XNet-Enhanced Deep BSDE Method and Numerical Analysis
For researchers solving high-dimensional semilinear parabolic PDEs, this work provides both theoretical guarantees for previously unsupported nonlinearities and a computationally cheaper architecture.
This paper extends the convergence theory of Deep BSDE methods to non-Lipschitz generators (e.g., Allen-Cahn and HJB equations) and introduces XNet, a shallow architecture that reduces computational cost while maintaining accuracy. Numerical experiments on 100-dimensional PDEs confirm the theory and show efficiency gains over standard feedforward networks.
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.