Colloque - Mario Ohlberger : Reduced Order Surrogate Models for PDE-Constrained Optimization and Inverse Problems

Yvon MadayChaire Informatique et sciences numériquesCollège de FranceAnnée 2025-2026Colloque : Aspects mathématiques et appliqués des méthodes de réduction de complexité - Mario Ohlberger : Reduced Order Surrogate Models for PDE-Constrained Optimization and Inverse ProblemsMario OhlbergerProfessor of applied mathematics and managing director of the Institute of Analysis and Numerics, University of Münster, GermanyRésuméClassically, model order reduction for parameterized systems is based on a so-called offline phase, where reduced approximation spaces are constructed and the reduced parameterized system is built, followed by an online phase, where the reduced system can be cheaply evaluated in a multi-query context. In this contribution, instead, we follow an active learning or enrichment approach where a multi-fidelity hierarchy of reduced order models is constructed on-the-fly while exploring a parameterized system. To this end we focus on learning-based reduction methods in the context of PDE constrained optimization and inverse problems and evaluate their overall efficiency. We discuss learning strategies, such as adaptive enrichment within a trust region optimization framework as well as a combination of reduced order models with machine learning approaches. Concepts of rigorous certification and convergence will be presented, as well as numerical experiments that demonstrate the efficiency of the proposed approaches.