When Do Extended Physics-Informed Neural Networks (XPINNs) Improve Generalization?
- National University of Singapore (Singapore); Brown University
- Brown University, Providence, RI (United States)
- National University of Singapore (Singapore)
Physics-informed neural networks (PINNs) have become a popular choice for solving high-dimensional partial differential equations (PDEs) due to their excellent approximation power and generalization ability. Recently, extended PINNs (XPINNs) based on domain decomposition methods have attracted considerable attention due to their effectiveness in modeling multiscale and multiphysics problems and their parallelization. However, theoretical understanding of their convergence and generalization properties remains unexplored. In this study, we take an initial step towards understanding how and when XPINNs outperform PINNs. Specifically, for general multilayer PINNs and XPINNs, we first provide a prior generalization bound via the complexity of the target functions in the PDE problem and a posterior generalization bound via the posterior matrix norms of the networks after optimization. Moreover, based on our bounds, we analyze the conditions under which XPINNs improve generalization. Concretely, our theory shows that the key building block of XPINN, namely, the domain decomposition, introduces a tradeoff for generalization. On the one hand, XPINNs decompose the complex PDE solution into several simple parts, which decreases the complexity needed to learn each part and boosts generalization. On the other hand, decomposition leads to less training data being available in each subdomain, and hence such a model is typically prone to overfitting and may become less generalizable. Empirically, we choose five PDEs to show when XPINNs perform better than, similar to, or worse than PINNs, hence demonstrating and justifying our new theory.
- Research Organization:
- Brown University, Providence, RI (United States)
- Sponsoring Organization:
- USDOE; OSD/AFOSR MURI
- Grant/Contract Number:
- SC0019453
- OSTI ID:
- 2282018
- Journal Information:
- SIAM Journal on Scientific Computing, Journal Name: SIAM Journal on Scientific Computing Journal Issue: 5 Vol. 44; ISSN 1064-8275
- Publisher:
- Society for Industrial and Applied Mathematics (SIAM)Copyright Statement
- Country of Publication:
- United States
- Language:
- English
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