Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2407.14566 |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866917729246642176 |
|---|---|
| author | Ouyang, Du Xiao, Jichang Wang, Xiaoqun |
| author_facet | Ouyang, Du Xiao, Jichang Wang, Xiaoqun |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_14566 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Generalization Error Analysis of Deep Backward Dynamic Programming for Solving Nonlinear PDEs Ouyang, Du Xiao, Jichang Wang, Xiaoqun Numerical Analysis 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. |
| title | Generalization Error Analysis of Deep Backward Dynamic Programming for Solving Nonlinear PDEs |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2407.14566 |