Deep learning algorithms have been widely used to solve linear Kolmogorov partial differential equations~(PDEs) in high dimensions, where the loss function is defined as a mathematical expectation. We propose to use the randomized quasi-Monte Carlo (RQMC) method instead of the Monte Carlo (MC) method for computing the loss function. In theory, we decompose the error from empirical risk minimization~(ERM) into the generalization error and the approximation error. Notably, the approximation error is independent of the sampling methods. We prove that the convergence order of the mean generalization error for the RQMC method is $O(n^{-1+\epsilon})$ for arbitrarily small $\epsilon>0$, while for the MC method it is $O(n^{-1/2+\epsilon})$ for arbitrarily small $\epsilon>0$. Consequently, we find that the overall error for the RQMC method is asymptotically smaller than that for the MC method as $n$ increases. Our numerical experiments show that the algorithm based on the RQMC method consistently achieves smaller relative $L^{2}$ error than that based on the MC method.

翻译：深度学习算法已被广泛应用于求解高维线性Kolmogorov偏微分方程（PDEs），其中损失函数定义为数学期望。我们提出使用随机拟蒙特卡洛（RQMC）方法替代蒙特卡洛（MC）方法计算损失函数。理论上，我们将经验风险最小化（ERM）的误差分解为泛化误差和逼近误差。值得注意的是，逼近误差与采样方法无关。我们证明，对于任意小的$\epsilon>0$，RQMC方法的平均泛化误差收敛阶为$O(n^{-1+\epsilon})$，而MC方法为$O(n^{-1/2+\epsilon})$。因此，随着$n$增大，RQMC方法的整体误差渐近小于MC方法。数值实验表明，基于RQMC方法的算法在相对$L^{2}$误差上始终优于基于MC方法的算法。