A Certified Adaptive Surrogate Hierarchy for Parametrized PDEs

  • Haasdonk, Bernard (University of Stuttgart)
  • Kleikamp, Hendrik (University of Münster)
  • Ohlberger, Mario (University of Münster)
  • Schindler, Felix (University of Münster)
  • Wenzel, Tizian (University of Stuttgart)

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We demonstrate recent advances in the context of a new certified, hierarchical and adaptive RB-ML-ROM surrogate model for time-dependent parametrized partial differential equations (PDEs) introduced in [1]. Based on a sufficiently accurate (but costly) full order model, we introduce an abstract hierarchy of certified reduced order models (ROMs), where each model in the hierarchy is used to generate training data for the next. If certification is available for each ROM, the hierarchy of models is automatically built and adapted on the fly to yield approximations of guaranteed prescribed accuracy, while being employed in some outer-loop context. As a particular choice, we consider (single-phase) reactive flow with long-time integration in porous media and employ standard certified Reduced Basis methods as a first layer ROM. As a second layer, we equip a machine-learning surrogate based on kernel methods or (deep) neural networks with error certification using the standard residual based estimate of the RB ROM. [1]: Haasdonk, B. and Kleikamp, H. and Ohlberger, M. and Schindler, F. and Wenzel, T. A new certified hierarchical and adaptive RB-ML-ROM surrogate model for parametrized PDEs. arXiv [math.NA] (2022) 2204.13454.