Deep BSDE: A Unified Framework for Solving High-Dimensional PDEs, FBSDEs, and Control Problems
Developing algorithms for solving high-dimensional partial differential equations (PDEs), forward-backward stochastic differential equations (FBSDEs), and control problems has been an exceedingly difficult task for a long time, due to the notorious difficulty known as the curse of dimensionality. In this talk we introduce the "deep BSDE method" as a unified framework to solve general high-dimensional PDEs and FBSDEs. Starting from the BSDE formulation, we approximate the unknown component by neural networks and design a proper loss function to optimize parameters. Numerical results of a variety of examples, including applications in game theory and eigenvalue problem, demonstrate that the proposed algorithm is quite effective in high-dimensions, in terms of both accuracy and speed. We furthermore provide a theoretical error analysis to illustrate the validity and property of the objective function.
Jiequn Han is an Instructor at the Department of Mathematics, Princeton University. His research draws inspiration from various disciplines of science and is devoted to solving high-dimensional problems arising from scientific computing. His current research interests mainly focus on solving high-dimensional partial differential equations and machine learning based-multiscale modeling.