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

Learning High-Dimensional McKean-Vlasov Forward-Backward Stochastic Differential Equations with General Distribution Dependence