Neural Networks in Infinite Dimensions

Modern science often needs to learn rules that transform one field into another, for example, turning input conditions into a full solution that evolves over space and time. Neural networks can achieve this, but we still lack a solid, transparent theory for very large or even infinite settings. This gap leads to practical issues: too much data may be needed, training can be costly, and accuracy is hard to predict. This project aims to build a rigorous yet usable foundation for learning such rules, also known as operators. We first design a new model family capable of capturing both local details and wave-like effects by explicitly controlling phase and amplitude in an interpretable way. To deepen the theory, we develop two complementary frameworks that combine insights from signal analysis and physics. These frameworks let us describe how neural networks capture complex patterns across space and time while respecting the physical laws that govern real systems. Within this setting, we study when learning is feasible, how much data is needed, and how the accuracy improves. At the heart of this analysis lies the concept of sparsity: representing complex behaviors using as few fundamental building blocks as possible. By studying how network architectures and activation functions influence accuracy and efficiency, we quantify the true representation cost of learning operators. These insights culminate in a unified framework that connects error bounds, data needs, and computational effort, together with prototype algorithms. If successful, this work will yield faster and more reliable models for time-dependent and parameter-dependent systems governed by partial differential equations and other operator-learning problems. It will help engineers and scientists make better decisions with less data and energy, bridging mathematical rigor and practical algorithmic design.