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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.<\/jats:p>","DOI":"10.1007\/s10444-026-10296-6","type":"journal-article","created":{"date-parts":[[2026,3,3]],"date-time":"2026-03-03T07:16:55Z","timestamp":1772522215000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Multi-fidelity learning of reduced order models for parabolic PDE constrained optimization"],"prefix":"10.1007","volume":"52","author":[{"ORCID":"https:\/\/orcid.org\/0009-0004-1909-3574","authenticated-orcid":false,"given":"Benedikt","family":"Klein","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6260-3574","authenticated-orcid":false,"given":"Mario","family":"Ohlberger","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2026,3,3]]},"reference":[{"issue":"3","key":"10296_CR1","doi-asserted-by":"publisher","first-page":"550","DOI":"10.1137\/16M1082469","volume":"60","author":"B Peherstorfer","year":"2018","unstructured":"Peherstorfer, B., Willcox, K., Gunzburger, M.: Survey of multifidelity methods in uncertainty propagation, inference, and optimization. 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