Abstract

Starting from surrogate models and PINN -based approaches, I will discuss how neural methods may help move beyond some classical frameworks in shape optimization, such as shape derivatives, level-set methods, and other smooth parametrizations of the domain. In particular, neural representations suggest new ways to handle topological changes, complex geometries, and free boundaries, where more rigid classical approaches may become restrictive.The price to pay is that, at present, the mathematical theory is still incomplete: compared with classical methods, there is no equally systematic understanding of convergence, stability, or approximation properties in the relevant variational settings. For this reason, the talk will emphasize perspectives rather than definitive answers.This talk is not intended as a survey of fully established results, but rather as an overview of possible directions.