@article{QuadeIseleAbel2020, author = {Quade, Markus and Isele, Thomas and Abel, Markus}, title = {Machine learning control}, series = {Physica : D, Nonlinear phenomena}, volume = {412}, journal = {Physica : D, Nonlinear phenomena}, publisher = {Elsevier}, address = {Amsterdam}, issn = {0167-2789}, doi = {10.1016/j.physd.2020.132582}, pages = {11}, year = {2020}, abstract = {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.}, language = {en} } @article{GoutQuadeShafietal.2017, author = {Gout, Julien and Quade, Markus and Shafi, Kamran and Niven, Robert K. and Abel, Markus}, title = {Synchronization control of oscillator networks using symbolic regression}, series = {Nonlinear Dynamics}, volume = {91}, journal = {Nonlinear Dynamics}, number = {2}, publisher = {Springer}, address = {Dordrecht}, issn = {0924-090X}, doi = {10.1007/s11071-017-3925-z}, pages = {1001 -- 1021}, year = {2017}, abstract = {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.}, language = {en} } @article{WaldripNivenAbeletal.2017, author = {Waldrip, S. H. and Niven, Robert K. and Abel, Markus and Schlegel, M.}, title = {Reduced-Parameter Method for Maximum Entropy Analysis of Hydraulic Pipe Flow Networks}, series = {Journal of hydraulic engineering}, volume = {144}, journal = {Journal of hydraulic engineering}, number = {2}, publisher = {American Society of Civil Engineers}, address = {Reston}, issn = {0733-9429}, doi = {10.1061/(ASCE)HY.1943-7900.0001379}, pages = {10}, year = {2017}, abstract = {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.}, language = {en} } @article{SawickiAbelSchoell2018, author = {Sawicki, Jakub and Abel, Markus and Sch{\"o}ll, Eckehard}, title = {Synchronization of organ pipes}, series = {The European physical journal : B, Condensed matter and complex systems}, volume = {91}, journal = {The European physical journal : B, Condensed matter and complex systems}, number = {2}, publisher = {Springer}, address = {New York}, issn = {1434-6028}, doi = {10.1140/epjb/e2017-80485-8}, pages = {9}, year = {2018}, abstract = {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.}, language = {en} } @article{QuadeAbelKutzetal.2018, author = {Quade, Markus and Abel, Markus and Kutz, J. Nathan and Brunton, Steven L.}, title = {Sparse identification of nonlinear dynamics for rapid model recovery}, series = {Chaos : an interdisciplinary journal of nonlinear science}, volume = {28}, journal = {Chaos : an interdisciplinary journal of nonlinear science}, number = {6}, publisher = {American Institute of Physics}, address = {Melville}, issn = {1054-1500}, doi = {10.1063/1.5027470}, pages = {10}, year = {2018}, abstract = {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.}, language = {en} } @article{CestnikAbel2019, author = {Cestnik, Rok and Abel, Markus}, title = {Inferring the dynamics of oscillatory systems using recurrent neural networks}, series = {Chaos : an interdisciplinary journal of nonlinear science}, volume = {29}, journal = {Chaos : an interdisciplinary journal of nonlinear science}, number = {6}, publisher = {American Institute of Physics}, address = {Melville}, issn = {1054-1500}, doi = {10.1063/1.5096918}, pages = {9}, year = {2019}, abstract = {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.}, language = {en} } @article{NivenAbelSchlegeletal.2019, author = {Niven, Robert K. and Abel, Markus and Schlegel, Michael and Waldrip, Steven H.}, title = {Maximum Entropy Analysis of Flow Networks: Theoretical Foundation and Applications}, series = {Entropy}, volume = {21}, journal = {Entropy}, number = {8}, publisher = {MDPI}, address = {Basel}, issn = {1099-4300}, doi = {10.3390/e21080776}, pages = {776}, year = {2019}, abstract = {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.}, language = {en} } @article{ParezanovicCordierSpohnetal.2016, author = {Parezanovic, Vladimir and Cordier, Laurent and Spohn, Andreas and Duriez, Thomas and Noack, Bernd R. and Bonnet, Jean-Paul and Segond, Marc and Abel, Markus and Brunton, Steven L.}, title = {Frequency selection by feedback control in a turbulent shear flow}, series = {Journal of fluid mechanics}, volume = {797}, journal = {Journal of fluid mechanics}, publisher = {Cambridge Univ. Press}, address = {New York}, issn = {0022-1120}, doi = {10.1017/jfm.2016.261}, pages = {247 -- 283}, year = {2016}, abstract = {Many previous studies have shown that the turbulent mixing layer under periodic forcing tends to adopt a lock-on state, where the major portion of the fluctuations in the flow are synchronized at the forcing frequency. The goal of this experimental study is to apply closed-loop control in order to provoke the lock-on state, using information from the flow itself. We aim to determine the range of frequencies for which the closed-loop control can establish the lock-on, and what mechanisms are contributing to the selection of a feedback frequency. In order to expand the solution space for optimal closed-loop control laws, we use the genetic programming control (CPC) framework. The best closed-loop control laws obtained by CPC are analysed along with the associated physical mechanisms in the mixing layer flow. The resulting closed-loop control significantly outperforms open-loop forcing in terms of robustness to changes in the free-stream velocities. In addition, the selection of feedback frequencies is not locked to the most amplified local mode, but rather a range of frequencies around it.}, language = {en} } @article{WinklerAbel2016, author = {Winkler, Michael and Abel, Markus}, title = {Optimized setup for two-dimensional convection experiments in thin liquid films}, series = {Review of scientific instruments : a monthly journal devoted to scientific instruments, apparatus, and techniques}, volume = {87}, journal = {Review of scientific instruments : a monthly journal devoted to scientific instruments, apparatus, and techniques}, publisher = {American Institute of Physics}, address = {Melville}, issn = {0034-6748}, doi = {10.1063/1.4950871}, pages = {11}, year = {2016}, abstract = {We present a novel experimental setup to investigate two-dimensional thermal convection in a freestanding thin liquid film. Such films can be produced in a controlled way on the scale of 5-1000 nm. Our primary goal is to investigate convection patterns and the statistics of reversals in Rayleigh-Benard convection with varying aspect ratio. Additionally, questions regarding the physics of liquid films under controlled conditions can be investigated, like surface forces, or stability under varying thermodynamical parameters. The film is suspended in a frame which can be adjusted in height and width to span an aspect ratio range of Gamma = 0.16-10. The top and bottom frame elements can be set to specific temperature within T = 15 degrees C to 55 degrees C. A thickness to area ratio of approximately 108 enables only two-dimensional fluid motion in the time scales relevant for turbulent motion. The chemical composition of the film is well-defined and optimized for film stability and reproducibility and in combination with carefully controlled ambient parameters allows the comparison to existing experimental and numerical data. Published by AIP Publishing.}, language = {en} } @article{QuadeAbelShafietal.2016, author = {Quade, Markus and Abel, Markus and Shafi, Kamran and Niven, Robert K. and Noack, Bernd R.}, title = {Prediction of dynamical systems by symbolic regression}, series = {Physical review : E, Statistical, nonlinear and soft matter physics}, volume = {94}, journal = {Physical review : E, Statistical, nonlinear and soft matter physics}, publisher = {American Society for Pharmacology and Experimental Therapeutics}, address = {Bethesda}, issn = {2470-0045}, doi = {10.1103/PhysRevE.94.012214}, pages = {15}, year = {2016}, abstract = {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.}, language = {en} }