Multi-fidelity learning of reduced order models for parabolic PDE constrained optimization

This article builds on the recently proposed RB-ML-ROM approach for parameterized parabolic PDEs and proposes a novel hierarchical trust region algorithm for solving parabolic PDE constrained optimization problems. Instead of using a traditional offline/online splitting approach for model order reduction, we adopt an active learning or enrichment strategy to construct a multi-fidelity hierarchy of reduced order models on-the-fly during the outer optimization loop. The multi-fidelity surrogate model consists of a full order model, a reduced order model and a machine learning model. The proposed hierarchical framework adaptively updates its hierarchy when querying parameters, utilizing a rigorous a posteriori error estimator in an error-aware trust region framework. Numerical experiments are given to demonstrate the efficiency of the proposed approach.