The deep BSDE method is a deep learning algorithm for solving high-dimensional nonlinear parabolic PDEs by reformulating them as backward stochastic differential equations (BSDEs) via the nonlinear Feynman-Kac formula.
Given a semilinear parabolic PDE for u(t,x) with terminal condition u(T,x) = g(x), the solution satisfies a BSDE: Y_t = u(t,X_t) and Z_t = sigma^T(t,X_t) grad u(t,X_t), where X_t is a forward diffusion process. The method discretizes time into N steps and approximates Z_{t_n} = sigma^T grad u(t_n, X_{t_n}) at each time step by a feedforward neural network phi_n(X_{t_n}). All sub-networks are stacked into one deep network and trained end-to-end by minimizing the terminal condition mismatch: Loss = E|g(X_T) - Y_T|^2, where Y_T is propagated forward through the discretized BSDE dynamics.
The gradient of the PDE solution plays the role of a “policy function” in the spirit of deep reinforcement learning. This reformulation avoids mesh-based discretization of the spatial domain, overcoming the curse of dimensionality. The method achieves sub-1% relative errors in 100 dimensions with runtimes of minutes on a laptop.
Algorithm
- Discretize [0,T] into N time steps
- Initialize learnable parameters: u(0,X_0) and grad u(0,X_0)
- At each time step t_n, approximate sigma^T grad u(t_n, .) by a sub-network phi_n
- Propagate forward: X_{n+1} = X_n + mu Delta t + sigma Delta W, Y_{n+1} = Y_n - f(t_n, X_n, Y_n, Z_n) Delta t + Z_n^T Delta W
- Minimize Loss = E|g(X_N) - Y_N|^2 via Adam SGD with batch size 64
Key Properties
- Overcomes curse of dimensionality for nonlinear parabolic PDEs
- Architecture: sub-network at each time step with 2 hidden layers of (d+10) neurons, ReLU, batch norm
- Loss function: terminal condition mismatch only (BSDE dynamics are hard-coded in forward propagation)
- Used as subroutine in deep fictitious play for solving N-player stochastic differential games
- Can be enhanced with asymptotic expansion control variates