Deep learning-based numerical methods for partial differential equations

The objective of this Final Year Project is to study deep learning-based numerical methods, with a focus on the Deep BSDE Solver, that can be applied on stochastic control problems, backward stochastic differential equations (BSDE) and partial differential equations (PDE) in high-dimensional space. Throughout this research, the project aims at constructing the deep learning-based algorithm & evaluating its performance through thirteen numerical experiments in both low- and high-dimensional space. In addition, the ‘non-explosion’ condition is raised based on the Deep BSDE Solver and further study is conducted on the restrictions it brings in for deep-learning based PDE solvers. In this paper, Chapter 2 focuses on the Deep BSDE Solver, which is one of the recent most influential deep learning-based PDE solvers. First, the key mathematical concepts, theorems and optimization algorithms used in the paper are presented in Section 2.1. Second, the deep BSDE solver is derived in details using the theorems and the deep neural network architecture of the algorithm is drafted. Section 2.3 then lists down the implementation methodology and evaluation criteria of the PDE solvers. Next, numerical experiments are conducted based on 13 semilinear parabolic PDE test cases, and the simulation results are evaluated through the proposed evaluation criteria for approximation accuracy and calculation speed. Lastly, the algorithm is further tested regarding to 2 significant parameters - dimension of the underlying space and the time horizon, which leads to the study on the ‘non-explosion condition’. Furthermore, in Chapter 3, three more deep learning-based PDE solvers - the Deep Splitting Method, the Forward-Backward Stochastic Neural Networks and the Hybrid-Now are introduced and reformulated to be applied on semilinear parabolic PDEs in terminal value form. The simulation results are compared across all four algorithms. Based on the reviewed algorithm architectures and simulation results, the algorithm with the optimal stability, best approximation accuracy, and fastest computation speed is concluded, and directions for further research - extension to second-order fully nonlinear PDEs and to other algorithms are concluded in Section 4.2.Bachelor of Science in Mathematical Sciences and Economic