TY - JOUR A1 - Quade, Markus A1 - Isele, Thomas A1 - Abel, Markus T1 - Machine learning control BT - explainable and analyzable methods JF - Physica : D, Nonlinear phenomena N2 - Recently, the term explainable AI came into discussion as an approach to produce models from artificial intelligence which allow interpretation. For a long time, symbolic regression has been used to produce explainable and mathematically tractable models. In this contribution, we extend previous work on symbolic regression methods to infer the optimal control of a dynamical system given one or several optimization criteria, or cost functions. In earlier publications, network control was achieved by automated machine learning control using genetic programming. Here, we focus on the subsequent path continuation analysis of the mathematical expressions which result from the machine learning model. In particular, we use AUTO to analyze the solution properties of the controlled oscillator system which served as our model. As a result, we show that there is a considerable advantage of explainable symbolic regression models over less accessible neural networks. In particular, the roadmap of future works may be to integrate such analyses into the optimization loop itself to filter out robust solutions by construction. KW - Explainable AI KW - Machine learning control KW - Dynamical systems KW - Synchronization control KW - Genetic programming Y1 - 2020 U6 - https://doi.org/10.1016/j.physd.2020.132582 SN - 0167-2789 SN - 1872-8022 VL - 412 PB - Elsevier CY - Amsterdam ER - TY - JOUR A1 - Niven, Robert K. A1 - Abel, Markus A1 - Schlegel, Michael A1 - Waldrip, Steven H. T1 - Maximum Entropy Analysis of Flow Networks: Theoretical Foundation and Applications JF - Entropy N2 - The concept of a "flow network"-a set of nodes and links which carries one or more flows-unites many different disciplines, including pipe flow, fluid flow, electrical, chemical reaction, ecological, epidemiological, neurological, communications, transportation, financial, economic and human social networks. This Feature Paper presents a generalized maximum entropy framework to infer the state of a flow network, including its flow rates and other properties, in probabilistic form. In this method, the network uncertainty is represented by a joint probability function over its unknowns, subject to all that is known. This gives a relative entropy function which is maximized, subject to the constraints, to determine the most probable or most representative state of the network. The constraints can include "observable" constraints on various parameters, "physical" constraints such as conservation laws and frictional properties, and "graphical" constraints arising from uncertainty in the network structure itself. Since the method is probabilistic, it enables the prediction of network properties when there is insufficient information to obtain a deterministic solution. The derived framework can incorporate nonlinear constraints or nonlinear interdependencies between variables, at the cost of requiring numerical solution. The theoretical foundations of the method are first presented, followed by its application to a variety of flow networks. KW - maximum entropy analysis KW - flow network KW - probabilistic inference Y1 - 2019 U6 - https://doi.org/10.3390/e21080776 SN - 1099-4300 VL - 21 IS - 8 SP - 776 PB - MDPI CY - Basel ER - TY - JOUR A1 - Cestnik, Rok A1 - Abel, Markus T1 - Inferring the dynamics of oscillatory systems using recurrent neural networks JF - Chaos : an interdisciplinary journal of nonlinear science N2 - We investigate the predictive power of recurrent neural networks for oscillatory systems not only on the attractor but in its vicinity as well. For this, we consider systems perturbed by an external force. This allows us to not merely predict the time evolution of the system but also study its dynamical properties, such as bifurcations, dynamical response curves, characteristic exponents, etc. It is shown that they can be effectively estimated even in some regions of the state space where no input data were given. We consider several different oscillatory examples, including self-sustained, excitatory, time-delay, and chaotic systems. Furthermore, with a statistical analysis, we assess the amount of training data required for effective inference for two common recurrent neural network cells, the long short-term memory and the gated recurrent unit. Published under license by AIP Publishing. Y1 - 2019 U6 - https://doi.org/10.1063/1.5096918 SN - 1054-1500 SN - 1089-7682 VL - 29 IS - 6 PB - American Institute of Physics CY - Melville ER - TY - JOUR A1 - Sawicki, Jakub A1 - Abel, Markus A1 - Schöll, Eckehard T1 - Synchronization of organ pipes JF - The European physical journal : B, Condensed matter and complex systems N2 - We investigate synchronization of coupled organ pipes. Synchronization and reflection in the organ lead to undesired weakening of the sound in special cases. Recent experiments have shown that sound interaction is highly complex and nonlinear, however, we show that two delay-coupled Van-der-Pol oscillators appear to be a good model for the occurring dynamical phenomena. Here the coupling is realized as distance-dependent, or time-delayed, equivalently. Analytically, we investigate the synchronization frequency and bifurcation scenarios which occur at the boundaries of the Arnold tongues. We successfully compare our results to experimental data. Y1 - 2018 U6 - https://doi.org/10.1140/epjb/e2017-80485-8 SN - 1434-6028 SN - 1434-6036 VL - 91 IS - 2 PB - Springer CY - New York ER - TY - JOUR A1 - Quade, Markus A1 - Abel, Markus A1 - Kutz, J. Nathan A1 - Brunton, Steven L. T1 - Sparse identification of nonlinear dynamics for rapid model recovery JF - Chaos : an interdisciplinary journal of nonlinear science N2 - Big data have become a critically enabling component of emerging mathematical methods aimed at the automated discovery of dynamical systems, where first principles modeling may be intractable. However, in many engineering systems, abrupt changes must be rapidly characterized based on limited, incomplete, and noisy data. Many leading automated learning techniques rely on unrealistically large data sets, and it is unclear how to leverage prior knowledge effectively to re-identify a model after an abrupt change. In this work, we propose a conceptual framework to recover parsimonious models of a system in response to abrupt changes in the low-data limit. First, the abrupt change is detected by comparing the estimated Lyapunov time of the data with the model prediction. Next, we apply the sparse identification of nonlinear dynamics (SINDy) regression to update a previously identified model with the fewest changes, either by addition, deletion, or modification of existing model terms. We demonstrate this sparse model recovery on several examples for abrupt system change detection in periodic and chaotic dynamical systems. Our examples show that sparse updates to a previously identified model perform better with less data, have lower runtime complexity, and are less sensitive to noise than identifying an entirely new model. The proposed abrupt-SINDy architecture provides a new paradigm for the rapid and efficient recovery of a system model after abrupt changes. Y1 - 2018 U6 - https://doi.org/10.1063/1.5027470 SN - 1054-1500 SN - 1089-7682 VL - 28 IS - 6 PB - American Institute of Physics CY - Melville ER - TY - JOUR A1 - Gout, Julien A1 - Quade, Markus A1 - Shafi, Kamran A1 - Niven, Robert K. A1 - Abel, Markus T1 - Synchronization control of oscillator networks using symbolic regression JF - Nonlinear Dynamics N2 - Networks of coupled dynamical systems provide a powerful way to model systems with enormously complex dynamics, such as the human brain. Control of synchronization in such networked systems has far-reaching applications in many domains, including engineering and medicine. In this paper, we formulate the synchronization control in dynamical systems as an optimization problem and present a multi-objective genetic programming-based approach to infer optimal control functions that drive the system from a synchronized to a non-synchronized state and vice versa. The genetic programming-based controller allows learning optimal control functions in an interpretable symbolic form. The effectiveness of the proposed approach is demonstrated in controlling synchronization in coupled oscillator systems linked in networks of increasing order complexity, ranging from a simple coupled oscillator system to a hierarchical network of coupled oscillators. The results show that the proposed method can learn highly effective and interpretable control functions for such systems. KW - Dynamical systems KW - Synchronization control KW - Genetic programming Y1 - 2017 U6 - https://doi.org/10.1007/s11071-017-3925-z SN - 0924-090X SN - 1573-269X VL - 91 IS - 2 SP - 1001 EP - 1021 PB - Springer CY - Dordrecht ER - TY - JOUR A1 - Waldrip, S. H. A1 - Niven, Robert K. A1 - Abel, Markus A1 - Schlegel, M. T1 - Reduced-Parameter Method for Maximum Entropy Analysis of Hydraulic Pipe Flow Networks JF - Journal of hydraulic engineering N2 - A maximum entropy (MaxEnt) method is developed to predict flow rates or pressure gradients in hydraulic pipe networks without sufficient information to give a closed-form (deterministic) solution. This methodology substantially extends existing deterministic flow network analysis methods. It builds on the MaxEnt framework previously developed by the authors. This study uses a continuous relative entropy defined on a reduced parameter set, here based on the external flow rates. This formulation ensures consistency between different representations of the same network. The relative entropy is maximized subject to observable constraints on the mean values of a subset of flow rates or potential differences, the frictional properties of each pipe, and physical constraints arising from Kirchhoff’s first and second laws. The new method is demonstrated by application to a simple one-loop network and a 1,123-node, 1,140-pipe water distribution network in the suburb of Torrens, Australian Capital Territory, Australia. KW - Maximum entropy method KW - Water distribution systems KW - Hydraulic networks KW - Pipe networks KW - Hydraulic models KW - Nonlinear analysis KW - Probability Y1 - 2017 U6 - https://doi.org/10.1061/(ASCE)HY.1943-7900.0001379 SN - 0733-9429 SN - 1943-7900 VL - 144 IS - 2 PB - American Society of Civil Engineers CY - Reston ER - TY - JOUR A1 - Fischer, Jost Leonhardt A1 - Bader, Rolf A1 - Abel, Markus T1 - Aeroacoustical coupling and synchronization of organ pipes JF - The journal of the Acoustical Society of America N2 - A synchronization experiment on two mutual interacting organ pipes is compared with a theoretical model which takes into account the coupling mechanisms by the underlying first principles of fluid mechanics and aeroacoustics. The focus is on the Arnold-tongue, a mathematical object in the parameter space of detuning and coupling strength which quantitatively captures the interaction of the synchronized sound sources. From the experiment, a nonlinearly shaped Arnold-tongue is obtained, describing the coupling of the synchronized pipe-pipe system. This is in contrast to the linear shaped Arnold-tongue found in a preliminary experiment of the coupled system pipe-loudspeaker. To understand the experimental result, a coarse-grained model of two nonlinear coupled self-sustained oscillators is developed. The model, integrated numerically, is in very good agreement with the synchronization experiment for separation distances of the pipes in the far field and in the intermediate field. The methods introduced open the door for a deeper understanding of the fundamental processes of sound generation and the coupling mechanisms on mutual interacting acoustic oscillators. (C) 2016 Acoustical Society of America. Y1 - 2016 U6 - https://doi.org/10.1121/1.4964135 SN - 0001-4966 SN - 1520-8524 VL - 140 SP - 2344 EP - 2351 PB - American Institute of Physics CY - Melville ER - TY - JOUR A1 - Waldrip, S. H. A1 - Niven, R. K. A1 - Abel, Markus A1 - Schlegel, M. T1 - Maximum Entropy Analysis of Hydraulic Pipe Flow Networks JF - Journal of hydraulic engineering KW - Maximum entropy method KW - Water distribution systems KW - Hydraulic networks KW - Pipe networks KW - Hydraulic models KW - Non-linear analysis KW - Probability Y1 - 2016 U6 - https://doi.org/10.1061/(ASCE)HY.1943-7900.0001126 SN - 0733-9429 SN - 1943-7900 VL - 142 SP - 332 EP - 347 PB - American Society of Civil Engineers CY - Reston ER - TY - JOUR A1 - Quade, Markus A1 - Abel, Markus A1 - Shafi, Kamran A1 - Niven, Robert K. A1 - Noack, Bernd R. T1 - Prediction of dynamical systems by symbolic regression JF - Physical review : E, Statistical, nonlinear and soft matter physics N2 - We study the modeling and prediction of dynamical systems based on conventional models derived from measurements. Such algorithms are highly desirable in situations where the underlying dynamics are hard to model from physical principles or simplified models need to be found. We focus on symbolic regression methods as a part of machine learning. These algorithms are capable of learning an analytically tractable model from data, a highly valuable property. Symbolic regression methods can be considered as generalized regression methods. We investigate two particular algorithms, the so-called fast function extraction which is a generalized linear regression algorithm, and genetic programming which is a very general method. Both are able to combine functions in a certain way such that a good model for the prediction of the temporal evolution of a dynamical system can be identified. We illustrate the algorithms by finding a prediction for the evolution of a harmonic oscillator based on measurements, by detecting an arriving front in an excitable system, and as a real-world application, the prediction of solar power production based on energy production observations at a given site together with the weather forecast. Y1 - 2016 U6 - https://doi.org/10.1103/PhysRevE.94.012214 SN - 2470-0045 SN - 2470-0053 VL - 94 PB - American Society for Pharmacology and Experimental Therapeutics CY - Bethesda ER -