This paper introduces a technique to accelerate the deep BSDE solver, a neural network-based method for high-dimensional backward stochastic differential equations (BSDEs), by incorporating asymptotic expansion (AE) as prior knowledge. The deep BSDE solver, proposed by E, Han, and Jentzen (2017), reformulates the BSDE as a control problem and uses deep learning to approximate the solution. However, it suffers from slow convergence and high loss functions, especially for non-smooth terminal conditions and drivers common in financial applications like funding valuation adjustments (FVA). The authors propose using the leading-order term of an asymptotic expansion of the BSDE around a linear driver to obtain an analytic approximation for the control variable Z (the delta hedging strategy). The neural network then learns only the residual Z^Res = Z - Z^AE, which is smaller and easier to approximate. This prior knowledge drastically reduces the loss function and accelerates convergence, as demonstrated on Bergman's model with different lending and borrowing rates, including high-dimensional examples up to 30 dimensions. The method is also extended to reflected BSDEs (RBSDEs) for American option pricing, where the reflecting process L_t is learned directly. Numerical experiments show that the AE-enhanced solver achieves faster convergence, lower loss, and more stable estimates compared to the baseline deep BSDE solver, even when hyperparameters like learning rate are tuned. The approach is particularly effective for problems with non-smooth payoffs and high dimensionality, making it a promising tool for practical quantitative finance applications.