Convergence analysis of wide shallow neural operators within the framework of Neural Tangent Kernel

Abstract: Neural operators are aiming at approximating operators mapping between Banach spaces of functions, achieving much success in the field of scientific computing. Compared to certain deep learning-based solvers, such as Physics-Informed Neural Networks (PINNs), Deep Ritz Method (DRM), neural operators can solve a class of Partial Differential Equations (PDEs). Although much work has been done to analyze the approximation and generalization error of neural operators, there is still a lack of analysis on their training error. In this work, we conduct the convergence analysis of gradient descent for the wide shallow neural operators and physics-informed shallow neural operators within the framework of Neural Tangent Kernel (NTK). The core idea lies on the fact that over-parameterization and random initialization together ensure that each weight vector remains near its initialization throughout all iterations, yielding the linear convergence of gradient descent. In this work, we demonstrate that under the setting of over-parametrization, gradient descent can find the global minimum regardless of whether it is in continuous time or discrete time.