# Data-driven modelling of non-linear systems by means of artificial neural network hybrids

Sebastian Herzog

Kumulative Dissertation
Datum der mündl. Prüfung:
2021-03-15

Erschienen:
2021-04-16

Betreuer:
Wörgötter, Florentin Prof. Dr.

Gutachter:
Wörgötter, Florentin Prof. Dr.

Gutachter:
Parlitz, Ulrich Apl. Prof. Dr.

Gutachter:
Kurths, Jürgen Prof. Dr. Dr.

Zum Verlinken/Zitieren: http://hdl.handle.net/21.11130/00-1735-0000-0008-57F5-2

## Dateien

Name:
PhDHerzog_eDiss.pdf

Größe:
69,6 MB

Format:
PDF

Diese Datei ist bis (einschließlich) 13.03.2022 gesperrt.

## Zusammenfassung

### Englisch

In the natural sciences, theory and experiment are in permanent interaction with each other. Experimental data provide impulses for new theories and theories suggest new experimental set-ups. Earlier, these two areas had been fairly balanced. However, the rapid increase in performance in semiconductor technology makes modern measurement methods possible. This leads currently to the accumulation of gigantic amounts of data that no human being can process by sight and thought alone. Thus, at the moment, the question of how such floods of data can ever be condensed into models , refined theories, and finally into knowledge is sometimes pushed into the background. The work presented in this thesis addresses this issue with a focus on data from non-linear systems. This study can be classified as belonging to the field of data-driven modelling with a methodological focus on hybrid artificial neural networks. The hybrid models presented here are combinations of artificial neural networks, stochastic graphical models and numerical solvers for ordinary differential equations with two goals: The first one is to predict the spatial–temporal dynamics, of the non-linear system, from which the data are coming, over a long time as accurately as possible. While the second one is to find more explicit representations for the data, which are easier to interpret. With respect to the prediction of the spatial–temporal dynamics of non-linear systems, the idea was to employ a well-known artificial neural network architecture that can encode the data in such a way that it can be well predicted by a stochastic graphical model. A specific advantage of this approach is that the artificial neural network has processing properties that can be used for tasks like state reconstruction. This artificial neural network+stochastic graphical model hybrid was evaluated on different non-linear systems and was able to achieve prediction horizons that exceeded the former state of the art, sometimes substantially, in all cases. The second hybrid, is a demonstration how an artificial neural network can be combined with with an numerical equation solver for ordinary differential equations. The goal was to characterise the underlying dynamics of a system as a vector field based on a predefined system of equations given by the user. This approach was applied (under the assumption of Hamilton equations), to the case of a multi agent system and was able to predict a vector field describing the motions of the agents. Therefore this hybrid approach is not trained to make a spatial-temporal prediction as accurately as possible, but to parameterize the derivative of the hidden states from the assumed equations (e.g. Hamilton equations) so that the predicted vector field represents the data with the smallest possible error. In particular, this approach can be used to validate whether assumptions about the physics (here we assumed that the data can be represented by the Hamilton equations) are applicable. Finally, a demonstration is given how to apply these two hybrids methods to real experimental data from a Rayleigh–B\'enard convection cell and the motion of Dictyostelium discoideum (a soil-dwelling amoeba) responding to electric fields of continuous current. For this purpose, the measured raw data must be transferred into a format that is suitable for training the hybrid system. This has been achieved with a newly-developed particle tracking method that is able to reconstruct even high particle densities and assemble them into two or three dimensional trajectories. These trajectories then serve as input for the here-introduced approaches. In both experimental cases, the two hybrid approaches were able to reproduce the date and make predictions that are easier to interpret than the reconstructed data, for example due to the fact that they are essentially noise free. In summary this thesis quantifies how different novel combinations of artificial neural networks with stochastic graphical models and numerical equation solvers for ordinary differential equations perform in predicting and explaining data from a variety of complex non-linear systems.**Keywords:**Data-driven modeling; Machine learning; Artificial neural network; Nonlinear systems; Chaos