AI 中文总结
该研究针对高维依赖黑塞矩阵的完全非线性抛物型偏微分方程，提出深度二阶随机残差方法（D2SRM），通过特定训练生成相关近似，建立理论并进行实验，结果表明随时间步长减小误差降低，还分离了误差界中的各项因素。
AI 中文摘要
我们引入了深度二阶随机残差方法（D2SRM）来处理高维、依赖黑塞矩阵的完全非线性抛物型偏微分方程。通过二阶布朗一步残差以及终端值和梯度惩罚，联合训练单个标量时空网络以生成解、梯度和黑塞矩阵的导数一致近似。对于具有单位扩散和足够弱黑塞耦合的全局利普希茨方程，我们在布朗占据空间中建立了适定性并发展了总体水平的收敛理论。在额外正则性条件下，后验估计通过时间步长及其总体目标来界定任何可允许候选的全喷流占据误差的平方。对于近似总体极小值，误差界分离了时间离散化、神经近似和总体次优性；当后两项为\(O(h)\)时，全喷流占据范数为\(O(h^{1/2})\)。在100维人工基准上的实验比较了终端处理，探测了已证明的小增益范围内外的黑塞耦合，并显示随着时间步长减小误差降低。代码可在指定网址获取。
英文摘要
We introduce the Deep Second-Order Stochastic Residual Method (D2SRM) for high-dimensional, Hessian-dependent fully nonlinear parabolic PDEs. A single scalar space--time network generates derivative-consistent approximations of the solution, gradient, and Hessian, which are trained jointly through second-order Brownian one-step residuals and terminal value and gradient penalties. For globally Lipschitz equations with identity diffusion and sufficiently weak Hessian coupling, we establish well-posedness in a Brownian occupation space and develop a population-level convergence theory. Under additional regularity, an a posteriori estimate bounds the squared full-jet occupation error of any admissible candidate by the time step and its population objective. For approximate population minimizers, the error bound separates time discretization, neural approximation, and population suboptimality; when the latter two terms are $O(h)$, the full-jet occupation norm is $O(h^{1/2})$. Experiments on a 100-dimensional manufactured benchmark compare terminal treatments, probe Hessian couplings inside and outside the proved small-gain range, and show decreasing errors as the time step decreases. The code is available at https://github.com/ZZHPKU/D2SRM.
