提出新方法求解复杂随机控制问题，精度显著提升。
Fractional-Boundary-Regularized Deep Galerkin Method for Variational Inequalities in Mixed Optimal Stopping and Control
- 用对偶变换将非线性方程转为线性，再引入分数边界正则化增强网络逼近能力
- 在抛物边界加入Sobolev-Slobodeckij范数，使网络解及其导数更稳定精确
- 通过原对偶关系自检，无解析解时也能建立可靠验证基准
混合最优停止与随机控制问题对应带有非线性哈密顿-雅可比-贝尔曼（HJB）算子的变分不等式，其数值求解长期困难且缺乏可靠基准。本文首先采用对偶方法将其转化为线性算子，进而提出分数边界正则化深度伽辽金法（FBR-DGM），在经典$L^2$损失基础上引入抛物边界上的Sobolev-Slobodeckij范数，强化解的正则性，显著提升网络近似及其导数的准确性。改进后的精度使得网络解可通过逆对偶变换还原为原始解。通过检验最优值、最优财富与最优控制之间的原对偶关系，可实现网络解的自洽性与稳定性验证，为无解析解情形提供了创新性的评估标准。
原文摘要 · Abstract (English)
Mixed optimal stopping and stochastic control problems define variational inequalities with non-linear Hamilton-Jacobi-Bellman (HJB) operators, whose numerical solution is notoriously difficult and lack of reliable benchmarks. We first use the dual approach to transform it into a linear operator, and then introduce a Fractional-Boundary-Regularized Deep Galerkin Method (FBR-DGM) that augments the classical $L^2$ loss with Sobolev-Slobodeckij norms on the parabolic boundary, enforcing regularity and yielding consistent improvements in the network approximation and its derivatives. The improved accuracy allows the network to be converted back to the original solution using the dual transform. The self-consistency and stability of the network can be tested by checking the primal-dual relationship among optimal value, optimal wealth, and optimal control, offering innovative benchmarks in the absence of analytical solutions.
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