Learning Families of Operators: Mathematical Foundations and Efficient Algorithms

Speakers: Adrien Weihs\n\nMany scientific computing problems involve families of related operators arising from variations in physical parameters, geometries, or governing equations. Multiple operator learning aims to approximate such families within a unified model. This talk develops mathematical and computational foundations for this setting from two complementary perspectives. The first is based on Multiple Neural Operators, a deep-learning framework for which we derive scaling laws for minimax approximation rates and generalization bounds. The second uses kernel methods within a general encoder-decoder framework, providing closed-form training, rigorous approximation guarantees for multi-input, multi-output operator learning, and substantially reduced computational costs. Numerical experiments on several families of parametric PDEs illustrate the accuracy and computational trade-offs of the different approaches.\n\nhttps://indico.math.cnrs.fr/event/17007/