can obtain the exact solution of the FP equations without data samples. Finally, the proposed method is compared in detail with physics-informed neural networks in terms of computational theory and computational effectiveness.