Encoded Forward Backward Stochastic Neural Network for High-Dimensional Backward Stochastic Differential Equations and Parabolic Partial Differential Equations

Backward stochastic differential equation (BSDE) provides probabilistic solutions for a class of parabolic partial differential equations (PDEs). DeepBSDE and FBSNN are two deep learning approaches for solving high-dimensional PDEs through approximating the solution of BSDEs. The conventional approach for learning functions defined on continuous domains is via fully-connected networks (FCNs) such that each input dimension is represented by a single neuron. In the current study, a new encoded FBSNN algorithm is proposed to enhance the efficiency and accuracy of approximating BSDEs using encoding and convolution. The input coordinates are encoded as tensors treated as images with multiple channels which can be processed efficiently by convolutional neural networks. The encoding mechanism enriches the input features such that the spatial and temporal features can be balanced. The encoded FBSNN algorithm provides a simple yet effective extension of the vanilla FBSNN algorithm such that BSDEs can be approximated more efficiently. The new algorithm is validated using the essentially high-dimensional Black-Scholes-Barenblatt and Hamilton-Jacobi-Bellman benchmark cases.