We consider the application of Koopman theory to nonlinear partial differential equations and data-driven spatio-temporal systems. We demonstrate that the observables chosen for constructing the ...Koopman operator are critical for enabling an accurate approximation to the nonlinear dynamics. If such observables can be found, then the dynamic mode decomposition (DMD) algorithm can be enacted to compute a finite-dimensional approximation of the Koopman operator, including its eigenfunctions, eigenvalues, and Koopman modes. We demonstrate simple rules of thumb for selecting a parsimonious set of observables that can greatly improve the approximation of the Koopman operator. Further, we show that the clear goal in selecting observables is to place the DMD eigenvalues on the imaginary axis, thus giving an objective function for observable selection. Judiciously chosen observables lead to physically interpretable spatio-temporal features of the complex system under consideration and provide a connection to manifold learning methods. Our method provides a valuable intermediate, yet interpretable, approximation to the Koopman operator that lies between the DMD method and the computationally intensive extended DMD (EDMD). We demonstrate the impact of observable selection, including kernel methods, and construction of the Koopman operator on several canonical nonlinear PDEs: Burgers’ equation, the nonlinear Schrödinger equation, the cubic-quintic Ginzburg-Landau equation, and a reaction-diffusion system. These examples serve to highlight the most pressing and critical challenge of Koopman theory: a principled way to select appropriate observables.
Implantable-cardioverter defibrillators (ICD) detect and terminate life-threatening ventricular tachyarrhythmia with electric shocks after they occur. This puts patients at risk if they are driving ...or in a situation where they can fall. ICD's shocks are also very painful and affect a patient's quality of life. It would be ideal if ICDs can accurately predict the occurrence of ventricular tachyarrhythmia and then issue a warning or provide preventive therapy. Our study explores the use of ICD data to automatically predict ventricular arrhythmia using heart rate variability (HRV). A 5 minute and a 10 second warning system are both developed and compared. The participants for this study consist of 788 patients who were enrolled in the ICD arm of the Sudden Cardiac Death-Heart Failure Trial (SCD-HeFT). Two groups of patient rhythms, regular heart rhythms and pre-ventricular-tachyarrhythmic rhythms, are analyzed and different HRV features are extracted. Machine learning algorithms, including random forests (RF) and support vector machines (SVM), are trained on these features to classify the two groups of rhythms in a subset of the data comprising the training set. These algorithms are then used to classify rhythms in a separate test set. This performance is quantified by the area under the curve (AUC) of the ROC curve. Both RF and SVM methods achieve a mean AUC of 0.81 for 5-minute prediction and mean AUC of 0.87-0.88 for 10-second prediction; an AUC over 0.8 typically warrants further clinical investigation. Our work shows that moderate classification accuracy can be achieved to predict ventricular tachyarrhythmia with machine learning algorithms using HRV features from ICD data. These results provide a realistic view of the practical challenges facing implementation of machine learning algorithms to predict ventricular tachyarrhythmia using HRV data, motivating continued research on improved algorithms and additional features with higher predictive power.
Celotno besedilo
Dostopno za:
DOBA, IZUM, KILJ, NUK, PILJ, PNG, SAZU, SIK, UILJ, UKNU, UL, UM, UPUK
There is a large body of functional data that supports the existence of subcellular compartmentation of the components of cyclic AMP action in the heart. Data from isolated perfused hearts and from ...purified ventricular myocytes imply a fixed and hormone-specific spatial relationship amongst components of cyclic AMP synthesis, response, and degradation. Available data demonstrate that within a cardiac myocyte, not all cyclic AMP gains access to all cyclic AMP-dependent protein kinase (PKA), that not all PKA interacts with all possible cellular substrates of PKA, and that only a subset of the myocyte's phosphodiesterases (PDEs) may degrade cyclic AMP after a given synthetic stimulus. Molecular mechanisms contributing to compartmentation are being discovered: localization of receptors, G proteins, and adenylyl cyclases in caveolar versus noncaveolar regions of the sarcolemma; localization of PKA by A-kinase anchoring proteins; localization of PKA substrates, PDE isoforms, and phosphoprotein phosphatases in discrete subcellular regions; and differential regulation of multiple isoforms of adenylyl cyclase, phosphoprotein phosphatase, and PDE in distinct subcellular compartments.
Celotno besedilo
Dostopno za:
DOBA, IZUM, KILJ, NUK, PILJ, PNG, SAZU, UILJ, UKNU, UL, UM, UPUK
Sparse model identification enables the discovery of nonlinear dynamical systems purely from data; however, this approach is sensitive to noise, especially in the low-data limit. In this work, we ...leverage the statistical approach of bootstrap aggregating (bagging) to robustify the sparse identification of the nonlinear dynamics (SINDy) algorithm. First, an ensemble of SINDy models is identified from subsets of limited and noisy data. The aggregate model statistics are then used to produce inclusion probabilities of the candidate functions, which enables uncertainty quantification and probabilistic forecasts. We apply this ensemble-SINDy (E-SINDy) algorithm to several synthetic and real-world datasets and demonstrate substantial improvements to the accuracy and robustness of model discovery from extremely noisy and limited data. For example, E-SINDy uncovers partial differential equations models from data with more than twice as much measurement noise as has been previously reported. Similarly, E-SINDy learns the Lotka Volterra dynamics from remarkably limited data of yearly lynx and hare pelts collected from 1900 to 1920. E-SINDy is computationally efficient, with similar scaling as standard SINDy. Finally, we show that ensemble statistics from E-SINDy can be exploited for active learning and improved model predictive control.
Despite recent achievements in the design and manufacture of advanced materials, the contributions from first-principles modeling and simulation have remained limited, especially in regards to ...characterizing how macroscopic properties depend on the heterogeneous microstructure. An improved ability to model and understand these multiscale and anisotropic effects will be critical in designing future materials, especially given rapid improvements in the enabling technologies of additive manufacturing and active metamaterials. In this review, we discuss recent progress in the data-driven modeling of dynamical systems using machine learning and sparse optimization to generate parsimonious macroscopic models that are generalizable and interpretable. Such improvements in model discovery will facilitate the design and characterization of advanced materials by improving efforts in (1) molecular dynamics, (2) obtaining macroscopic constitutive equations, and (3) optimization and control of metamaterials.
We demonstrate the effective use of randomized methods for linear algebra to perform network-based analysis of complex vortical flows. Network theoretic approaches can reveal the connectivity ...structures among a set of vortical elements and analyze their collective dynamics. These approaches have recently been generalized to analyze high-dimensional turbulent flows, for which network computations can become prohibitively expensive. In this work, we propose efficient methods to approximate network quantities, such as the leading eigendecomposition of the adjacency matrix, using randomized methods. Specifically, we use the Nyström method to approximate the leading eigenvalues and eigenvectors, achieving significant computational savings and reduced memory requirements. The effectiveness of the proposed technique is demonstrated on two high-dimensional flow fields: two-dimensional flow past an airfoil and two-dimensional turbulence. We find that quasi-uniform column sampling outperforms uniform column sampling, while both feature the same computational complexity.
Celotno besedilo
Dostopno za:
DOBA, IZUM, KILJ, NUK, PILJ, PNG, SAZU, SIK, UILJ, UKNU, UL, UM, UPUK
Finite-time Lyapunov exponents (FTLEs) provide a powerful approach to compute time-varying analogs of invariant manifolds in unsteady fluid flow fields. These manifolds are useful to visualize the ...transport mechanisms of passive tracers advecting with the flow. However, many vehicles and mobile sensors are not passive, but are instead actuated according to some intelligent trajectory planning or control law; for example, model predictive control and reinforcement learning are often used to design energy-efficient trajectories in a dynamically changing background flow. In this work, we investigate the use of FTLE on such controlled agents to gain insight into optimal transport routes for navigation in known unsteady flows. We find that these controlled FTLE (cFTLE) coherent structures separate the flow field into different regions with similar costs of transport to the goal location. These separatrices are functions of the planning algorithm's hyper-parameters, such as the optimization time horizon and the cost of actuation. Computing the invariant sets and manifolds of active agent dynamics in dynamic flow fields is useful in the context of robust motion control, hyperparameter tuning, and determining safe and collision-free trajectories for autonomous systems. Moreover, these cFTLE structures provide insight into effective deployment locations for mobile agents with actuation and energy constraints to traverse the ocean or atmosphere.
PDF
