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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2502.06238 |
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Table of Contents:
- Semilinear parabolic partial differential equations (PDEs) are fundamental to modeling complex dynamical systems across scientific domains. The Deep Backward Stochastic Differential Equation (BSDE) method is a promising approach for high-dimensional PDEs; however, existing convergence results apply only to globally Lipschitz generators, excluding important cases such as Allen--Cahn and Hamilton--Jacobi--Bellman (HJB) equations. This paper presents both a theoretical and a computational advance for Deep BSDE methods. Theoretically, we establish the convergence theory for non--Lipschitz generators--covering Allen--Cahn equations with cubic nonlinearity and HJB equations with quadratic gradient growth--based on a bounded double--well lemma and a truncated-BSDE analysis within the Bouchard--Touzi--Zhang theory. Computationally, we instantiate the framework with XNet, a shallow architecture with $\mathcal O(L)$ parameters that preserves strong approximation while substantially reducing optimization and computational cost. Numerical experiments on 100--dimensional PDEs corroborate the predicted convergence behavior and demonstrate significant efficiency gains over standard feedforward implementations.