Reduced-Order Surrogate Models for PDE-Constrained Optimization and Inverse Problems

Abstract

Traditionally, model order reduction for parameterized systems relies on a so-called offline phase, in which reduced approximation spaces are constructed and the reduced parameterized system is built, followed by an online phase, where the reduced system can be evaluated efficiently in a multi-query context.

In this work, however, we adopt an active learning or enrichment approach in which 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, along with numerical experiments demonstrating the efficiency of the proposed approaches.

Mario Ohlberger

Mario Ohlberger is a full professor of applied mathematics and managing director of the Institute of Analysis and Numerics at the University of Münster. He is a member of the Center for Data Science and Complexity as well as the Center for Multiscale Theory and Computation.

Since 2019, he has been one of the spokespersons for the Cluster of Excellence Mathematics Münster: Dynamics – Geometry – Structure.

His research focuses on numerical analysis for partial differential equations, with a particular emphasis on error control and adaptivity, model order reduction and surrogate modeling, numerical multiscale methods, software development, scientific computing, and scientific machine learning.