Deep-Control BSDE: Layerwise Brownian-Weighted Regression for High-Dimensional Semilinear PDEs

Abstract

High-dimensional semilinear parabolic partial differential equations arise in stochastic control, financial engineering, and uncertainty quantification, but classical spatial discretizations suffer from the curse of dimensionality. Motivated by Gaussian perturbation and conditional regression in denoising score matching, we propose Deep-Control BSDE (\DCBSDE{}), a layerwise control-regression method for Markovian backward stochastic differential equations. From an implicit Euler scheme, we identify the discrete ideal control $z_n^\pi=\mathcal M_nu_{n+1}^\pi$ as the conditional projection coefficient of the successor value response onto the one-step Brownian increment. At each time level, the method freezes the successor value function, approximates the resulting Brownian conditional-moment target using finitely many branches, regresses the control, and then fits the value through the implicit BSDE relation. A frozen linear-response baseline and antithetic pairing reduce finite-branch fluctuations while preserving the conditional target, and a within-layer correction coordinates the value and control approximations. We establish backward stability through a contraction property of the Brownian projection and derive conditional consistency when discretization, local learning, numerical, and finite-branch errors vanish jointly. Experiments across six benchmarks demonstrate that \DCBSDE{} achieves a favorable overall balance among value accuracy, control accuracy, and dynamic consistency while exhibiting generally stable performance across random seeds. These results support the use of \DCBSDE{} in applications requiring reliable joint approximation of the value and control processes.