Deep Identification of Nonlinear Systems in Koopman Form

Iacob, Lucian Cristian; I., Beintema, Gerben; Schoukens, Maarten; Tóth, Roland

doi:10.1109/cdc45484.2021.9682946

Cited by 6 publications

(2 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Autoencoders are relevant in dynamic systems since their scope is ideally to map information vectors into themselves, obtaining a lower-dimensional data representation in a hidden layer that can be interpreted as the system state in a latent space. Recent work that exploit this idea using linear or nonlinear embeddings can be found in [207,149,200,206]. Other recent approaches exploit deep variational autoencoders to estimate a probability distribution over the latent space [167].…”

Section: Introductionmentioning

confidence: 99%

Deep networks for system identification: a Survey

Pillonetto¹,

Aravkin²,

Gedon³

et al. 2023

Preprint

View full text Add to dashboard Cite

Deep learning is a topic of considerable current interest. The availability of massive data collections and powerful software resources has led to an impressive amount of results in many application areas that reveal essential but hidden properties of the observations. System identification learns mathematical descriptions of dynamic systems from input-output data and can thus benefit from the advances of deep neural networks to enrich the possible range of models to choose from. For this reason, we provide a survey of deep learning from a system identification perspective. We cover a wide spectrum of topics to enable researchers to understand the methods, providing rigorous practical and theoretical insights into the benefits and challenges of using them. The main aim of the identified model is to predict new data from previous observations. This can be achieved with different deep learning based modelling techniques and we discuss architectures commonly adopted in the literature, like feedforward, convolutional, and recurrent networks. Their parameters have to be estimated from past data trying to optimize the prediction performance. For this purpose, we discuss a specific set of first-order optimization tools that is emerged as efficient. The survey then draws connections to the well-studied area of kernel-based methods. They control the data fit by regularization terms that penalize models not in line with prior assumptions. We illustrate how to cast them in deep architectures to obtain deep kernel-based methods. The success of deep learning also resulted in surprising empirical observations, like the counter-intuitive behaviour of models with many parameters. We discuss the role of overparameterized models, including their connection to kernels, as well as implicit regularization mechanisms which affect generalization, specifically the interesting phenomena of benign overfitting and double-descent. Finally, we highlight numerical, computational and software aspects in the area with the help of applied examples.1 This research has been partially supported by the Progetto di Ateneo CPDA147754/14-New statistical learning approach for multi-agents adaptive estimation and coverage control, the project Deep probabilistic regression -new models and learning algorithms (contract number: 2021-04301), funded by the Swedish Research Council, the Wallenberg AI, Autonomous Systems and Software Program (WASP) funded by Knut and Alice Wallenberg Foundation and by Kjell och Märta Beijer Foundation. This paper was not presented at any IFAC meeting. Corresponding author Gianluigi Pillonetto Ph.

show abstract

Section: Introductionmentioning

confidence: 99%

Deep networks for system identification: a Survey

Pillonetto¹,

Aravkin²,

Gedon³

et al. 2023

Preprint

View full text Add to dashboard Cite

show abstract

“…However, the main problem is that the choice of the observables is heuristic and there are no guarantees on the quality of the resulting model. To tackle this, one solution is to use data-driven techniques to learn the lifting from data, in order to circumvent the manual selection of observables (Lusch et al, 2018), (Iacob et al, 2021). Nevertheless, this is still an approximation and the questions on how to embed the nonlinear system into an exact linear finite dimensional lifted representation and when this is possible at all are still open.…”

Section: Introductionmentioning

confidence: 99%

Finite Dimensional Koopman Form of Polynomial Nonlinear Systems

Iacob¹,

Schoukens²,

Tóth³

2023

Preprint

View full text Add to dashboard Cite

The Koopman framework is a popular approach to transform a finite dimensional nonlinear system into an infinite dimensional, but linear model through a lifting process, using so-called observable functions. While there is an extensive theory on infinite dimensional representations in the operator sense, there are few constructive results on how to select the observables to realize them. When it comes to the possibility of finite Koopman representations, which are highly important form a practical point of view, there is no constructive theory. Hence, in practice, often a data-based method and ad-hoc choice of the observable functions is used. When truncating to a finite number of basis, there is also no clear indication of the introduced approximation error. In this paper, we propose a systematic method to compute the finite dimensional Koopman embedding of a specific class of polynomial nonlinear systems in continuous-time such that, the embedding, without approximation, can fully represent the dynamics of the nonlinear system.

show abstract