The BSDE solvers for high-dimensional PDEs and BSDEs

Abstract

Conventional numerical methods for high-dimensional parabolic partial differential equations (PDEs) suffer from the notorious "curse of dimensionality". Inspired by the FCNN-based deep BSDE solver in E et al. (2017) and Han et al. (2018), this thesis presents a CNN3H-based deep BSDE solver and a CNN2H-based BSDE solver by converting the fully connected neural networks (FCNNs) to the convolutional neural networks with 3 hidden layers (CNN3Hs) or with 2 hidden layers (CNN2Hs), and a linear BSDE solver by replacing the FCNNs with linear combinations. We also employ the connection between PDEs and backward stochastic differential equations (BSDEs), i.e. the Feynman-Kac formula. Owing to fewer parameters, the proposed BSDE solvers demonstrate higher efficiency than the FCNN-based deep BSDE solver without sacrificing accuracy when solving some 100-dimensional and 1000-dimensional PDEs.