Neural Operator for Scientific Computing

Citation

Li, Zongyi (2025) Neural Operator for Scientific Computing. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/fz9s-fq86. https://resolver.caltech.edu/CaltechTHESIS:06032025-033640769

Abstract

Scientific computing, which aims to accurately simulate complex physical phenomena, often requires substantial computational resources. By viewing data as continuous functions, we leverage the smoothness structures of function spaces to enable efficient large-scale simulations. We introduce the neural operator, a universal machine learning framework designed to approximate solution operators in infinite-dimensional spaces, achieving scalable physical simulations. The thesis begins with the introduction and definition of neural operators. Chapters 2-4 discuss architecture designs of neural operators including graph neural operator, multipole neural operator, and Fourier neural operator. Chapters 5-7 discuss physics-based learning techniques such as dissipative loss, physics-informed loss, and scale consistency loss. Chapters 8-10 discuss geometric neural operators with various boundary shapes, including latent space embedding, learned deformation, and optimal transport. Chapters 11-12 discuss further applications of neural operator in weather forecast and carbon capture storage.

Thesis Files