A new Certified Hierarchical and Adaptive RB-ML-ROM Surrogate Model for Parametrized PDEs

Abstract

In this talk, we present a new surrogate modeling technique for efficient approximation of solutions and output quantities of parametrized partial differential equations [1]. The model is hierarchical as it is built on a full order model (FOM), reduced order model (ROM) and machine-learning (ML) model chain. The model is adaptive in the sense that the ROM and ML model are adapted on-the-fly during a sequence of parametric requests to the model. To allow for a certification of the model hierarchy, as well as to control the adaptation process, we employ rigorous a posteriori error estimates for the ROM and ML models, see [3]. The model is therefore able to fulfill fixed or adaptively chosen error tolerances for every requested parameter. In particular, we provide an example of an ML-based model that allows for rigorous analytical quality statements. Numerical experiments showcase the efficiency of our approach in different scenarios, for instance a parameter optimization problem and uncertainty quantification. Here, the ROM is instantiated by Reduced Basis Methods [2] and the ML model is given by a neural network, similar to [4], or a VKOGA kernel model, see [5]. However, a wide range of ML algorithms is applicable within the modeling chain while maintaining the certification of the results.