Approximation of functionals on Korobov spaces with Fourier Functional Networks

Abstract

Learning from functional data with deep neural networks has become increasingly useful, and numerous neural network architectures have been developed to tackle high-dimensional problems raised in practical domains. Despite the impressive practical achievements, theoretical foundations underpinning the ability of neural networks to learn from functional data largely remain unexplored. In this paper, we investigate the approximation capacity of a functional neural network, called Fourier Functional Network, consisting of Fourier neural operators and deep convolutional neural networks with a great reduction in parameters. We establish rates of approximating by Fourier Functional Networks nonlinear continuous functionals defined on Korobov spaces of periodic functions. Finally, our results demonstrate dimension-independent convergence rates, which overcomes the curse of dimension.