Deep Learning for High-Dimensional PDEs on Manifold with Neumann and Dirichlet Boundary Conditions

Abstract.

The deep backward stochastic differential equation (BSDE) method is a deep learning algorithm for solving high-dimensional partial differential equations (PDEs) in the whole space, which is trained using the sampled trajectories of microscopic stochastic processes; however, generally it is incapable of solving the problems on a manifold or with boundary constraints. To overcome this issue, we propose a strategy compliant with BSDE theory in this paper, termed the Constraint-Informed BSDE method. The key idea of the Constraint-Informed BSDE method is to first construct appropriate stochastic processes based on the physical implications of boundary conditions and manifold constraints, and further derive the BSDEs that the solutions to the corresponding PDEs satisfy. Specifically, we derive the BSDEs satisfied by the solutions of PDEs with Neumann and Dirichlet boundary conditions on multifold sphere and in high-dimensional domain. Finally, we demonstrate the performance of the Constraint-Informed BSDE method through extensive numerical experiments. Since there is no need to explicitly incorporate boundary conditions into the loss function, Constraint-Informed BSDE method exhibits sufficient accuracy, stability, and strong robustness.

Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://github.com/WANGH950/CIBSDE and in the supplementary materials ( M182682_Supplementary_Materials.zip [82.5MB]). [Formula: see text]