@article{21199,
  abstract     = {{As in almost every other branch of science, the major advances in data
science and machine learning have also resulted in significant improvements
regarding the modeling and simulation of nonlinear dynamical systems. It is
nowadays possible to make accurate medium to long-term predictions of highly
complex systems such as the weather, the dynamics within a nuclear fusion
reactor, of disease models or the stock market in a very efficient manner. In
many cases, predictive methods are advertised to ultimately be useful for
control, as the control of high-dimensional nonlinear systems is an engineering
grand challenge with huge potential in areas such as clean and efficient energy
production, or the development of advanced medical devices. However, the
question of how to use a predictive model for control is often left unanswered
due to the associated challenges, namely a significantly higher system
complexity, the requirement of much larger data sets and an increased and often
problem-specific modeling effort. To solve these issues, we present a universal
framework (which we call QuaSiModO:
Quantization-Simulation-Modeling-Optimization) to transform arbitrary
predictive models into control systems and use them for feedback control. The
advantages of our approach are a linear increase in data requirements with
respect to the control dimension, performance guarantees that rely exclusively
on the accuracy of the predictive model, and only little prior knowledge
requirements in control theory to solve complex control problems. In particular
the latter point is of key importance to enable a large number of researchers
and practitioners to exploit the ever increasing capabilities of predictive
models for control in a straight-forward and systematic fashion.}},
  author       = {{Peitz, Sebastian and Bieker, Katharina}},
  journal      = {{Automatica}},
  publisher    = {{Elsevier}},
  title        = {{{On the Universal Transformation of Data-Driven Models to Control Systems}}},
  doi          = {{10.1016/j.automatica.2022.110840}},
  volume       = {{149}},
  year         = {{2023}},
}

@article{27426,
  abstract     = {{Regularization is used in many different areas of optimization when solutions
are sought which not only minimize a given function, but also possess a certain
degree of regularity. Popular applications are image denoising, sparse
regression and machine learning. Since the choice of the regularization
parameter is crucial but often difficult, path-following methods are used to
approximate the entire regularization path, i.e., the set of all possible
solutions for all regularization parameters. Due to their nature, the
development of these methods requires structural results about the
regularization path. The goal of this article is to derive these results for
the case of a smooth objective function which is penalized by a piecewise
differentiable regularization term. We do this by treating regularization as a
multiobjective optimization problem. Our results suggest that even in this
general case, the regularization path is piecewise smooth. Moreover, our theory
allows for a classification of the nonsmooth features that occur in between
smooth parts. This is demonstrated in two applications, namely support-vector
machines and exact penalty methods.}},
  author       = {{Gebken, Bennet and Bieker, Katharina and Peitz, Sebastian}},
  journal      = {{Journal of Global Optimization}},
  number       = {{3}},
  pages        = {{709--741}},
  title        = {{{On the structure of regularization paths for piecewise differentiable regularization terms}}},
  doi          = {{10.1007/s10898-022-01223-2}},
  volume       = {{85}},
  year         = {{2023}},
}

@article{20731,
  abstract     = {{We present a novel algorithm that allows us to gain detailed insight into the effects of sparsity in linear and nonlinear optimization, which is of great importance in many scientific areas such as image and signal processing, medical imaging, compressed sensing, and machine learning (e.g., for the training of neural networks). Sparsity is an important feature to ensure robustness against noisy data, but also to find models that are interpretable and easy to analyze due to the small number of relevant terms. It is common practice to enforce sparsity by adding the ℓ1-norm as a weighted penalty term. In order to gain a better understanding and to allow for an informed model selection, we directly solve the corresponding multiobjective optimization problem (MOP) that arises when we minimize the main objective and the ℓ1-norm simultaneously. As this MOP is in general non-convex for nonlinear objectives, the weighting method will fail to provide all optimal compromises. To avoid this issue, we present a continuation method which is specifically tailored to MOPs with two objective functions one of which is the ℓ1-norm. Our method can be seen as a generalization of well-known homotopy methods for linear regression problems to the nonlinear case. Several numerical examples - including neural network training - demonstrate our theoretical findings and the additional insight that can be gained by this multiobjective approach.}},
  author       = {{Bieker, Katharina and Gebken, Bennet and Peitz, Sebastian}},
  journal      = {{IEEE Transactions on Pattern Analysis and Machine Intelligence}},
  number       = {{11}},
  pages        = {{7797--7808}},
  publisher    = {{IEEE}},
  title        = {{{On the Treatment of Optimization Problems with L1 Penalty Terms via Multiobjective Continuation}}},
  doi          = {{10.1109/TPAMI.2021.3114962}},
  volume       = {{44}},
  year         = {{2022}},
}

@article{16290,
  abstract     = {{The control of complex systems is of critical importance in many branches of science, engineering, and industry, many of which are governed by nonlinear partial differential equations. Controlling an unsteady fluid flow is particularly important, as flow control is a key enabler for technologies in energy (e.g., wind, tidal, and combustion), transportation (e.g., planes, trains, and automobiles), security (e.g., tracking airborne contamination), and health (e.g., artificial hearts and artificial respiration). However, the high-dimensional, nonlinear, and multi-scale dynamics make real-time feedback control infeasible. Fortunately, these high- dimensional systems exhibit dominant, low-dimensional patterns of activity that can be exploited for effective control in the sense that knowledge of the entire state of a system is not required. Advances in machine learning have the potential to revolutionize flow control given its ability to extract principled, low-rank feature spaces characterizing such complex systems.We present a novel deep learning modelpredictive control framework that exploits low-rank features of the flow in order to achieve considerable improvements to control performance. Instead of predicting the entire fluid state, we use a recurrent neural network (RNN) to accurately predict the control relevant quantities of the system, which are then embedded into an MPC framework to construct a feedback loop. In order to lower the data requirements and to improve the prediction accuracy and thus the control performance, incoming sensor data are used to update the RNN online. The results are validated using varying fluid flow examples of increasing complexity.}},
  author       = {{Bieker, Katharina and Peitz, Sebastian and Brunton, Steven L. and Kutz, J. Nathan and Dellnitz, Michael}},
  issn         = {{0935-4964}},
  journal      = {{Theoretical and Computational Fluid Dynamics}},
  pages        = {{577–591}},
  title        = {{{Deep model predictive flow control with limited sensor data and online learning}}},
  doi          = {{10.1007/s00162-020-00520-4}},
  volume       = {{34}},
  year         = {{2020}},
}