17 Conclusion and outlook
A tool to capture and predict the behavior of nonlinear complex and chaotic dynamical systems within a range of some model parameter values \(\vec{\beta}\) is presented. The tool is called control-oriented Cluster-based Network Modeling (CNMc). It could be shown that CNMccontrol-oriented Cluster-based Network Modeling is able to capture and make predictions for the well-known Lorenz system (Lorenz 1963). With having removed one of the major limitations in the first attempt of CNMccontrol-oriented Cluster-based Network Modeling (Pierzyna 2021), the introduced version of CNMccontrol-oriented Cluster-based Network Modeling is not limited to any dimension anymore. Furthermore, the restriction of the dynamical system to exhibit a circular trajectory is removed. Since these two limitations could be removed, the presented CNMccontrol-oriented Cluster-based Network Modeling can be applied to any general dynamical system. To outline this fact, 10 different dynamical systems are implemented by default in CNMccontrol-oriented Cluster-based Network Modeling. Some of these dynamical systems were used to evaluate CNMccontrol-oriented Cluster-based Network Modeling performance. It could be observed that CNMccontrol-oriented Cluster-based Network Modeling is not only able to deal with the Lorenz system but also with more complicated systems. The objective is to represent the characteristic behavior of general dynamical systems that could be fulfilled on all tested systems.
The third limitation which could be removed is the unacceptably high computational time with Non-negative Matrix Factorization (NMF). It could be highlighted that Singular Value Decomposition (SVD) returns the decomposition within seconds, instead of hours, without adding any inaccuracies. Moreover, SVDSingular Value Decomposition does not require a parameter study. Executing NMFNon-negative Matrix Factorization once is already computational more expensive than SVDSingular Value Decomposition, but with a parameter study, NMFNon-negative Matrix Factorization becomes even more unsatisfactory in the application. By having removed these 3 major limitations, CNMccontrol-oriented Cluster-based Network Modeling can be applied to any dynamical system within a reasonable computational time on a regular laptop. Nevertheless, CNMccontrol-oriented Cluster-based Network Modeling contains algorithms, which highly benefit from computational power. Thus, faster outputs are achieved with clusters. Also, with having replaced the B-spline interpolation through linear interpolation, the predicted trajectories can be visually depicted appropriately without the Another important introduced advancement is that the B-spline interpolation was replaced by linear interpolation. This allows to avoid unreasonably high interpolation errors (oscillations) of the trajectory and enables an appropriate visualization.
CNMccontrol-oriented Cluster-based Network Modeling Is written from scratch in a modular way such that implementing it into existing code, replacing employed algorithms with others is straightforward or used as a black-box function. All important parameters can be adjusted via one file (settings.py). Helpful post-processing features are part of CNMccontrol-oriented Cluster-based Network Modeling and can also be controlled with settings.py. Overall CNMccontrol-oriented Cluster-based Network Modeling includes a high number of features, e.g., a log file, storing results at desired steps, saving plots as HTML files which allow extracting further information about the outcome, the ability to execute multiple models consequentially, and activating and disabling each step of CNMccontrol-oriented Cluster-based Network Modeling. All displayed outputs in this thesis were generated with CNMccontrol-oriented Cluster-based Network Modeling. Finally, one limitation which remains shall be mentioned. The used SVDSingular Value Decomposition code receives sparse matrices, however, it returns a dense matrix. The consequence is that with high model orders \(L\), quickly multiple hundreds of gigabytes of RAM are required. The maximal \(L\) which could be achieved on the laptop of the author, which has 16 GB RAM, is \(L=7\).
As an outlook, a new SVDSingular Value Decomposition algorithm should be searched for or written from scratch. The demand for the new SVDSingular Value Decomposition solver is that it must receive sparse matrices and also returns the solution in form of sparse matrices. With that \(L\) could be increased, i.e., \(L>7\). In this thesis, it could be shown that CNMccontrol-oriented Cluster-based Network Modeling can handle chaotic systems well. Thus, the next step could be, replacing the current data generation step, where differential equations are solved, with actual CFDComputational Fluid Dynamics data as input. Hence, the objective would be to apply CNMccontrol-oriented Cluster-based Network Modeling to real CFDComputational Fluid Dynamics data to predict flow fields.