DMS Applied and Computational Mathematics Seminar

Speaker: Dr. Zhongqiang Zhang (Worcester Polytechnic Institute, Worcester Massachusetts) 

Title: Solving Fokker-Planck Equations in High Dimensions Using Tensor Neural Networks 

Abstract: We solve high-dimensional Fokker-Planck equations on the whole space by using tensor neural networks. The tensor neural networks consist of a sum of tensor products of one-dimensional feedforward networks or a linear combination of several selected radial basis functions. These networks allow us to exploit auto-differentiation in major Python packages efficiently. Furthermore, using radial basis functions can fully avoid auto-differentiation, which is very expensive in high dimensions. We then use the physics-informed neural networks and stochastic gradient descent methods to learn the tensor networks. One essential step is to determine a proper numerical support for the Fokker-Planck equation. We demonstrate numerically that the tensor neural networks in physics-informed machine learning are efficient for Fokker-Planck equations from two to ten dimensions.