A Class of Logistic Functions for Approximating State-Inclusive Koopman Operators

An outstanding challenge in nonlinear systems theory is identification or learning of a given nonlinear system's Koopman operator directly from data or models. Advances in extended dynamic mode decomposition approaches and machine learning methods have enabled data-driven discovery of Koopman operators, for both continuous and discrete-time systems. Since Koopman operators are often infinite-dimensional, they are approximated in practice using finite-dimensional systems. The fidelity and convergence of a given finite-dimensional Koopman approximation is a subject of ongoing research. In this paper we introduce a class of Koopman observable functions that confer an approximate closure property on their corresponding finite-dimensional approximations of the Koopman operator. We derive error bounds for the fidelity of this class of observable functions, as well as identify two key learning parameters which can be used to tune performance. We illustrate our approach on two classical nonlinear system models: the Van Der Pol oscillator and the bistable toggle switch.Comment: 8 page

Thermoacoustic instability - a dynamical system and time domain analysis

This study focuses on the Rijke tube problem, which includes features relevant to the modeling of thermoacoustic coupling in reactive flows: a compact acoustic source, an empirical model for the heat source, and nonlinearities. This thermo-acoustic system features a complex dynamical behavior. In order to synthesize accurate time-series, we tackle this problem from a numerical point-of-view, and start by proposing a dedicated solver designed for dealing with the underlying stiffness, in particular, the retarded time and the discontinuity at the location of the heat source. Stability analysis is performed on the limit of low-amplitude disturbances by means of the projection method proposed by Jarlebring (2008), which alleviates the linearization with respect to the retarded time. The results are then compared to the analytical solution of the undamped system, and to Galerkin projection methods commonly used in this setting. This analysis provides insight into the consequences of the various assumptions and simplifications that justify the use of Galerkin expansions based on the eigenmodes of the unheated resonator. We illustrate that due to the presence of a discontinuity in the spatial domain, the eigenmodes in the heated case, predicted by using Galerkin expansion, show spurious oscillations resulting from the Gibbs phenomenon. By comparing the modes of the linear to that of the nonlinear regime, we are able to illustrate the mean-flow modulation and frequency switching. Finally, time-series in the fully nonlinear regime, where a limit cycle is established, are analyzed and dominant modes are extracted. The analysis of the saturated limit cycles shows the presence of higher frequency modes, which are linearly stable but become significant through nonlinear growth of the signal. This bimodal effect is not captured when the coupling between different frequencies is not accounted for.Comment: Submitted to Journal of Fluid Mechanic

A Computational Study of a Data Assimilation Algorithm for the Two-dimensional Navier--Stokes Equations

We study the numerical performance of a continuous data assimilation (downscaling) algorithm, based on ideas from feedback control theory, in the context of the two-dimensional incompressible Navier--Stokes equations. Our model problem is to recover an unknown reference solution, asymptotically in time, by using continuous-in-time coarse-mesh nodal-point observational measurements of the velocity field of this reference solution (subsampling), as might be measured by an array of weather vane anemometers. Our calculations show that the required nodal observation density is remarkably less that what is suggested by the analytical study; and is in fact comparable to the {\it number of numerically determining Fourier modes}, which was reported in an earlier computational study by the authors. Thus, this method is computationally efficient and performs far better than the analytical estimates suggest

A Data-Driven Approximation of the Koopman Operator: Extending Dynamic Mode Decomposition

The Koopman operator is a linear but infinite dimensional operator that governs the evolution of scalar observables defined on the state space of an autonomous dynamical system, and is a powerful tool for the analysis and decomposition of nonlinear dynamical systems. In this manuscript, we present a data driven method for approximating the leading eigenvalues, eigenfunctions, and modes of the Koopman operator. The method requires a data set of snapshot pairs and a dictionary of scalar observables, but does not require explicit governing equations or interaction with a "black box" integrator. We will show that this approach is, in effect, an extension of Dynamic Mode Decomposition (DMD), which has been used to approximate the Koopman eigenvalues and modes. Furthermore, if the data provided to the method are generated by a Markov process instead of a deterministic dynamical system, the algorithm approximates the eigenfunctions of the Kolmogorov backward equation, which could be considered as the "stochastic Koopman operator" [1]. Finally, four illustrative examples are presented: two that highlight the quantitative performance of the method when presented with either deterministic or stochastic data, and two that show potential applications of the Koopman eigenfunctions

Discrete-Time Recurrent Neural Network and Its Application to Compression of Infra-Red Spectrum

We study the discrete-time recurrent neural network that derived from the Leaky-integrator model and its application to compression of infra-red spec- trum. Our results show that the discrete-time Leaky-integrator recurrent neural network (RNN) model can be used to approximate the continuous-time model and inherit its dynamical characters if a proper step size is chosen. Moreover, the discrete-time Leaky-integrator RNN model is absolutely stable. By developing the double discrete integral method and employing the state space search algorithm for the discrete-time recurrent neural network model, we demonstrate with quality spectra regenerated from the compressed data how to compress the infra-red spectrum effectively. The information we stored is the parameters of the system and its initial states. The method offers an ideal setting to carry out the recurrent neural network approach to chaotic cases of data compression