arXiv subjects

Bingni W. Brunton

Publications and source records attributed to Bingni W. Brunton.

29 records · Page 2Linked to original sources

arXiv2020

Phase-based control of periodic fluid flows

Fluid flows play a central role in scientific and technological development, and many of these flows are characterized by a dominant oscillation, such as the vortex shedding in the wake of nearly all transportation vehicles. The ability to control vortex shedding is critical to improve the aerodynamic performance of these unsteady fluid flow systems. This goal requires precise characterization of how perturbations affect the long-time phase of the oscillatory flow, as well as the ability to control transient behaviors. In this work, we develop an energy-efficient flow control strategy to rapidly alter the oscillation phase of time-periodic fluid flows, leveraging theory developed for periodic biological systems. First, we perform a phase-sensitivity analysis to construct a reduced-order model for the response of the flow oscillation to impulsive control inputs at various phases. Next, we introduce two control strategies for real-time phase control based on the phase-sensitivity function: 1) optimal phase control, obtained by solving the Euler-Lagrange equations as a two-point boundary value problem, and 2) model-predictive control (MPC). Our approach is demonstrated for two unsteady flow systems, the incompressible laminar flow past a circular cylinder and the flow past an airfoil. We show that effective phase control may be achieved with several actuation strategies, including blowing and rotary control. Moreover, our control approach uses realistic measurements of the lift force on the body, rather than requiring high-dimensional measurements of the full flow field.

physics.flu-dyn↗

arXiv2020

Phase-consistent dynamic mode decomposition from multiple overlapping spatial domains

Dynamic mode decomposition (DMD) provides a principled approach to extract physically interpretable spatial modes from time-resolved flow field data, along with a linear model for how the amplitudes of these modes evolve in time. Recently, DMD has been extended to work with more realistic data that is under-resolved either in time or space, or with data collected in the same spatial domain over multiple independent time windows. In this work, we develop an extension to DMD to synthesize globally consistent modes from velocity fields collected independently in multiple partially overlapping spatial domains. We propose a tractable optimization to identify modes that span multiple windows and align their phases to be consistent in the overlapping regions. First, we demonstrate this approach on data from direct numerical simulation, where it is possible to split the data into overlapping domains and benchmark against ground-truth modes. We consider the laminar flow past a cylinder as an example with distinct frequencies, along with the spatially developing mixing layer, which exhibits a frequency spectrum that evolves continuously as the measurement window moves downstream. Next, we analyze experimental velocity fields from PIV in six overlapping domains in the wake of a cross-flow turbine. On the numerical examples, we demonstrate the robustness of this approach to increasing measurement noise and decreasing size of the overlap regions. In all cases, it is possible to obtain a phase-aligned, composite reconstruction of the full time-resolved flow field from the phase-consistent modes over the entire domain.

physics.flu-dyn↗

arXiv2019

Centering Data Improves the Dynamic Mode Decomposition

Dynamic mode decomposition (DMD) is a data-driven method that models high-dimensional time series as a sum of spatiotemporal modes, where the temporal modes are constrained by linear dynamics. For nonlinear dynamical systems exhibiting strongly coherent structures, DMD can be a useful approximation to extract dominant, interpretable modes. In many domains with large spatiotemporal data---including fluid dynamics, video processing, and finance---the dynamics of interest are often perturbations about fixed points or equilibria, which motivates the application of DMD to centered (i.e. mean-subtracted) data. In this work, we show that DMD with centered data is equivalent to incorporating an affine term in the dynamic model and is not equivalent to computing a discrete Fourier transform. Importantly, DMD with centering can always be used to compute eigenvalue spectra of the dynamics. However, in many cases DMD without centering cannot model the corresponding dynamics, most notably if the dynamics have full effective rank. Additionally, we generalize the notion of centering to extracting arbitrary, but known, fixed frequencies from the data. We corroborate these theoretical results numerically on three nonlinear examples: the Lorenz system, a surveillance video, and brain recordings. Since centering the data is simple and computationally efficient, we recommend it as a preprocessing step before DMD; furthermore, we suggest that it can be readily used in conjunction with many other popular implementations of the DMD algorithm.

math.DS↗

arXiv2018

Data-driven Spatiotemporal Modal Decomposition for Time Frequency Analysis

We propose a new solution to the blind source separation problem that factors mixed time-series signals into a sum of spatiotemporal modes, with the constraint that the temporal components are intrinsic mode functions (IMF's). The key motivation is that IMF's allow the computation of meaningful Hilbert transforms of non-stationary data, from which instantaneous time-frequency representations may be derived. Our spatiotemporal intrinsic mode decomposition (STIMD) method leverages spatial correlations to generalize the extraction of IMF's from one-dimensional signals, commonly performed using the empirical mode decomposition (EMD), to multi-dimensional signals. Further, this data-driven method enables future-state prediction. We demonstrate STIMD on several synthetic examples, comparing it to common matrix factorization techniques, namely singular value decomposition (SVD), independent component analysis (ICA), and dynamic mode decomposition (DMD). We show that STIMD outperforms these methods at reconstruction and extracting interpretable modes. Next, we apply STIMD to analyze two real-world datasets, gravitational wave data and neural recordings from the rodent hippocampus.

math.NA↗

arXiv2018

Neural-inspired sensors enable sparse, efficient classification of spatiotemporal data

Sparse sensor placement is a central challenge in the efficient characterization of complex systems when the cost of acquiring and processing data is high. Leading sparse sensing methods typically exploit either spatial or temporal correlations, but rarely both. This work introduces a new sparse sensor optimization that is designed to leverage the rich spatiotemporal coherence exhibited by many systems. Our approach is inspired by the remarkable performance of flying insects, which use a few embedded strain-sensitive neurons to achieve rapid and robust flight control despite large gust disturbances. Specifically, we draw on nature to identify targeted neural-inspired sensors on a flapping wing to detect body rotation. This task is particularly challenging as the rotational twisting mode is three orders-of-magnitude smaller than the flapping modes. We show that nonlinear filtering in time, built to mimic strain-sensitive neurons, is essential to detect rotation, whereas instantaneous measurements fail. Optimized sparse sensor placement results in efficient classification with approximately ten sensors, achieving the same accuracy and noise robustness as full measurements consisting of hundreds of sensors. Sparse sensing with neural inspired encoding establishes a new paradigm in hyper-efficient, embodied sensing of spatiotemporal data and sheds light on principles of biological sensing for agile flight control.

math.OC↗

arXiv2017

Data-Driven Sparse Sensor Placement for Reconstruction

Optimal sensor placement is a central challenge in the design, prediction, estimation, and control of high-dimensional systems. High-dimensional states can often leverage a latent low-dimensional representation, and this inherent compressibility enables sparse sensing. This article explores optimized sensor placement for signal reconstruction based on a tailored library of features extracted from training data. Sparse point sensors are discovered using the singular value decomposition and QR pivoting, which are two ubiquitous matrix computations that underpin modern linear dimensionality reduction. Sparse sensing in a tailored basis is contrasted with compressed sensing, a universal signal recovery method in which an unknown signal is reconstructed via a sparse representation in a universal basis. Although compressed sensing can recover a wider class of signals, we demonstrate the benefits of exploiting known patterns in data with optimized sensing. In particular, drastic reductions in the required number of sensors and improved reconstruction are observed in examples ranging from facial images to fluid vorticity fields. Principled sensor placement may be critically enabling when sensors are costly and provides faster state estimation for low-latency, high-bandwidth control. MATLAB code is provided for all examples.

math.OC↗

arXiv2016

Chaos as an Intermittently Forced Linear System

Understanding the interplay of order and disorder in chaotic systems is a central challenge in modern quantitative science. We present a universal, data-driven decomposition of chaos as an intermittently forced linear system. This work combines Takens' delay embedding with modern Koopman operator theory and sparse regression to obtain linear representations of strongly nonlinear dynamics. The result is a decomposition of chaotic dynamics into a linear model in the leading delay coordinates with forcing by low energy delay coordinates; we call this the Hankel alternative view of Koopman (HAVOK) analysis. This analysis is applied to the canonical Lorenz system, as well as to real-world examples such as the Earth's magnetic field reversal, and data from electrocardiogram, electroencephalogram, and measles outbreaks. In each case, the forcing statistics are non-Gaussian, with long tails corresponding to rare events that trigger intermittent switching and bursting phenomena; this forcing is highly predictive, providing a clear signature that precedes these events. Moreover, the activity of the forcing signal demarcates large coherent regions of phase space where the dynamics are approximately linear from those that are strongly nonlinear.

math.DS↗

arXiv2016

Multistep Model for Predicting Upper-Limb 3D Isometric Force Application from Pre-Movement Electrocorticographic Features

Neural correlates of movement planning onset and direction may be present in human electrocorticography in the signal dynamics of both motor and non-motor cortical regions. We use a three-stage model of jPCA reduced-rank hidden Markov model (jPCA-RR-HMM), regularized shrunken-centroid discriminant analysis (RDA), and LASSO regression to extract direction-sensitive planning information and movement onset in an upper-limb 3D isometric force task in a human subject. This mode achieves a relatively high true positive force-onset prediction rate of 60% within 250ms, and an above-chance 36% accuracy (17% chance) in predicting one of six planned 3D directions of isometric force using pre-movement signals. We also find direction-distinguishing information up to 400ms before force onset in the pre-movement signals, captured by electrodes placed over the limb-ipsilateral dorsal premotor regions. This approach can contribute to more accurate decoding of higher-level movement goals, at earlier timescales, and inform sensor placement. Our results also contribute to further understanding of the spatiotemporal features of human motor planning.

q-bio.NC↗

arXiv2015

Koopman invariant subspaces and finite linear representations of nonlinear dynamical systems for control

In this work, we explore finite-dimensional linear representations of nonlinear dynamical systems by restricting the Koopman operator to an invariant subspace. The Koopman operator is an infinite-dimensional linear operator that evolves observable functions of the state-space of a dynamical system [Koopman 1931, PNAS]. Dominant terms in the Koopman expansion are typically computed using dynamic mode decomposition (DMD). DMD uses linear measurements of the state variables, and it has recently been shown that this may be too restrictive for nonlinear systems [Williams et al. 2015, JNLS]. Choosing nonlinear observable functions to form an invariant subspace where it is possible to obtain linear models, especially those that are useful for control, is an open challenge. Here, we investigate the choice of observable functions for Koopman analysis that enable the use of optimal linear control techniques on nonlinear problems. First, to include a cost on the state of the system, as in linear quadratic regulator (LQR) control, it is helpful to include these states in the observable subspace, as in DMD. However, we find that this is only possible when there is a single isolated fixed point, as systems with multiple fixed points or more complicated attractors are not globally topologically conjugate to a finite-dimensional linear system, and cannot be represented by a finite-dimensional linear Koopman subspace that includes the state. We then present a data-driven strategy to identify relevant observable functions for Koopman analysis using a new algorithm to determine terms in a dynamical system by sparse regression of the data in a nonlinear function space [Brunton et al. 2015, arxiv]; we show how this algorithm is related to DMD. Finally, we demonstrate how to design optimal control laws for nonlinear systems using techniques from linear optimal control on Koopman invariant subspaces.

math.DS↗

arXiv2015

Unsupervised decoding of long-term, naturalistic human neural recordings with automated video and audio annotations

Fully automated decoding of human activities and intentions from direct neural recordings is a tantalizing challenge in brain-computer interfacing. Most ongoing efforts have focused on training decoders on specific, stereotyped tasks in laboratory settings. Implementing brain-computer interfaces (BCIs) in natural settings requires adaptive strategies and scalable algorithms that require minimal supervision. Here we propose an unsupervised approach to decoding neural states from human brain recordings acquired in a naturalistic context. We demonstrate our approach on continuous long-term electrocorticographic (ECoG) data recorded over many days from the brain surface of subjects in a hospital room, with simultaneous audio and video recordings. We first discovered clusters in high-dimensional ECoG recordings and then annotated coherent clusters using speech and movement labels extracted automatically from audio and video recordings. To our knowledge, this represents the first time techniques from computer vision and speech processing have been used for natural ECoG decoding. Our results show that our unsupervised approach can discover distinct behaviors from ECoG data, including moving, speaking and resting. We verify the accuracy of our approach by comparing to manual annotations. Projecting the discovered cluster centers back onto the brain, this technique opens the door to automated functional brain mapping in natural settings.

q-bio.NC↗

arXiv2014

Extracting spatial-temporal coherent patterns in large-scale neural recordings using dynamic mode decomposition

There is a broad need in the neuroscience community to understand and visualize large-scale recordings of neural activity, big data acquired by tens or hundreds of electrodes simultaneously recording dynamic brain activity over minutes to hours. Such dynamic datasets are characterized by coherent patterns across both space and time, yet existing computational methods are typically restricted to analysis either in space or in time separately. Here we report the adaptation of dynamic mode decomposition (DMD), an algorithm originally developed for the study of fluid physics, to large-scale neuronal recordings. DMD is a modal decomposition algorithm that describes high-dimensional dynamic data using coupled spatial-temporal modes; the resulting analysis combines key features of performing principal components analysis (PCA) in space and power spectral analysis in time. The algorithm scales easily to very large numbers of simultaneously acquired measurements. We validated the DMD approach on sub-dural electrode array recordings from human subjects performing a known motor activation task. Next, we leveraged DMD in combination with machine learning to develop a novel method to extract sleep spindle networks from the same subjects. We suggest that DMD is generally applicable as a powerful method in the analysis and understanding of large-scale recordings of neural activity.

q-bio.NC↗