@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{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} } @misc{WaldripNivenAbeletal.2017, author = {Waldrip, Steven H. and Niven, Robert K. and Abel, Markus and Schlegel, Michael}, title = {Consistent maximum entropy representations of pipe flow networks}, series = {AIP conference proceedings}, volume = {1853}, journal = {AIP conference proceedings}, number = {1}, publisher = {American Institute of Physics}, address = {Melville}, isbn = {978-0-7354-1527-0}, issn = {0094-243X}, doi = {10.1063/1.4985365}, year = {2017}, abstract = {The maximum entropy method is used to predict flows on water distribution networks. This analysis extends the water distribution network formulation of Waldrip et al. (2016) Journal of Hydraulic Engineering (ASCE), by the use of a continuous relative entropy defined on a reduced parameter set. This reduction in the parameters that the entropy is defined over ensures consistency between different representations of the same network. The performance of the proposed reduced parameter method is demonstrated with a one-loop network case study.}, language = {en} } @misc{WaldripNivenAbeletal.2017, author = {Waldrip, Steven H. and Niven, Robert K. and Abel, Markus and Schlegel, Michael}, title = {Maximum entropy analysis of transport networks}, series = {AIP conference proceedings}, volume = {1853}, journal = {AIP conference proceedings}, number = {1}, publisher = {American Institute of Physics}, address = {Melville}, isbn = {978-0-7354-1527-0}, issn = {0094-243X}, doi = {10.1063/1.4985364}, pages = {8}, year = {2017}, abstract = {The maximum entropy method is used to derive an alternative gravity model for a transport network. The proposed method builds on previous methods which assign the discrete value of a maximum entropy distribution to equal the traffic flow rate. The proposed method however, uses a distribution to represent each flow rate. The proposed method is shown to be able to handle uncertainty in a more elegant way and give similar results to traditional methods. It is able to incorporate more of the observed data through the entropy function, prior distribution and integration limits potentially allowing better inferences to be made.}, language = {en} } @article{KaiserNoackCordieretal.2014, author = {Kaiser, Eurika and Noack, Bernd R. and Cordier, Laurent and Spohn, Andreas and Segond, Marc and Abel, Markus and Daviller, Guillaume and Osth, Jan and Krajnovic, Sinisa and Niven, Robert K.}, title = {Cluster-based reduced-order modelling of a mixing layer}, series = {Journal of fluid mechanics}, volume = {754}, journal = {Journal of fluid mechanics}, publisher = {Cambridge Univ. Press}, address = {New York}, issn = {0022-1120}, doi = {10.1017/jfm.2014.355}, pages = {365 -- 414}, year = {2014}, 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} } @misc{KaiserNoackCordieretal.2014, author = {Kaiser, Eurika and Noack, Bernd R. and Cordier, Laurent and Spohn, Andreas and Segond, Marc and Abel, Markus and Daviller, Guillaume and Osth, Jan and Krajnovic, Sinisa and Niven, Robert K.}, title = {Cluster-based reduced-order modelling of a mixing layer}, series = {Postprints der Universit{\"a}t Potsdam : Mathematisch Naturwissenschaftliche Reihe}, journal = {Postprints der Universit{\"a}t Potsdam : Mathematisch Naturwissenschaftliche Reihe}, number = {605}, issn = {1866-8372}, doi = {10.25932/publishup-41611}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-416113}, pages = {365 -- 414}, year = {2014}, abstract = {We propose a novel cluster-based reduced-order modelling (CROM) strategy for unsteady flows. CROM combines the cluster analysis pioneered in Gunzburger's group (Burkardt, Gunzburger \& Lee, Comput. Meth. Appl. Mech. Engng, vol. 196, 2006a, pp. 337-355) and transition matrix models introduced in fluid dynamics in Eckhardt's group (Schneider, Eckhardt \& Vollmer, Phys. Rev. E, vol. 75, 2007, art. 066313). CROM constitutes a potential alternative to POD models and generalises the Ulam-Galerkin method classically used in dynamical systems to determine a finite-rank approximation of the Perron-Frobenius operator. The proposed strategy processes a time-resolved sequence of flow snapshots in two steps. First, the snapshot data are clustered into a small number of representative states, called centroids, in the state space. These centroids partition the state space in complementary non-overlapping regions (centroidal Voronoi cells). Departing from the standard algorithm, the probabilities of the clusters are determined, and the states are sorted by analysis of the transition matrix. Second, the transitions between the states are dynamically modelled using a Markov process. Physical mechanisms are then distilled by a refined analysis of the Markov process, e. g. using finite-time Lyapunov exponent (FTLE) and entropic methods. This CROM framework is applied to the Lorenz attractor (as illustrative example), to velocity fields of the spatially evolving incompressible mixing layer and the three-dimensional turbulent wake of a bluff body. For these examples, CROM is shown to identify non-trivial quasi-attractors and transition processes in an unsupervised manner. CROM has numerous potential applications for the systematic identification of physical mechanisms of complex dynamics, for comparison of flow evolution models, for the identification of precursors to desirable and undesirable events, and for flow control applications exploiting nonlinear actuation dynamics.}, 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} }