  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 on 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 multi-layer 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 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.

Machine Learning;Numerical Analysis;Dynamical Systems;Numerical Analysis;Machine Learning