用于解决高维半线性PDE的新的高效近似方案:深 BSDE 求解器的控制变量法 (A new efficient approximation scheme for solving high-dimensional semilinear PDEs: control variate method for Deep BSDE solver)

This paper introduces a new approximation scheme for solving high-dimensional semilinear partial differential equations (PDEs) and backward stochastic differential equations (BSDEs). First, we decompose a target semilinear PDE (BSDE) into two parts, namely "dominant" linear and "small" nonlinear PDEs. Then, we apply a Deep BSDE solver with a new control variate method to solve those PDEs, where approximations based on an asymptotic expansion technique are effectively applied to the linear part and also used as control variates for the nonlinear part. Moreover, our theoretical result indicates that errors of the proposed method become much smaller than those of the original Deep BSDE solver. Finally, we show numerical experiments to demonstrate the validity of our method, which is consistent with the theoretical result in this paper.

翻译：本文引入了解决高维半线性半线性部分方程式( PDEs) 和后向随机差分方程式( BSDEs) 的新近似方案。首先, 我们将目标半线性PDE( BSDE) 分解成两部分, 即“ 主导” 线性和“ 小型” 非线性 PDEs 。然后, 我们应用深BSDE 解答器, 采用新的控制变异方法来解决这些 PDE, 即基于无症状扩展技术的近似值有效应用于线性部分, 并用作非线性部分的控制变量。此外, 我们的理论结果表明, 拟议方法的错误比原始深线性 BSDE 解算器的错误要小得多。最后, 我们展示了数字实验来证明我们的方法的有效性, 这与本文的理论结果是一致的。