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<p>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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathcal {O}(L)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <mi>L</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> 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.</p>

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XNet-Enhanced Deep BSDE Method and Numerical Analysis

  • 期刊论文

  • 发表日期：

    2026年3月24日

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  • Xiaotao Zheng,
  • Xingye Yue,
  • Zhihong Xia,
  • Xin Li

摘要

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 \(\mathcal {O}(L)\) 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.