  This paper introduces a new approximation scheme for solving high-dimensional
semilinear partial differential equations (PDEs) and backward stochastic
differential equations (BSDEs). First, we decompose a target semilinear PDE
(BSDE) into two parts, namely "dominant" linear and "small" nonlinear PDEs.
Then, we employ a Deep BSDE solver with a new control variate method to solve
those PDEs, where approximations based on an asymptotic expansion technique are
effectively applied to the linear part and also used as control variates for
the nonlinear part. Moreover, our theoretical result indicates that errors of
the proposed method become much smaller than those of the original Deep BSDE
solver. Finally, we show numerical experiments to demonstrate the validity of
our method, which is consistent with the theoretical result in this paper.
