Generalization analysis of operator learning by deep neural networks

Deep learning methods based on neural networks have demonstrated remarkable practical success in various domains, such as computer vision and natural language processing. Concurrently, operator learning has emerged as a widely adopted framework for addressing the challenge of learning nonlinear continuous mappings between infinite-dimensional function spaces. However, theoretical investigations in this area are still in their early stages. There is a pressing need for a comprehensive learning theory that specifically focuses on deep neural networks and their ability to learn nonlinear operators. In this project, our primary objective is to conduct rigorous mathematical analyses and establish a learning theory for deep operator neural networks, with a particular emphasis on functional data analysis. We shall first propose a novel functional neural network architecture that combines a discretization map and a deep neural network to approximate nonlinear and Fréchet smooth functionals. By deriving explicit rates of approximation, we aim to enhance our understanding of the approximation capabilities of the proposed functional neural network and highlight its advantages over conventional methods. Next, we plan to develop the learning theory of functional neural networks by investigating the consistency and error bounds of the deep functional learning algorithms they generate, particularly in the presence of heavy-tailed noises. Our goal is to explore how these networks can handle challenging data scenarios and provide robust learning guarantees. Then, we shall introduce a novel operator neural network designed for learning operators between high-dimensional input spaces and infinite-dimensional output spaces. By leveraging deep convolutional neural networks and a reconstruction map, we aim to mitigate the curse of dimensionality inherent in such learning tasks. Finally, we shall establish error bounds and convergence rates for the learning process of operators between infinite-dimensional function spaces using deep operator neural networks. These networks are composed of three key components: a discretization map, a neural network, and a reconstruction map. These analyses will contribute to our understanding of the performance and limitations of these networks when applied to operator learning problems.