Deep Operator Learning Approximation and Distributed Applications

Explore the theoretical foundations and practical applications of neural operators in this 38-minute conference talk from IPAM's Scientific Machine Learning Workshop. Delve into deep learning architectures specifically designed to approximate operators—mappings between infinite-dimensional function spaces—and their widespread applications in solving partial differential equation problems, including predicting solutions from initial or boundary conditions. Examine the analysis of error convergence and generalization properties that apply to a broad class of commonly used neural operators, addressing key theoretical questions that have remained unresolved despite their empirical success. Discover how these theoretical developments inform the design of distributed and federated learning algorithms that leverage neural operator approximation structures to tackle two critical practical challenges: handling heterogeneous and multiscale input functions, and extending frameworks to multi-operator learning settings for generalization to previously unseen tasks. Review numerical evidence supporting these applications and gain insights into the intersection of theoretical analysis and practical implementation in scientific machine learning.