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This thesis aims in collecting and generalizing facts about complex valued neural networks, extends feedforward neural networks to the complex valued case and applies complex valued neural networks to a set of real-world systems models. A brief review on real valued neural networks is presented to introduce notations and terminology, followed by a then a brief review of complex analysis. Complex valued system identification is extensively addressed in this thesis and novel. The difficulties of the complex valued minimization problem are explained and ways to overcome these problems are addressed. The generalization of real-valued feedforward neural networks to complex valued neural networks is introduced and this theory is then generalized to recurrent complex valued neural network in general. New interpretations of the error function are highlighted which lead to a novel recurrent architecture, which allows for continuous time modeling. The developed methods are applied to two artificial chaotic time series datasets (i.e., the logistics map and the Lorenz system) to benchmark the novel approach introduced in this thesis. Complex recurrent neural networks are applied to several real-world applications including (i) nonlinear transformer modeling, (ii) financial time series forecasting, (iii) the simulation of neuron synchronization in a real brain model, and (iv) a novel approach to solve the Binding problem. These applications clearly demonstrate the advantages of complex valued neural networks compared to the traditional real-valued approach. Practical aspects for the implementation of complex-valued neural networks and some fine points to consider in benchmarking studies for regression problems are discussed. Diese Arbeit verallgemeinert komplexwertige neuronale Netze, erweitert vorwärts gerichtete neuronale Netze auf die komplexe Zahlendomäne und wendet komplexwertige neuronale Netze auf mehrere realistische Systemmodelle an. Um Notation und Terminologie zu definieren, werden einführend komplexe Analysis und reellwertige neuronale Netze besprochen. Erstmalig wird eine komplexwertige Systemidentifikation umfassend behandelt. Schwierigkeiten des komplexwertigen Minimisierungsproblems werden erläutert und Auswege daraus beschrieben. Die Verallgemeinerung reellwertiger vorwärts gerichteter neuronaler Netze auf komplexwertige neuronale Netze wird eingeführt und diese Theorie wird dann auf rekurrente komplexwertige neuronale Netze ausgeweitet. Neue Interpretationen der Fehlerfunktion werden erarbeitet, die zu einer neuen rekurrenten Architektur führt, die eine kontinuierliche Zeitmodellierung erlaubt. |
