Theoretical foundations of physics-informed neural networks and deep neural operators

This chapter presents a brief review of the theoretical foundations of physics-informed neural networks (PINNs) and deep neural operators. PINN is one of the most popular deep learning approaches for solving both forward and inverse problems of partial differential equations (PDEs). It provides seamless ways of embedding laws of physics into deep neural networks (DNNs) by leveraging auto-differentiation. At the same time, operator learning emerged as a new learning paradigm for learning nonlinear operators, particularly ones relevant to PDEs. Deep Operator Network (DeepONet) is one of the first pioneering models whose architecture is inspired by the universal approximation theorem. DeepONets can generate reliable real-time responses when they are pretrained with a large amount of data pairs of inputs and outputs. Topics to be covered include mathematical formulations, approximation error estimates, approximation theory of DNNs, and training/optimization methods.