Numerical approximation methods for the Koopman operator have advanced considerably in the last few years. In particular, data-driven approaches such as dynamic mode decomposition (DMD) and its generalization, the extended-DMD (EDMD), are becoming increasingly popular in practical applications. The EDMD improves upon the classical DMD by the inclusion of a flexible choice of dictionary of observables which spans a finite dimensional subspace on which the Koopman operator can be approximated. This enhances the accuracy of the solution reconstruction and broadens the applicability of the Koopman formalism. Although the convergence of the EDMD has been established, applying the method in practice requires a careful choice of the observables to improve convergence with just a finite number of terms. This is especially difficult for high dimensional and highly nonlinear systems. In this paper, we employ ideas from machine learning to improve upon the EDMD method. We develop an iterative approximation algorithm which couples the EDMD with a trainable dictionary represented by an artificial neural network. Using the Duffing oscillator and the Kuramoto Sivashinsky partical differential equation as examples, we show that our algorithm can effectively and efficiently adapt the trainable dictionary to the problem at hand to achieve good reconstruction accuracy without the need to choose a fixed dictionary a priori. Furthermore, to obtain a given accuracy, we require fewer dictionary terms than EDMD with fixed dictionaries. This alleviates an important shortcoming of the EDMD algorithm and enhances the applicability of the Koopman framework to practical problems.

Concise, accurate descriptions of physical systems through their conserved quantities abound in the natural sciences. In data science, however, current research often focuses on regression problems, without routinely incorporating additional assumptions about the system that generated the data. Here, we propose to explore a particular type of underlying structure in the data: Hamiltonian systems, where an "energy" is conserved. Given a collection of observations of such a Hamiltonian system over time, we extract phase space coordinates and a Hamiltonian function of them that acts as the generator of the system dynamics. The approach employs an auto-encoder neural network component to estimate the transformation from observations to the phase space of a Hamiltonian system. An additional neural network component is used to approximate the Hamiltonian function on this constructed space, and the two components are trained simultaneously. As an alternative approach, we also demonstrate the use of Gaussian processes for the estimation of such a Hamiltonian. After two illustrative examples, we extract an underlying phase space as well as the generating Hamiltonian from a collection of movies of a pendulum. The approach is fully data-driven, and does not assume a particular form of the Hamiltonian function.

A core prediction of the vesicular transport model is that COPI vesicles are responsible for trafficking anterograde cargoes forward. In this study, we test this prediction by examining the properties and requirements of inter-Golgi transport within fused cells, which requires mobile carriers in order for exchange of constituents to occur. We report that both small soluble and membrane-bound secretory cargo and exogenous Golgi resident glycosyl-transferases are exchanged between separated Golgi. Large soluble aggregates, which traverse individual stacks, do not transfer between Golgi, implying that small cargoes (which can fit in a typical transport vesicle) are transported by a different mechanism. Super-resolution microscopy reveals that the carriers of both anterograde and retrograde cargoes are the size of COPI vesicles, contain coatomer, and functionally require ARF1 and coatomer for transport. The data suggest that COPI vesicles traffic both small secretory cargo and steady-state Golgi resident enzymes among stacked cisternae that are stationary.DOI: http://dx.doi.org/10.7554/eLife.01296.001

We present a new microscopic ODE-based model for pedestrian dynamics: the Gradient Navigation Model. The model uses a superposition of gradients of distance functions to directly change the direction of the velocity vector. The velocity is then integrated to obtain the location. The approach differs fundamentally from force based models needing only three equations to derive the ODE system, as opposed to four in, e.g., the Social Force Model. Also, as a result, pedestrians are no longer subject to inertia. Several other advantages ensue: Model induced oscillations are avoided completely since no actual forces are present. The derivatives in the equations of motion are smooth and therefore allow the use of fast and accurate high order numerical integrators. At the same time, existence and uniqueness of the solution to the ODE system follow almost directly from the smoothness properties. In addition, we introduce a method to calibrate parameters by theoretical arguments based on empirically validated assumptions rather than by numerical tests.These parameters, combined with the accurate integration, yield simulation results with no collisions of pedestrians. Several empirically observed system phenomena emerge without the need to recalibrate the parameter set for each scenario: obstacle avoidance, lane formation, stop-and-go waves and congestion at bottlenecks. The density evolution in the latter is shown to be quantitatively close to controlled experiments. Likewise, we observe a dependence of the crowd velocity on the local density that compares well with benchmark fundamental diagrams.

A novel direction of arrival (DOA) technique is presented which constructs estimates of the relative delay mixing parameters associated with each signal by taking the ratio of time-frequency representations of two mixtures. The technique is based on the Degenerate Unmixing and Estimation Technique (DUET) [l]. If the sources are W-disjoint orthogonal, meaning that only one signal is active in the timefrequency plane a t a given time-frequency, then the ratio only depends on the mixing parameters of one source. The ratio can thus be used to generate estimates of the mixing parameters and these estimates can be clustered to determine both the number of sources present in the mixtures and their associated mixing parameters. The method allows for the estimation of the DOA for many sources using only two receive antennas, whereas traditional techniques require N antennas to estimate N -1 angles of arrival.Simulation results are presented and compared to MUSIC, ESPRIT, and other DOA estimation techniques. I N T R O D U C T I O NThe goal of accurately estimating the arrival angle of a signal on an antenna array is long standing in the field of signal processing. Direction of arrival estimation is important for such tasks as tracking the signal emitter and smart antenna array processing for interference reduction in mobile wireless systems.Most DOA techniques require N antennas to estimate N -1 angles of arrival. A notable exception to the N -1 angles of arrival rule uses forth-order cumulants to estimate three time delays from two mixtures[2]. One advantage of the technique presented here is that it requires only two antenna elements to estimate the arrival angle of an arbitrary number of sources. This reduction in the required number of antenna elements is made by assuming the sources are W-disjoint orthogonal.This paper applies the work on the Degenerate Unmixing and Estimation Technique(DUET) on W-disjoint orthogonal signals originally proposed in [3] to wireless signals. W-disjoint orthogonal signals have disjoint support for their time-frequency representation. For example, multiple M-ary frequency shift keyed signals are W-disjoint orthogonal, except for the occasional hit when two or more signals transmit a t the same frequency a t the same time. Another (perhaps surprising) example of W-disjoint orthogonal signals is speech. Tests show that voice data satisfies the Wdisjoint orthogonality constraint closely enough to allow accurate angle of arrival estimation and blind separation [3,4].In essence, the W-disjoint orthogonal assumption assumes that all signals are instantaneously separated in the frequency domain. Thus the technique presented herein would not work, for example, when the signals are sinusoids modulated a t exactly the same frequency as the signals in that case would not be W-disjoint orthogonal.Note that one could employ a bank of narrow bandpass filters to create a number of narrowband signal channels and use the DOA estimation schemes described in the above overview literature. In this case, with ...

Mesoporous silica MCM-41 is prepared, for which the inner surfaces are modified by 3-(aminopropyl)triethoxysilane (APTES) in a controlled manner. Nitrogen gas adsorpition yields a pore diameter of 2.2 nm for the APTES functionalized MCM-41.2H nuclear magnetic resonance (NMR) and broadband dielectric spectroscopy (BDS) provide detailed and consistent insights into the temperature-dependent reorientation dynamics of water in this confinement. We find that a liquid water species becomes accompanied by a solid water species when cooling through ~210 K, as indicated by an onset of bimodal2H spin-lattice relaxation. The reorientation of the liquid water species is governed by pronounced dynamical heterogeneity in the whole temperature range. Its temperature dependence shows a mild dynamic crossover when the solid water species emerges and, hence, the volume accessible to the liquid water species further shrinks. Therefore, we attribute this variation in the temperature dependence to a change from bulk-like behavior towards interface-dominated dynamics. Below this dynamic crossover,2H line-shape and stimulted-echo studies show that water reorientation becomes anisotropic upon cooling, suggesting that these NMR approaches, but also BDS measurements do no longer probe the structural (α) relaxation, but rather a secondary (β) relaxation of water at sufficiently low temperatures. Then, another dynamic crossover at ~180 K can be rationalized in terms of a change of the temperature dependence of theβrelaxation in response to a glassy freezing of theαrelaxation of confined water. Comparing these results for APTES modied MCM-41 with previous findings for mesoporous silica with various pore diameters, we obtain valuable information about the dependence of water dynamics in restricted geometries on the size of the nanoscopic confinements and the properties of the inner surfaces.

Matching dynamical systems, through different forms of conjugacies and equivalences, has long been a fundamental concept, and a powerful tool, in the study and classification of nonlinear dynamic behavior (e.g. through normal forms). In this paper we will argue that the use of the Koopman operator and its spectrum is particularly well suited for this endeavor, both in theory, but also especially in view of recent data-driven algorithm developments. We believe, and document through illustrative examples, that this can nontrivially extend the use and applicability of the Koopman spectral theoretical and computational machinery beyond modeling and prediction, towards what can be considered as a systematic discovery of "Cole-Hopf-type" transformations for dynamics.

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