Prediction and control of nonlinear dynamical systems using machine learning

Künstliche Intelligenz und Machine Learning erfreuen sich in Folge der rapide gestiegenen Rechenleistung immer größerer Popularität. Sei es autonomes Fahren, Gesichtserkennung, bildgebende Diagnostik in der Medizin oder Robotik – die Anwendungsvielfalt scheint keine Grenzen zu kennen. Um jedoch systematischen Bias und irreführende Ergebnisse zu vermeiden, ist ein tiefes Verständnis der Methoden und ihrer Sensitivitäten von Nöten. Anhand der Vorhersage chaotischer Systeme mit Reservoir Computing – einem künstlichen rekurrenten neuronalem Netzwerk – wird im Rahmen dieser Dissertation beleuchtet, wie sich verschiedene Eigenschaften des Netzwerks auf die Vorhersagekraft und Robustheit auswirken. Es wird gezeigt, wie sich die Variabilität der Vorhersagen – sowohl was die exakte zukünftige Trajektorie betrifft als auch das statistische Langzeitverhalten (das "Klima") des Systems – mit geeigneter Parameterwahl signifikant reduzieren lässt. Die Nichtlinearität der Aktivierungsfunktion spielt hierbei eine besondere Rolle, weshalb ein Skalierungsparameter eingeführt wird, um diese zu kontrollieren. Des Weiteren werden differenzielle Eigenschaften des Netzwerkes untersucht und gezeigt, wie ein kontrolliertes Entfernen der "richtigen" Knoten im Netzwerk zu besseren Vorhersagen führt und die Größe des Netzwerkes stark reduziert werden kann bei gleichzeitig nur moderater Verschlechterung der Ergebnisse. Dies ist für Hardware Realisierungen von Reservoir Computing wie zum Beispiel Neuromorphic Computing relevant, wo möglichst kleine Netzwerke von Vorteil sind. Zusätzlich werden unterschiedliche Netzwerktopologien wie Small World Netzwerke und skalenfreie Netzwerke beleuchtet. Mit den daraus gewonnenen Erkenntnissen für bessere Vorhersagen von nichtlinearen dynamischen Systemen wird eine neue Kontrollmethode entworfen, die es ermöglicht, dynamische Systeme flexibel in verschiedenste Zielzustände zu lenken. Hierfür wird – anders als bei vielen bisherigen Ansätzen – keine Kenntnis der zugrundeliegenden Gleichungen des Systems erfordert. Ebenso wird nur eine begrenzte Datenmenge verlangt, um Reservoir Computing hinreichend zu trainieren. Zudem ist es nicht nur möglich, chaotisches Verhalten in einen periodischen Zustand zu zwingen, sondern auch eine Kontrolle auf komplexere Zielzustände wie intermittentes Verhalten oder ein spezifischer anderer chaotischer Zustand. Dies ermöglicht eine Vielzahl neuer potenzieller realer Anwendungen, von personalisierten Herzschrittmachern bis hin zu Kontrollvorrichtungen für Raketentriebwerke zur Unterbindung von kritischen Verbrennungsinstabilitäten. Als Schritt zur Weiterentwicklung von Reservoir Computing zu einem verbesserten hybriden System, das nicht nur rein datenbasiert arbeitet, sondern auch physikalische Zusammenhänge berücksichtigt, wird ein Ansatz vorgestellt, um lineare und nichtlinearen Kausalitätsstrukturen zu separieren. Dies kann verwendet werden, um Systemgleichungen oder Restriktionen für ein hybrides System zur Vorhersage oder Kontrolle abzuleiten.Artificial intelligence and machine learning are becoming increasingly popular as a result of the rapid increase in computing power. Be it autonomous driving, facial recognition, medical imaging diagnostics or robotics – the variety of applications seems to have no limits. However, to avoid systematic bias and misleading results, a deep understanding of the methods and their sensitivities is needed. Based on the prediction of chaotic systems with reservoir computing – an artificial recurrent neural network – this dissertation sheds light on how different properties of the network affect the predictive power and robustness. It is shown how the variability of the predictions – both in terms of the exact short-term predictions and the long-term statistical behaviour (the "climate") of the system – can be significantly reduced with appropriate parameter choices. The nonlinearity of the activation function plays a special role here, thus a scaling parameter is introduced to control it. Furthermore, differential properties of the network are investigated and it is shown how a controlled removal of the right nodes in the network leads to better predictions, whereas the size of the network can be greatly reduced while only moderately degrading the results. This is relevant for hardware realizations of reservoir computing such as neuromorphic computing, where networks that are as small as possible are advantageous. Additionally, different network topologies such as small world networks and scale-free networks are investigated. With the insights gained for better predictions of nonlinear dynamical systems, a new control method is designed that allows dynamical systems to be flexibly forced into a wide variety of dynamical target states. For this – unlike many previous approaches – no knowledge of the underlying equations of the system is required. Further, only a limited amount of data is needed to sufficiently train reservoir computing. Moreover, it is possible not only to force chaotic behavior to a periodic state, but also to control for more complex target states such as intermittent behavior or a specific different chaotic state. This enables a variety of new potential real-world applications, from personalized cardiac pacemakers to control devices for rocket engines to suppress critical combustion instabilities. As a step toward advancing reservoir computing to an improved hybrid system that is not only purely data-based but also takes into account physical relationships, an approach is presented to separate linear and nonlinear causality structures. This can be used to derive system equations or constraints for a hybrid prediction or control system

Linear and nonlinear market correlations: characterizing financial crises and portfolio optimization

Pearson correlation and mutual information based complex networks of the day-to-day returns of US S&P500 stocks between 1985 and 2015 have been constructed in order to investigate the mutual dependencies of the stocks and their nature. We show that both networks detect qualitative differences especially during (recent) turbulent market periods thus indicating strongly fluctuating interconnections between the stocks of different companies in changing economic environments. A measure for the strength of nonlinear dependencies is derived using surrogate data and leads to interesting observations during periods of financial market crises. In contrast to the expectation that dependencies reduce mainly to linear correlations during crises we show that (at least in the 2008 crisis) nonlinear effects are significantly increasing. It turns out that the concept of centrality within a network could potentially be used as some kind of an early warning indicator for abnormal market behavior as we demonstrate with the example of the 2008 subprime mortgage crisis. Finally, we apply a Markowitz mean variance portfolio optimization and integrate the measure of nonlinear dependencies to scale the investment exposure. This leads to significant outperformance as compared to a fully invested portfolio.Comment: 12 pages, 11 figures, Phys. Rev. E, accepte

Controlling dynamical systems to complex target states using machine learning: next-generation vs. classical reservoir computing

Controlling nonlinear dynamical systems using machine learning allows to not only drive systems into simple behavior like periodicity but also to more complex arbitrary dynamics. For this, it is crucial that a machine learning system can be trained to reproduce the target dynamics sufficiently well. On the example of forcing a chaotic parametrization of the Lorenz system into intermittent dynamics, we show first that classical reservoir computing excels at this task. In a next step, we compare those results based on different amounts of training data to an alternative setup, where next-generation reservoir computing is used instead. It turns out that while delivering comparable performance for usual amounts of training data, next-generation RC significantly outperforms in situations where only very limited data is available. This opens even further practical control applications in real world problems where data is restricted.Comment: IJCNN 202

Controlling nonlinear dynamical systems into arbitrary states using machine learning

Controlling nonlinear dynamical systems is a central task in many different areas of science and engineering. Chaotic systems can be stabilized (or chaotified) with small perturbations, yet existing approaches either require knowledge about the underlying system equations or large data sets as they rely on phase space methods. In this work we propose a novel and fully data driven scheme relying on machine learning (ML), which generalizes control techniques of chaotic systems without requiring a mathematical model for its dynamics. Exploiting recently developed ML-based prediction capabilities, we demonstrate that nonlinear systems can be forced to stay in arbitrary dynamical target states coming from any initial state. We outline and validate our approach using the examples of the Lorenz and the Rössler system and show how these systems can very accurately be brought not only to periodic, but even to intermittent and different chaotic behavior. Having this highly flexible control scheme with little demands on the amount of required data on hand, we briefly discuss possible applications ranging from engineering to medicine

Reducing network size and improving prediction stability of reservoir computing

Reservoir computing is a very promising approach for the prediction of complex nonlinear dynamical systems. Besides capturing the exact short-term trajectories of nonlinear systems, it has also proved to reproduce its characteristic long-term properties very accurately. However, predictions do not always work equivalently well. It has been shown that both short- and long-term predictions vary significantly among different random realizations of the reservoir. In order to gain an understanding on when reservoir computing works best, we investigate some differential properties of the respective realization of the reservoir in a systematic way. We find that removing nodes that correspond to the largest weights in the output regression matrix reduces outliers and improves overall prediction quality. Moreover, this allows to effectively reduce the network size and, therefore, increase computational efficiency. In addition, we use a nonlinear scaling factor in the hyperbolic tangent of the activation function. This adjusts the response of the activation function to the range of values of the input variables of the nodes. As a consequence, this reduces the number of outliers significantly and increases both the short- and long-term prediction quality for the nonlinear systems investigated in this study. Our results demonstrate that a large optimization potential lies in the systematical refinement of the differential reservoir properties for a given dataset.Comment: 11 pages, 8 figures, published in Chao

Evaluation of the South Asian Monsoon in the ECHAM/MESSy

The South Asian Monsoon has been evaluated in the ECHAM/MESSy Atmo-spheric Chemistry (EMAC) model in comparison to observations and models par-ticipating in the 5th Phase of the Coupled Model Intercomparison Project (CMIP5).Existing diagnostics of the Earth System Model Evaluation Tool (ESMValTool)have been applied to historical CMIP5 simulations and to an EMAC timeslice ex-periment representing the year 2000 to investigate precipitation in the South AsiaMonsoon period. The EMAC simulation generally overestimates precipitation inSouth Asia during the local summer and winter seasons compared to observations,and also the global precipitation intensity, which is calculated as summer minuswinter difference. This bias in precipitation intensity results in an overestimationof the global monsoon domains, which are areas that exceed a precipitation inten-sity of 2.5 mm per day. Compared to the CMIP5 historical simulations, not only theCMIP5 multi-model mean but also each individual model is more skillful in termsof pattern correlation index. Possible reasons for biases in the EMAC simulationcould for example arise from biases in circulation or the representation of clouds.The simulation of the 850 hPa low-level wind was better represented than the pre-cipitation intensity, although some biases compared to meteorological reanalysisexist. Diagnostics have been newly developed and implemented into the ESMVal-Tool to examine the representation of Cloud Radiative Effects (CRE) as externalsolar radiation is the main driver for monsoon. Deviations from observations arefound in particular for the simulated shortwave radiation, which could be relatedto problems in the representation of low-level clouds. The biases found in theparticular EMAC simulation evaluated here could partly be due to the boundaryconditions used, in particular the prescribed sea surface temperatures (SST) thatare taken from a CMIP5 model simulation with the CMCC model. In addition, acoupled ocean could possibly partially compensate for the biases in precipitation.Additional work is required to investigate this further. For example, an EMACsimulation with observed SSTs or a coupled ocean could be evaluated to betterunderstand the reasons for the biases found her

Good and bad predictions: Assessing and improving the replication of chaotic attractors by means of reservoir computing

The prediction of complex nonlinear dynamical systems with the help of machine learning techniques has become increasingly popular. In particular, reservoir computing turned out to be a very promising approach especially for the reproduction of the long-term properties of a nonlinear system. Yet, a thorough statistical analysis of the forecast results is missing. Using the Lorenz and Rössler system, we statistically analyze the quality of prediction for different parametrizations - both the exact short-term prediction as well as the reproduction of the longterm properties (the climate) of the system as estimated by the correlation dimension and largest Lyapunov exponent. We find that both short- and long-term predictions vary significantly among the realizations. Thus, special care must be taken in selecting the good predictions as realizations, which deliver better short-term prediction also tend to better resemble the long-term climate of the system. Instead of only using purely random Erdös-Renyi networks, we also investigate the benefit of alternative network topologies such as small world or scalefree networks and show which effect they have on the prediction quality. Our results suggest that the overall performance with respect to the reproduction of the climate of both the Lorenz and Rössler system is worst for scale-free networks. For the Lorenz system, there seems to be a slight benefit of using small world networks, while for the Rössler system, small world and Erdös-Renyi networks performed equivalently well. In general, the observation is that reservoir computing works for all network topologies investigated here

Complex network-based analysis of nonlinear dependencies in multidimensional financial time series

Cross-correlation and mutual information based complex networks of the day-to-day returns of US S&P500 stocks between 1985 and 2015 have been constructed in order to investigate the mutual dependencies of the stocks and their nature. We show that both networks detect qualitative differences especially during (recent) turbulent market periods thus indicating strongly fluctuating interconnections between the stocks of different companies in changing economic environments. A measure for the strength of nonlinear dependencies has been derived using surrogate data and led to interesting observations during periods of financial market crisis. In contrast to the prevailing view that dependencies reduce mainly to linear correlations during crisis it turned out that (at least in the crisis after 2008) nonlinear effects are significantly increasing. Finally, we apply a Markowitz mean variance portfolio optimization and integrate the measure of nonlinear dependencies to scale the investment exposure. This leads to significant outperformance as compared to a fully invested portfolio

Controlling dynamical systems to complex target states using machine learning: next-generation vs. classical reservoir computing

Controlling nonlinear dynamical systems using machine learning allows to not only drive systems into simple behavior like periodicity but also to more complex arbitrary dynamics. For this, it is crucial that a machine learning system can be trained to reproduce the target dynamics sufficiently well. On the example of forcing a chaotic parametrization of the Lorenz system into intermittent dynamics, we show first that classical reservoir computing excels at this task. In a next step, we compare those results based on different amounts of training data to an alternative setup, where next-generation reservoir computing is used instead. It turns out that while delivering comparable performance for usual amounts of training data, next-generation RC significantly outperforms in situations where only very limited data is available. This opens even further practical control applications in real world problems where data is restricted

Nonlinear correlations in financial markets: A complex network based analysis

It is common practice in finance to quantify correlations among financial time series in terms of their linear Pearson correlation coefficient. Knowing that financial time series show intermittent behavior being reminiscent of turbulence and leading to the well-known fat tails in the probability distribution as well as nonlinearities that also significantly show up as deviations from randomness in the distribution of Fourier phases, it is justified to assume that also nonlinear correlations among financial time series may be present. Therefore, Pearson correlation and mutual information based complex networks of the day-to-day returns of US S&P500 stocks between 1985 and 2015 have been constructed in order to investigate the mutual dependencies of the stocks and their nature. By deriving a measure for the strength of nonlinear correlations using surrogate data we show that a significant amount of information is lost when only relying on linear correlations measures.Our studies revealed that in contrast to the expectation that dependencies reduce mainly to linear correlations during crises, at least in the 2008 crisis nonlinear effects are significantly increasing. More specifically, we find that there are different types of financial crises in terms of nonlinear effects. Our results indicate that during the 2008 crisis nonlinear effects were significantly stronger than in preceding crises like the bursting of the Dot-com bubble. Furthermore, we found that major political events seem to have no significant mid- and long-term impact on market correlation structure in contrast to financial crises. In addition, it turns out that the concept of centrality within a network could potentially be used as some kind of an early warning indicator for abnormal market behavior as we also demonstrate with the example of the 2008 subprime mortgage crisis. Finally we developed a practical application in the field of portfolio optimization. We showed that scaling the investment exposure based on the strength of nonlinear correlations leads to an outperformance compared to a fully invested portfolio with static exposure. Furthermore, we obtained first interesting results on the relationship of linear and nonlinear correlations with causality measure