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We propose a very general framework for deriving rigorous bounds on theapproximation error for physics-informed neural networks (PINNs) and operatorlearning architectures such as DeepONets and FNOs as well as forphysics-informed operator learning. These bounds guarantee that PINNs and(physics-informed) DeepONets or FNOs will efficiently approximate theunderlying solution or solution operator of generic partial differentialequations (PDEs). Our framework utilizes existing neural network approximationresults to obtain bounds on more involved learning architectures for PDEs. Weillustrate the general framework by deriving the first rigorous bounds on theapproximation error of physics-informed operator learning and by showing thatPINNs (and physics-informed DeepONets and FNOs) mitigate the curse ofdimensionality in approximating nonlinear parabolic PDEs.

Generic bounds on the approximation error for physics-informed (and) operator learning