Forward–Backward Stochastic Neural Networks: Deep Learning of High-Dimensional Partial Differential Equations

Classical numerical methods for solving partial differential equations suffer from the curse of dimensionality mainly due to their reliance on meticulously generated spatio-temporal grids. Inspired by modern deep learning-based techniques for solving forward and inverse problems associated with partial differential equations, we circumvent the tyranny of numerical discretization by devising an algorithm that is scalable to high dimensions. In particular, we approximate the unknown solution by a deep neural network which essentially enables us to benefit from the merits of automatic differentiation. To train the aforementioned neural network, we leverage the well-known connection between high-dimensional partial differential equations and forward–backward stochastic differential equations. In fact, independent realizations of a standard Brownian motion will act as training data. We test the effectiveness of our approach for a couple of benchmark problems spanning a number of scientific domains, including Black–Scholes–Barenblatt and Hamilton–Jacobi–Bellman equations, both in 100 dimensions.