Generalization error analysis of deep backward dynamic programming for solving nonlinear PDEs

We explore the application of the quasi-Monte Carlo (QMC) method in deep backward dynamic programming (DBDP) [1] for numerically solving high-dimensional nonlinear partial differential equations (PDEs). Our study focuses on examining the generalization error as a component of the total error in the DBDP framework, discovering that the rate of convergence for the generalization error is influenced by the choice of sampling methods. Specifically, for a given batch size m, the generalization error under QMC methods exhibits a convergence rate of $O\left(m^{-1+\epsilon }\right)$, where $\epsilon >0$ can be made arbitrarily small. This rate is notably more favorable than that of the traditional Monte Carlo (MC) methods, which is $O\left(m^{-1/2+\epsilon }\right)$. Our theoretical analysis shows that the generalization error under QMC methods achieves a higher order of convergence than their MC counterparts. Numerical experiments demonstrate that QMC indeed surpasses MC in delivering solutions that are both more precise and stable.