Multi-Task Neural Operators and Autoregressive Methods for PDEs
Hayden Schaeffer, University of California, Los Angeles
11 May 2026
Learning collections of solution operators for nonlinear PDEs remains challenging due to high dimensionality, multiscale phenomena, and limited observational data. Recent advances span both PDE foundation models (utilizing transformer-based or autoregressive spatiotemporal architecture) and multi-operator neural operator methods, which leverage shared structure across tasks. When trained on a wide range of datasets, these approaches enable zero-shot and few-shot generalization across parameter regimes. In this talk, we present both perspectives, highlighting their strengths in capturing complex dependencies, achieving high accuracy, and generalizing beyond the training distribution. We further present recent theoretical results establishing statistical generalization guarantees for multi-task and multiple operator learning, including explicit approximation-estimation tradeoffs and scaling laws that characterize dependence on dataset size, model capacity, and hierarchical sampling. Focusing on incompressible and compressible flows, we provide empirical and theoretical insights into model accuracy, robustness, and scalability on large datasets.
About the speaker
Hayden Schaeffer is the Director of Applied Mathematics and a Professor of Mathematics at the University of California, Los Angeles. His research is in mathematical foundations of AI and scientific machine learning, optimization, and dynamics. He has received an NSF CAREER award, an AFOSR Young Investigator Award, an NSF Mathematical Sciences Postdoctoral Research Fellowship, a UC President’s Postdoctoral Fellowship, and an NDSEG Fellowship.