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Physics-informed neural networks (PINNs) have become a popular choice forsolving high-dimensional partial differential equations (PDEs) due to theirexcellent approximation power and generalization ability. Recently, ExtendedPINNs (XPINNs) based on domain decomposition methods have attractedconsiderable attention due to their effectiveness in modeling multiscale andmultiphysics problems and their parallelization. However, theoreticalunderstanding on their convergence and generalization properties remainsunexplored. In this study, we take an initial step towards understanding howand when XPINNs outperform PINNs. Specifically, for general multi-layer PINNsand XPINNs, we first provide a prior generalization bound via the complexity ofthe target functions in the PDE problem, and a posterior generalization boundvia the posterior matrix norms of the networks after optimization. Moreover,based on our bounds, we analyze the conditions under which XPINNs improvegeneralization. Concretely, our theory shows that the key building block ofXPINN, namely the domain decomposition, introduces a tradeoff forgeneralization. On the one hand, XPINNs decompose the complex PDE solution intoseveral simple parts, which decreases the complexity needed to learn each partand boosts generalization. On the other hand, decomposition leads to lesstraining data being available in each subdomain, and hence such model istypically prone to overfitting and may become less generalizable. Empirically,we choose five PDEs to show when XPINNs perform better than, similar to, orworse than PINNs, hence demonstrating and justifying our new theory.

When Do Extended Physics-Informed Neural Networks (XPINNs) Improve Generalization?