Generalization Error Analysis of Deep Backward Dynamic Programming for Solving Nonlinear PDEs
<|reference_start|>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) (Hure et al. 2020) 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 $O(m^{-1+\varepsilon})$, where $\varepsilon$ can be made arbitrarily small. This rate is notably more favorable than that of the traditional Monte Carlo (MC) methods, which is $O(m^{-1/2+\varepsilon})$. 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.<|reference_end|>
@article{ouyang2024generalization,
    title={Generalization Error Analysis of Deep Backward Dynamic Programming for
  Solving Nonlinear PDEs},
    author={Du Ouyang, Jichang Xiao and Xiaoqun Wang},
    journal={arXiv preprint arXiv:2407.14566},
    year={2024},
    archivePrefix={arXiv},
    eprint={2407.14566},
    primaryClass={math.NA cs.NA}
}