Spectral analysis of the Koopman operator for partial differential equations.

We provide an overview of the Koopman-operator analysis for a class of partial differential equations describing relaxation of the field variable to a stable stationary state. We introduce Koopman eigenfunctionals of the system and use the notion of conjugacy to develop spectral expansion of the Koopman operator. For linear systems such as the diffusion equation, the Koopman eigenfunctionals can be expressed as linear functionals of the field variable. The notion of inertial manifolds is shown to correspond to joint zero level sets of Koopman eigenfunctionals, and the notion of isostables is defined as the level sets of the slowest decaying Koopman eigenfunctional. Linear diffusion equation, nonlinear Burgers equation, and nonlinear phase-diffusion equation are analyzed as examples.

Predicting the Critical Number of Layers for Hierarchical Support Vector Regression.

Hierarchical support vector regression (HSVR) models a function from data as a linear combination of SVR models at a range of scales, starting at a coarse scale and moving to finer scales as the hierarchy continues. In the original formulation of HSVR, there were no rules for choosing the depth of the model. In this paper, we observe in a number of models a phase transition in the training error-the error remains relatively constant as layers are added, until a critical scale is passed, at which point the training error drops close to zero and remains nearly constant for added layers. We introduce a method to predict this critical scale a priori with the prediction based on the support of either a Fourier transform of the data or the Dynamic Mode Decomposition (DMD) spectrum. This allows us to determine the required number of layers prior to training any models.

On learning Hamiltonian systems from data.

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 autoencoder 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 jointly. 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.

Dynamic autoinoculation and the microbial ecology of a deep water hydrocarbon irruption.

The irruption of gas and oil into the Gulf of Mexico during the Deepwater Horizon event fed a deep sea bacterial bloom that consumed hydrocarbons in the affected waters, formed a regional oxygen anomaly, and altered the microbiology of the region. In this work, we develop a coupled physical-metabolic model to assess the impact of mixing processes on these deep ocean bacterial communities and their capacity for hydrocarbon and oxygen use. We find that observed biodegradation patterns are well-described by exponential growth of bacteria from seed populations present at low abundance and that current oscillation and mixing processes played a critical role in distributing hydrocarbons and associated bacterial blooms within the northeast Gulf of Mexico. Mixing processes also accelerated hydrocarbon degradation through an autoinoculation effect, where water masses, in which the hydrocarbon irruption had caused blooms, later returned to the spill site with hydrocarbon-degrading bacteria persisting at elevated abundance. Interestingly, although the initial irruption of hydrocarbons fed successive blooms of different bacterial types, subsequent irruptions promoted consistency in the structure of the bacterial community. These results highlight an impact of mixing and circulation processes on biodegradation activity of bacteria during the Deepwater Horizon event and suggest an important role for mixing processes in the microbial ecology of deep ocean environments.

Ergodic theory and visualization. II. Fourier mesochronic plots visualize (quasi)periodic sets.

We present an application and analysis of a visualization method for measure-preserving dynamical systems introduced by I. Mezic and A. Banaszuk [Physica D 197, 101 (2004)], based on frequency analysis and Koopman operator theory. This extends our earlier work on visualization of ergodic partition [Z. Levnajic and I. Mezic, Chaos 20, 033114 (2010)]. Our method employs the concept of Fourier time average [I. Mezic and A. Banaszuk, Physica D 197, 101 (2004)], and is realized as a computational algorithms for visualization of periodic and quasi-periodic sets in the phase space. The complement of periodic phase space partition contains chaotic zone, and we show how to identify it. The range of method's applicability is illustrated using well-known Chirikov standard map, while its potential in illuminating higher-dimensional dynamics is presented by studying the Froeschlé map and the Extended Standard Map.

A computational physiology approach to personalized treatment models: the beneficial effects of slow breathing on the human cardiovascular system.

Heart rate variability biofeedback intervention involves slow breathing at a rate of ∼6 breaths/min (resonance breathing) to maximize respiratory and baroreflex effects on heart period oscillations. This intervention has wide-ranging clinical benefits and is gaining empirical support as an adjunct therapy for biobehavioral disorders, including asthma and depression. Yet, little is known about the system-level cardiovascular changes that occur during resonance breathing or the extent to which individuals differ in cardiovascular benefit. This study used a computational physiology approach to dynamically model the human cardiovascular system at rest and during resonance breathing. Noninvasive measurements of heart period, beat-to-beat systolic and diastolic blood pressure, and respiration period were obtained from 24 healthy young men and women. A model with respiration as input was parameterized to better understand how the cardiovascular processes that control variability in heart period and blood pressure change from rest to resonance breathing. The cost function used in model calibration corresponded to the difference between the experimental data and model outputs. A good match was observed between the data and model outputs (heart period, blood pressure, and corresponding power spectral densities). Significant improvements in several modeled cardiovascular functions (e.g., blood flow to internal organs, sensitivity of the sympathetic component of the baroreflex, ventricular elastance) were observed during resonance breathing. Individual differences in the magnitude and nature of these dynamic responses suggest that computational physiology may be clinically useful for tailoring heart rate variability biofeedback interventions for the needs of individual patients.

A Koopman operator-based prediction algorithm and its application to COVID-19 pandemic and influenza cases.

Future state prediction for nonlinear dynamical systems is a challenging task. Classical prediction theory is based on a, typically long, sequence of prior observations and is rooted in assumptions on statistical stationarity of the underlying stochastic process. These algorithms have trouble predicting chaotic dynamics, "Black Swans" (events which have never previously been seen in the observed data), or systems where the underlying driving process fundamentally changes. In this paper we develop (1) a global and local prediction algorithm that can handle these types of systems, (2) a method of switching between local and global prediction, and (3) a retouching method that tracks what predictions would have been if the underlying dynamics had not changed and uses these predictions when the underlying process reverts back to the original dynamics. The methodology is rooted in Koopman operator theory from dynamical systems. An advantage is that it is model-free, purely data-driven and adapts organically to changes in the system. While we showcase the algorithms on predicting the number of infected cases for COVID-19 and influenza cases, we emphasize that this is a general prediction methodology that has applications far outside of epidemiology.

Control of soft robots with inertial dynamics.

Soft robots promise improved safety and capability over rigid robots when deployed near humans or in complex, delicate, and dynamic environments. However, infinite degrees of freedom and the potential for highly nonlinear dynamics severely complicate their modeling and control. Analytical and machine learning methodologies have been applied to model soft robots but with constraints: quasi-static motions, quasi-linear deflections, or both. Here, we advance the modeling and control of soft robots into the inertial, nonlinear regime. We controlled motions of a soft, continuum arm with velocities 10 times larger and accelerations 40 times larger than those of previous work and did so for high-deflection shapes with more than 110° of curvature. We leveraged a data-driven learning approach for modeling, based on Koopman operator theory, and we introduce the concept of the static Koopman operator as a pregain term in optimal control. Our approach is rapid, requiring less than 5 min of training; is computationally low cost, requiring as little as 0.5 s to build the model; and is design agnostic, learning and accurately controlling two morphologically different soft robots. This work advances rapid modeling and control for soft robots from the realm of quasi-static to inertial, laying the groundwork for the next generation of compliant and highly dynamic robots.

Applied Koopmanism.

A majority of methods from dynamical system analysis, especially those in applied settings, rely on Poincaré's geometric picture that focuses on "dynamics of states." While this picture has fueled our field for a century, it has shown difficulties in handling high-dimensional, ill-described, and uncertain systems, which are more and more common in engineered systems design and analysis of "big data" measurements. This overview article presents an alternative framework for dynamical systems, based on the "dynamics of observables" picture. The central object is the Koopman operator: an infinite-dimensional, linear operator that is nonetheless capable of capturing the full nonlinear dynamics. The first goal of this paper is to make it clear how methods that appeared in different papers and contexts all relate to each other through spectral properties of the Koopman operator. The second goal is to present these methods in a concise manner in an effort to make the framework accessible to researchers who would like to apply them, but also, expand and improve them. Finally, we aim to provide a road map through the literature where each of the topics was described in detail. We describe three main concepts: Koopman mode analysis, Koopman eigenquotients, and continuous indicators of ergodicity. For each concept, we provide a summary of theoretical concepts required to define and study them, numerical methods that have been developed for their analysis, and, when possible, applications that made use of them. The Koopman framework is showing potential for crossing over from academic and theoretical use to industrial practice. Therefore, the paper highlights its strengths in applied and numerical contexts. Additionally, we point out areas where an additional research push is needed before the approach is adopted as an off-the-shelf framework for analysis and design.

Multiscale Adaptive Search.

We present a continuous-space multiscale adaptive search (MAS) algorithm for single or multiple searchers that finds a stationary target in the presence of uncertainty in sensor diameter. The considered uncertainty simulates the influence of the changing environment and terrain as well as adversarial actions that can occur in practical applications. When available, information about the foliage areas and a priori distribution of the target position is included in the MAS algorithm. By adapting to various uncertainties, MAS algorithm reduces the median search time to find the target with a probability of detection of at least PD and a probability of false alarm of at most PFA. We prove that MAS algorithm discovers the target with the desired performance bounds PD and PFA. The unique features of the MAS algorithm are realistic second-order dynamics of the mobile sensors that guarantees uniform coverage of the surveyed area and a two-step Neyman-Pearson-based decision-making process. Computer simulations show that MAS algorithm performs significantly better than lawnmower-type search and billiard-type random search. Our tests suggest that the median search time in the MAS algorithm may be inversely proportional to the number of participating searchers. As opposed to lawnmower search, the median search time in the MAS algorithm depends only logarithmically on the magnitude of uncertainty.

Uniform resonant chaotic mixing in fluid flows.

Laminar flows can produce particle trajectories that are chaotic, with nearby tracers separating exponentially in time. For time-periodic, two-dimensional flows and steady three-dimensional (3D) flows, enhancements in mixing due to chaotic advection are typically limited by impenetrable transport barriers that form at the boundaries between ordered and chaotic mixing regions. However, for time-dependent 3D flows, it has been proposed theoretically that completely uniform mixing is possible through a resonant mechanism called singularity-induced diffusion; this is thought to be the case even if the time-dependent and 3D perturbations are infinitesimally small. It is important to establish the conditions for which uniform mixing is possible and whether or not those conditions are met in flows that typically occur in nature. Here we report experimental and numerical studies of mixing in a laminar vortex flow that is weakly 3D and weakly time-periodic. The system is an oscillating horizontal vortex chain (produced by a magnetohydrodynamic technique) with a weak vertical secondary flow that is forced spontaneously by Ekman pumping--a mechanism common in vortical flows with rigid boundaries, occurring in many geophysical, industrial and biophysical flows. We observe completely uniform mixing, as predicted by singularity-induced diffusion, but only for oscillation periods close to typical circulation times.

Ergodic theory and visualization. I. Mesochronic plots for visualization of ergodic partition and invariant sets.

We present a computational study of a visualization method for invariant sets based on ergodic partition theory, first proposed by Mezic (Ph.D. thesis, Caltech, 1994) and Mezic and Wiggins [Chaos 9, 213 (1999)]. The algorithms for computation of the time averages of observables on phase space are developed and used to provide an approximation of the ergodic partition of the phase space. We term the graphical representation of this approximation--based on time averages of observables--a mesochronic plot (from Greek: meso--mean, chronos--time). The method is useful for identifying low-dimensional projections (e.g., two-dimensional slices) of invariant structures in phase spaces of dimensionality bigger than two. We also introduce the concept of the ergodic quotient space, obtained by assigning a point to every ergodic set, and provide an embedding method whose graphical representation we call the mesochronic scatter plot. We use the Chirikov standard map as a well-known and dynamically rich example in order to illustrate the implementation of our methods. In addition, we expose applications to other higher dimensional maps such as the Froéschle map for which we utilize our methods to analyze merging of resonances and, the three-dimensional extended standard map for which we study the conjecture on its ergodicity [I. Mezic, Physica D 154, 51 (2001)]. We extend the study in our next paper [Z. Levnajic and I. Mezic, e-print arXiv:0808.2182] by investigating the visualization of periodic sets using harmonic time averages. Both of these methods are related to eigenspace structure of the Koopman operator [I. Mezic and A. Banaszuk, Physica D 197, 101 (2004)].

Exponentially decaying modes and long-term prediction of sea ice concentration using Koopman mode decomposition.

Sea ice cover in the Arctic and Antarctic is an important indicator of changes in the climate, with important environmental, economic and security consequences. The complexity of the spatio-temporal dynamics of sea ice makes it difficult to assess the temporal nature of the changes-e.g. linear or exponential-and their precise geographical loci. In this study, Koopman Mode Decomposition (KMD) is applied to satellite data of sea ice concentration for the Northern and Southern hemispheres to gain insight into the temporal and spatial dynamics of the sea ice behavior and to predict future sea ice behavior. We observe spatial modes corresponding to the mean and annual variation of Arctic and Antarctic sea ice concentration and observe decreases in the mean sea ice concentration from early to later periods, as well as corresponding shifts in the locations that undergo significant annual variation in sea ice concentration. We discover exponentially decaying spatial modes in both hemispheres and discuss their precise spatial extent, and also perform predictions of future sea ice concentration. The Koopman operator-based, data-driven decomposition technique gives insight into spatial and temporal dynamics of sea ice concentration not apparent in traditional approaches.

Electrokinetic mixing in electrode-embedded multiwell plates to improve the diffusion limited kinetics of biosensing platforms.

Rapid and accurate biosensing with low concentrations of the analytes is usually challenged by the diffusion limited reaction kinetics. Thus, as a remedy, long incubation times or excess amounts of the reagents are employed to ensure the reactions to go to completion. Therefore, mixing becomes both a serious problem and necessity to overcome that diffusion limitation and homogenize the samples, especially for the biochemical reactions that take place in multiwell plates. Because the current mixing platforms such as shakers/vortexers, sonicators, magnetic stirrers and acoustic mixers have disadvantages including, but not limited to, being invasive/harfmul to the samples, causing the samples to splash out or stick to the walls of the wells and allowing foreign compartments to enter the solutions in the wells. Here we propose a noninvasive and safer (considering the risk of sample loss) technology that provides electrokinetic-mixing (EKM) of the reagents placed in electrode-embedded multiwell plates where the incubation times, or in other words, the time required for the desired molecules to meet in stationary solutions, can be reduced substantially. In order to demonstrate the power of this innovation, in this specific case, a simple Förster resonance energy transfer (FRET) based quenching bioplatform was adopted, where a molecular beacon DNA (MB) modified with sulfhydryl (-SH) and fluorescein (FITC) dye at opposite terminals was incubated with 10 nm sized gold nanoparticles (AuNPs) in the wells of an electrode-embedded multiwell plate, in which a printed circuit board (PCB) was attached at the bottom to control the liquid flows by EKM. When the MB binds to AuNPs through thiolate chemistry in the solution, FITC dye comes in close proximity to the AuNP surface and the emission is quenched via FRET principle. Thus, this quenching percentage over time was our comparison parameter for the mixing and no mixing cases to demonstrate the impact of mixing on the quenching kinetics. This reaction was conducted with different concentrations of AuNPs to observe the impact of mixing on MB quenching kinetics when the concentrations of the AuNPs were increased. Total quenching efficiency could go up to 90% in the presence of the AuNPs and it took about 60 min to reach stability. When the EKM was involved, fluorescence quenching time for the MBs could be reduced by up to 4.1 times. Thus, it was demonstrated that this technology may improve the kinetics of the diffusion limited biological reactions take place in multiwell plates substantially so that it may be adopted in various different sensing platforms for rapid measurements.

Search strategy in a complex and dynamic environment: the MH370 case.

Search and detection of objects on the ocean surface is a challenging task due to the complexity of the drift dynamics and lack of known optimal solutions for the path of the search agents. This challenge was highlighted by the unsuccessful search for Malaysian Flight 370 (MH370) which disappeared on March 8, 2014. In this paper, we propose an improvement of a search algorithm rooted in the ergodic theory of dynamical systems which can accommodate complex geometries and uncertainties of the drifting search areas on the ocean surface. We illustrate the effectiveness of this algorithm in a computational replication of the conducted search for MH370. We compare the algorithms using many realizations with random initial positions, and analyze the influence of the stochastic drift on the search success. In comparison to conventional search methods, the proposed algorithm leads to an order of magnitude improvement in success rate over the time period of the actual search operation. Simulations of the proposed search control also indicate that the initial success rate of finding debris increases in the event of delayed search commencement. This is due to the existence of convergence zones in the search area which leads to local aggregation of debris in those zones and hence reduction of the effective size of the area to be searched.

Koopman Mode Analysis of agent-based models of logistics processes.

Modern logistics processes and systems can feature extremely complicated dynamics. Agent Based Modeling is emerging as a powerful modeling tool for design, analysis and control of such logistics systems. However, the complexity of the model itself can be overwhelming and mathematical meta-modeling tools are needed that aggregate information and enable fast and accurate decision making and control system design. Here we present Koopman Mode Analysis (KMA) as such a tool. KMA uncovers exponentially growing, decaying or oscillating collective patterns in dynamical data. We apply the methodology to two problems, both of which exhibit a bifurcation in dynamical behavior, but feature very different dynamics: Medical Treatment Facility (MTF) logistics and ship fueling (SF) logistics. The MTF problem features a transition between efficient operation at low casualty rates and inefficient operation beyond a critical casualty rate, while the SF problem features a transition between short mission life at low initial fuel levels and sustained mission beyond a critical initial fuel level. Both bifurcations are detected by analyzing the spectrum of the associated Koopman operator. Mathematical analysis is provided justifying the use of the Dynamic Mode Decomposition algorithm in punctuated linear decay dynamics that is featured in the SF problem.

Electrokinetic Mixing for Improving the Kinetics of an HbA1c Immunoassay.

The efficiency of the diagnostic platforms utilizing ELISA technique or immunoassays depends highly on incubation times of the recognition elements or signaling molecules and volume of the patient samples. In conventional immunoassays, long incubation times and excess amounts of the recognition and signaling molecules are used. The technology proposed here uses electrokinetic mixing of the reagents involved in a sandwich immunoassay based diagnostic assay in electrode-enabled microwell plates in such a way that the incubation times and volumes can be reduced substantially. The integration of the electrodes at the bottom of the conventional microwell plates ensures that the motions of the liquid flows in the wells can be controlled through the application of high frequency AC current along these electrodes. The strategy to generate chaotic mixing by modification of standard multiwell plates, enables its use in high throughput screening, in contrast to microfluidic channel-based technologies that are difficult to incorporate into conventional plates. An immunoassay for detection of glycated hemoglobin (HbA1c) that can reveal a patient's average level of blood sugar from the past 2-3 months instead of just measuring the current levels and thereby constitutes a reliable diabetes monitoring platform was chosen as a pilot assay for technology demonstration. The overall incubation time for the assay was reduced by approximately a factor of five when electrokinetic mixing was employed. Furthermore, when the quantity of the reagents was reduced by half, almost no distinguishable signals could be obtained with conventional immunoassay, while electrokinetic mixing still facilitated acquisition of signals while varying concentration of the glycated hemoglobin. There was also a substantial difference in the signal intensities especially for the low concentrations of the HbA1c obtained from electrokinetic mixing assisted and conventional immunoassay when the quantity of the reagents and incubation times were kept constant, which is also an indication of the increase in bioassay efficiency. The electrokinetic mixing technique has the potential to improve the efficiency of immunoassay based diagnostic platforms with reduced assay time and reagent amounts, leading to higher throughput analysis of clinical samples. It may also open new avenues in point of care diagnostic devices, where kinetics and sampling size/volume play a critical role.

An agent-based model of urban insurgence: Effect of gathering sites and Koopman mode analysis.

The paper investigates the effect of preferential gathering sites on urban insurgency in an agent-based model (ABM). The ABM model was proposed in earlier work and has been validated using FBI data. There is a nonlinear tradeoff between the local density of citizens due to the number of preferential gathering sites and the ability of law enforcement officers (LEOs) to adequately patrol that leads to a non-monotonic behavior in the number of large scale outburst of insurgency with respect to the number of gathering sites. The inclusion of a moderate number of sites decreases the number of large-scale outbursts. Having no gathering sites or a large number of gathering sites has a dilutive effect on the number of large-scale outbursts. Thus, this non-monotonicity indicates that a small number of organized units produces a larger insurgency effect than a larger number of distributed units. It is also shown, using Koopman mode analysis, that the spatial morphology of agents due to the gathering sites gives rise to temporal organization of the model dynamics; there is a prominent quasi-periodic component in the number of active and intimidated citizens and in the spatial distribution of the LEOs.

The Translational Value of Psychophysiology Methods and Mechanisms: Multilevel, Dynamic, Personalized.

OBJECTIVE: It has been nearly 15 years since Kazdin and Nock published methodological and research recommendations for understanding mechanisms of change in child and adolescent therapy. Their arguments and enthusiasm for research on mechanisms of behavior change (MOBCs) resonated across disciplines and disorders, as it shined a light on the crucial importance of understanding how and for whom treatments instigate behavior change and how therapeutic mechanisms might be extended to "situations and settings of everyday life." Initial efforts focused on how psychotherapy works and linear models, yet the use of theory to guide the study of mechanisms, and laboratory experiments to manipulate them, is broadly applicable. METHOD: This article considers dynamic physiological processes that support behavior change. Specifically, it examines the utility of psychophysiological methods to measure and promote behavior change. Moreover, it embeds the baroreflex mechanism, a well-defined heart-brain feedback loop, within the theories and strategies of MOBC research. RESULTS AND CONCLUSION: Individuals' subjective and expressive experience of change does not always align with their physiological reactivity. Thus, behavior change may be best understood when concurrently assessed across multiple biobehavioral levels. Further, behavior is initiated in the moment, often before conscious deliberation, suggesting that multilevel behavior change research may benefit from real-time methodological designs. Last, substance use trajectories vary widely, suggesting that different MOBCs are more or less active in individuals depending on their personal constituency and the functional need that their substance use serves; thus, methods that are amenable to personalized modeling approaches are important.

An ultrashort mixing length micromixer: the shear superposition micromixer.

We report for the first time a laminar high-performance continuous micromixing process of two fluids over a length of 200 microns in under 10 milliseconds achieved by an optimization of the control parameters amplitude and frequency in the mixing device denoted as 'Shear Superposition Micromixer'. We improve mixing time by approximately 5 orders of magnitude over diffusion-limited mixing. The data indicate that rapid mixing is a result of the combined action of Taylor-Aris dispersion in the main and secondary microchannels and unsteady vortex motion that occurs at finite Reynolds number, which occurs above a threshold amplitude and frequency. The mixing performance is quantified using micron-resolution particle image velocimetry (micro-PIV) and computational fluid dynamics (CFD) simulations.