  One of the core problems in mean-field control and mean-field games is to
solve the corresponding McKean-Vlasov forward-backward stochastic differential
equations (MV-FBSDEs). Most existing methods are tailored to special cases in
which the mean-field interaction only depends on expectation or other moments
and thus inadequate to solve problems when the mean-field interaction has full
distribution dependence.
  In this paper, we propose a novel deep learning method for computing
MV-FBSDEs with a general form of mean-field interactions. Specifically, built
on fictitious play, we recast the problem into repeatedly solving standard
FBSDEs with explicit coefficient functions. These coefficient functions are
used to approximate the MV-FBSDEs' model coefficients with full distribution
dependence, and are updated by solving another supervising learning problem
using training data simulated from the last iteration's FBSDE solutions. We use
deep neural networks to solve standard BSDEs and approximate coefficient
functions in order to solve high-dimensional MV-FBSDEs. Under proper
assumptions on the learned functions, we prove that the convergence of the
proposed method is free of the curse of dimensionality (CoD) by using a class
of integral probability metrics previously developed in [Han, Hu and Long,
arXiv:2104.12036]. The proved theorem shows the advantage of the method in high
dimensions. We present the numerical performance in high-dimensional MV-FBSDE
problems, including a mean-field game example of the well-known Cucker-Smale
model whose cost depends on the full distribution of the forward process.
