  We propose and study a scheme combining the finite element method and machine
learning techniques for the numerical approximations of coupled nonlinear
forward-backward stochastic partial differential equations (FBSPDEs) with
homogeneous Dirichlet boundary conditions. Precisely, we generalize the
pioneering work of Dunst and Prohl [SIAM J. Sci. Comp., 38(2017), 2725--2755]
by considering general nonlinear and nonlocal FBSPDEs with more inclusive
coupling; self-contained proofs are provided and different numerical techniques
for the resulting finite-dimensional equations are adopted. For such FBSPDEs,
we first prove the existence and uniqueness of the strong solution as well as
of the weak solution. Then the finite element method in the spatial domain
leads to approximations of FBSPDEs by finite-dimensional forward-backward
stochastic differential equations (FBSDEs) which are numerically computed by
using some deep learning-based schemes. The convergence analysis is addressed
for the spatial discretization of FBSPDEs, and the numerical examples,
including both decoupled and coupled cases, indicate that our methods are quite
efficient.
