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：深度学习算法已广泛应用于求解高维线性Kolmogorov偏微分方程（PDE），其损失函数定义为数学期望。本文提出采用随机化拟蒙特卡罗（RQMC）方法替代蒙特卡罗（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与近似误差，其中近似误差与采样方法无关。我们证明，对于任意小$\epsilon>0$，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方法的总体误差渐近小于MC方法。数值实验表明，基于RQMC方法的算法相较于基于MC方法的算法，始终能获得更小的相对$L^{2}$误差。