4 research outputs found
Be greedy and learn: efficient and certified algorithms for parametrized optimal control problems
We consider parametrized linear-quadratic optimal control problems and
provide their online-efficient solutions by combining greedy reduced basis
methods and machine learning algorithms. To this end, we first extend the
greedy control algorithm, which builds a reduced basis for the manifold of
optimal final time adjoint states, to the setting where the objective
functional consists of a penalty term measuring the deviation from a desired
state and a term describing the control energy. Afterwards, we apply machine
learning surrogates to accelerate the online evaluation of the reduced model.
The error estimates proven for the greedy procedure are further transferred to
the machine learning models and thus allow for efficient a posteriori error
certification. We discuss the computational costs of all considered methods in
detail and show by means of two numerical examples the tremendous potential of
the proposed methodology
Adaptive machine learning-based surrogate modeling to accelerate PDE-constrained optimization in enhanced oil recovery
In this contribution, we develop an efficient surrogate modeling framework for simulation-based optimization of enhanced oil recovery, where we particularly focus on polymer flooding. The computational approach is based on an adaptive training procedure of a neural network that directly approximates an input-output map of the underlying PDE-constrained optimization problem. The training process thereby focuses on the construction of an accurate surrogate model solely related to the optimization path of an outer iterative optimization loop. True evaluations of the objective function are used to finally obtain certified results. Numerical experiments are given to evaluate the accuracy and efficiency of the approach for a heterogeneous five-spot benchmark problem.publishedVersio
A new Certified Hierarchical and Adaptive RB-ML-ROM Surrogate Model for Parametrized PDEs
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.Comment: 27 pages, 5 figure
Ontologies for Models and Algorithms in Applied Mathematics and Related Disciplines
In applied mathematics and related disciplines, the
modeling-simulation-optimization workflow is a prominent scheme, with
mathematical models and numerical algorithms playing a crucial role. For these
types of mathematical research data, the Mathematical Research Data Initiative
has developed, merged and implemented ontologies and knowledge graphs. This
contributes to making mathematical research data FAIR by introducing semantic
technology and documenting the mathematical foundations accordingly. Using the
concrete example of microfracture analysis of porous media, it is shown how the
knowledge of the underlying mathematical model and the corresponding numerical
algorithms for its solution can be represented by the ontologies.Comment: Preprint of a Conference Paper to appear in the Proceeding of the
17th International Conference on Metadata and Semantics Researc