Generalization Theory for Multi-Task Operator Learning and PDE Foundation Models 

Recent advances in multi-task operator learning and PDE foundation models have demonstrated strong empirical success in learning families of solution operators across diverse physical regimes. In this talk, we focus on the theoretical foundations of these approaches, presenting recent statistical generalization guarantees for multi-task operator learning, including explicit approximation-estimation tradeoffs and scaling laws that characterize the dependence of prediction error on model capacity, dataset size, and sampling. These results provide a rigorous framework for understanding generalization to previously unseen parameter-function-location triples and for designing scalable operator-learning methods for scientific machine learning.