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

TL;DR

This paper introduces a hierarchical, adaptive surrogate modeling framework combining full order, reduced order, and machine learning models for parametrized PDEs, with rigorous error certification and on-the-fly adaptation.

Contribution

It proposes a novel certified hierarchical surrogate model integrating ROM and ML with adaptive on-line updates and rigorous error estimates for parametrized PDEs.

Findings

Efficient approximation demonstrated on Monte Carlo and optimization problems.

Adaptive model chain improves accuracy during parametric requests.

Rigorous a posteriori error estimates enable certification of the surrogate model.

Abstract

We present a new surrogate modeling technique for efficient approximation of input-output maps governed by parametrized PDEs. 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. In particular, we provide an example of an ML-based model that allows for rigorous analytical quality statements. We demonstrate the efficiency of the modeling chain on a Monte Carlo and a parameter-optimization example. Here, the ROM is instantiated by Reduced Basis Methods and the ML model is given by a neural network or a VKOGA kernel model.