Learning Families of Operators: Mathematical Foundations and Efficient Algorithms
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Salle de conférence LJAD
Many 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.