  We propose a very general framework for deriving rigorous bounds on the
approximation error for physics-informed neural networks (PINNs) and operator
learning architectures such as DeepONets and FNOs as well as for
physics-informed operator learning. These bounds guarantee that PINNs and
(physics-informed) DeepONets or FNOs will efficiently approximate the
underlying solution or solution operator of generic partial differential
equations (PDEs). Our framework utilizes existing neural network approximation
results to obtain bounds on more involved learning architectures for PDEs. We
illustrate the general framework by deriving the first rigorous bounds on the
approximation error of physics-informed operator learning and by showing that
PINNs (and physics-informed DeepONets and FNOs) mitigate the curse of
dimensionality in approximating nonlinear parabolic PDEs.
