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We present a new surrogate modeling technique for efficient approximation ofinput-output maps governed by parametrized PDEs. The model is hierarchical asit is built on a full order model (FOM), reduced order model (ROM) andmachine-learning (ML) model chain. The model is adaptive in the sense that theROM and ML model are adapted on-the-fly during a sequence of parametricrequests to the model. To allow for a certification of the model hierarchy, aswell as to control the adaptation process, we employ rigorous a posteriorierror estimates for the ROM and ML models. In particular, we provide an exampleof an ML-based model that allows for rigorous analytical quality statements. Wedemonstrate the efficiency of the modeling chain on a Monte Carlo and aparameter-optimization example. Here, the ROM is instantiated by Reduced BasisMethods and the ML model is given by a neural network or a VKOGA kernel model.

A new certified hierarchical and adaptive RB-ML-ROM surrogate model for parametrized PDEs