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

They have received considerable attention in the literature and interesting connections to partial differential equations have been obtained (see e.g., [3] and the references therein). The key feature of backward stochastic differential equations is the random terminal condition that the solution is required to satisfy. These equations are referred to as forward-backward stochastic differential equations, if the randomness in the terminal condition is coming from the state of a forward stochastic differential equation. The solution to a forward-backward stochastic differential equation can be written as a deterministic function of time and the state process. Under suitable regularity assumptions, this function can be shown to be the solution of a parabolic partial differential equation [3]. A forward-backward stochastic differential equation is called uncoupled if the solution of the backward equation does not enter the dynamics of the forward equation and coupled if it does. The corresponding parabolic partial differential equation is semi-linear in case the forward-backward stochastic differential equation is uncoupled and quasi-linear if it is coupled.