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Modeling and Control of Dynamical Systems with Reservoir Computing

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Modeling and Control of Dynamical Systems with Reservoir Computing Modeling and Control of Dynamical Systems with Reservoir Computing DISSERTATION Presented in Partial Fullfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University BY DANIEL CANADAY, MS GRADUATE PROGRAM IN PHYSICS THE OHIO STATE UNIVERSITY 2019 COMMITTEE MEMBERS: DANIEL J. GAUTHIER,ADVISER GREGORY LAFYATIS RICHARD FURNSTAHL MIKHAIL BELKIN Copyright by Daniel Canaday 2019 Abstract There is currently great interest in applying artificial neural networks to a host of commer- cial and industrial tasks. Such networks with a layered, feedforward structure are currently deployed in technologies ranging from facial recognition software to self-driving cars. They are favored by a large portion of machine learning experts for a number of reasons. Namely: they possess a documented ability to generalize to unseen data and handle large data sets; there exists a number of well-understood training algorithms and integrated software packages for implementing them; and they have rigorously proven expressive power making them capable of approximating any bounded, static map arbitrarily well. Within the last couple of decades, reservoir computing has emerged as a method for train- ing a different type of artificial neural network known as a recurrent neural network. Unlike layered, feedforward neural networks, recurrent neural networks are non-trivial dynamical systems that exhibit time-dependence and dynamical memory. In addition to being more bi- ologically plausible, they more naturally handle time-dependent tasks such as predicting the load on an electrical grid or efficiently controlling a complicated industrial process. Fully- trained recurrent neural networks have high expressive power and are capable of emulating broad classes of dynamical systems. However, despite many recent insights, reservoir com- puting remains relatively young as a field. It remains unclear what fundamental properties yield a well-performing reservoir computer. In practice, this results in their design being left to domain experts, despite the actual training process being remarkably simple to implement. In this thesis, I describe a number of numerical and experimental results that expand the understanding and application of reservoir computing techniques. I develop an algorithm for controlling unknown dynamical systems with layers of reservoir computers. I demonstrate this algorithm by stabilizing a range of complex behavior in simulated Lorenz and Mackey-Glass systems. I additionally control an experimental, chaotic circuit with fast fluctuations. Using my technique, I demonstrate control within the measured noise level for some trajectories. iii This control algorithm is executed on a lightweight, readily-available platform with a 1 MHz closed-loop controller. I also develop a reservoir computing scheme with autonomous, Boolean networks capable of processing complex, real-valued data. I show that this system is capable of emulating, in real time, a benchmark chaotic time-series with high precision and a record-breaking speed of 160 million predictions per second. Finally, I present a technique for obtaining efficient, low dimensional reservoir comput- ers. I demonstrate with numerical examples that the efficient reservoir computers can predict a benchmark time-series more accurately than standard reservoir computers 25 times larger. Through a linear analysis, I find that these efficient reservoirs prefer specific topologies over the random, unstructured reservoir computers that are currently standard. iv Dedication This thesis is dedicated to my parents, my sister, and my wife. v Acknowledgements Although the results presented in this thesis are my own, none of it would be possible without the professional collaboration and personal support of many people. I would first like to acknowledge the support and guidance of my advisor, Prof. Daniel J. Gauthier. I have greatly benefited from his wide expertise, his ability to communicate clearly, and his willingness to engage with students such as myself. He has taught me through example the importance of having excellent presentation and networking skills, some of which I hope have rubbed off on me these past several years. I would like to also acknowledge the many useful scientific discussions with our many collaborators, including Prof. Edward Ott, Prof. Brian Hunt, Prof. Michelle Girvan, and Dr. Andrew Pomerance. These interactions helped clarify many important and difficult concepts for me, as well as seed the ideas that became the projects discussed in this thesis. I would like to thank the support of my committee members Prof. Greg Lafyatis, Prof. Richard Furnstahl, and Prof. Mikhail Belkin. They have all been helpful in navigating the candidacy and defense processes. I appreciate their thoughtful questions during our meetings and their willingness to take the time to read my thesis. I would like to also thank the support of Prof. Nandini Trivedi, Prof. Yuan-Ming Lu, and Prof. Lou DiMauro, who have all advised me at some point in my academic career at The Ohio State University. I would also like to acknowledge Kris Dunlap, who was always willing to answer my many questions throughout graduate school. I want to thank my previous and current office-mates–particularly Kathryn Nicolich and Taimur Islam–who helped break up my workday with interesting conversations, as well as provided emotional support through our shared graduate school experience. I also want to thank my house-mates Michael Darcy, Brendan McCullian, and Noah Charles for all of their support. I am very lucky to have made such good friends in graduate school. vi Most importantly, I want to thank my family for their unwavering love and support. My parents Cheryl Canaday and Marcus Canaday have always been my most vocal supporters, and for that I am forever grateful. Visits from my sister Emily Canaday are always wonderful. My wife Alexandra Cisek has provided constant emotional support that has been critical to making it through to graduation. Finally, I gratefully knowledge the financial support of U.S. Army Research Office Grant No. W911NF-12-1-0099, the Army STTR Program Office Contract No. W31P4Q-19-C-0014, Potomac Research, LLC, and The Ohio State University. vii Vita Bachelor of Science, Mathematics and Physics . 2010-2014 The Ohio State University Master of Science, Physics . 2014-2017 The Ohio State University Data Science Internship . 2019 Potomac Research, LLC Publications D. Canaday, A. Griffith, and D.J. Gauthier, ‘Rapid Time Series Prediction with a Hardware- Based Reservoir Computer,’ Chaos 28, 123119 (2018). Field of Study Major Field: Physics viii Contents Abstract iii Dedication v Acknowledgements vi Vita viii List of Figures xiii List of Tables xxiv 1 Introduction 1 1.1 Novel Contribution and Outline............................4 2 Foundations of Reservoir Computing8 2.1 Dynamical Systems....................................8 2.1.1 Types of Dynamical Systems.......................... 10 2.1.2 Delay Embedding................................ 11 2.2 Machine Learning..................................... 12 2.2.1 Performance Measures.............................. 13 2.2.2 Hyperparameters................................. 14 2.3 Artificial Neural Networks................................ 14 2.3.1 Feedforward ANNs............................... 15 ix 2.3.2 Training...................................... 18 2.3.3 The Problem of RNNs.............................. 18 2.4 The Reservoir Computing "Trick"............................ 19 2.4.1 The Echo State Network............................. 20 2.4.2 Matrix Generation................................ 21 2.4.3 Hyperparameter Selection............................ 22 2.4.4 Traing an ESN................................... 24 2.5 Necessary Properties of RC............................... 25 2.5.1 Generalized Synchronization.......................... 25 2.5.2 Separability.................................... 27 2.5.3 Approximation.................................. 28 2.6 Conclusions........................................ 28 3 Control of Unknown Systems with Deep Reservoir Computing 30 3.1 Problem Formulation................................... 32 3.2 Single Layer Reservoir Controller............................ 33 3.2.1 Choosing vtrain .................................. 36 3.2.2 Hyperparameter Considerations–Mackey-Glass System.......... 36 3.3 Adding Controller Layers................................ 42 3.3.1 Deep Hyperparameters............................. 42 3.4 Numerical Results–Lorenz System........................... 44 3.4.1 Unstable Steady States.............................. 45 3.4.2 Additional Layers................................ 47 3.4.3 Lorenz Origin................................... 48 3.4.4 Known Fixed Points............................... 49 3.4.5 Ellipses Near Attractor.............................. 49 3.4.6 Synchronization.................................. 52 x 3.5 Experimental Circuit................................... 54 3.5.1 FPGA-Accelerated Controller.......................... 56 3.5.2 Control Results.................................. 57 3.6 Conclusions........................................ 63 4 Reservoir Computing with Autonomous, Boolean Networks 66 4.1 Challenges of Real-Time Prediction........................... 67 4.1.1 Physical RC.................................... 68 4.1.2 Real-Time Prediction with Optical RC..................... 69 4.2 Field-Programmable Gate Arrays............................ 70 4.2.1 Synchronous versus Autonomous
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