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

Analysis of the Generalization Error of deep learning based on Randomized Quasi-Monte Carlo for Solving Linear Kolmogorov PDEs