Generalization Capabilities of Machine Learning for Solving Multiple Partial Differential Equations

This project develops a rigorous theoretical foundation for multi-operator learning, providing a mathematical framework to understand how neural networks can efficiently learn across collections of complex physical systems. Artificial intelligence (AI) research, and in particular deep learning, has made recent advances in scientific computing, where empirical results outpace our theoretical understanding of why they work and how to design them reliably. This project addresses these questions by establishing theoretical rates and scalings that describe how model size, data, and problem structure govern accuracy and generalization, and by quantifying when a single model can perform effectively across multiple problems. By leveraging structures and mathematical properties, the project aims to identify the mechanisms that enable efficient learning in high-dimensional settings. These advances are important for AI since they provide principles for designing models that are both accurate and resource-efficient, improving predictability, interpretability, and robustness. The project will also contribute to workforce development by training researchers in applied mathematics, applied analysis, and AI techniques. The project establishes a mathematical framework for analyzing approximation and generalization properties of neural networks when applied to collections of nonlinear partial differential equations. The aim is to derive explicit rates for approximation and generalization errors in general prediction settings. The approach will also incorporate structural assumptions to explain gains in sample efficiency and the emergence of algebraic rates in high-dimensional settings. In addition, the project develops theory-driven principles for the design of neural network architectures tailored to multi-operator learning, incorporating data-dependent priors and shared representations. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.