Koopman learning with episodic memory.

Koopman operator theory has found significant success in learning models of complex, real-world dynamical systems, enabling prediction and control. The greater interpretability and lower computational costs of these models, compared to traditional machine learning methodologies, make Koopman learning an especially appealing approach. Despite this, little work has been performed on endowing Koopman learning with the ability to leverage its own failures.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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 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.

Ergodicity-Based Cooperative Multiagent Area Coverage via a Potential Field.

This paper considers a problem of area coverage where the objective is to achieve given coverage density by use of multiple mobile agents. We present an ergodicity-based coverage algorithm which enables a centralized feedback control for multiagent system based on radial basis function (RBF) representation of the ergodicity problem and a solution of an appropriately designed stationary heat equation for the potential field. The heat equation uses a source term that depends on the difference between the given goal density distribution and the current coverage density (time average of RBFs along trajectories).

Programmable Potentials: Approximate N-body potentials from coarse-level logic.

This paper gives a systematic method for constructing an N-body potential, approximating the true potential, that accurately captures meso-scale behavior of the chemical or biological system using pairwise potentials coming from experimental data or ab initio methods. The meso-scale behavior is translated into logic rules for the dynamics. Each pairwise potential has an associated logic function that is constructed using the logic rules, a class of elementary logic functions, and AND, OR, and NOT gates.

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. Mezić and A. Banaszuk [Physica D 197, 101 (2004)], based on frequency analysis and Koopman operator theory.

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.

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.

Model reduction for agent-based social simulation: coarse-graining a civil violence model.

Agent-based modeling (ABM) constitutes a powerful computational tool for the exploration of phenomena involving emergent dynamic behavior in the social sciences. This paper demonstrates a computer-assisted approach that bridges the significant gap between the single-agent microscopic level and the macroscopic (coarse-grained population) level, where fundamental questions must be rationally answered and policies guiding the emergent dynamics devised. Our approach will be illustrated through an agent-based model of civil violence.

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.

The redistribution of power: neurocardiac signaling, alcohol and gender.

Human adaptability involves interconnected biological and psychological control processes that determine how successful we are in meeting internal and environmental challenges. Heart rate variability (HRV), the variability in consecutive R-wave to R-wave intervals (RRI) of the electrocardiogram, captures synergy between the brain and cardiovascular control systems that modulate adaptive responding. Here we introduce a qualitatively new dimension of adaptive change in HRV quantified as a redistribution of spectral power by applying the Wasserstein distance with exponent 1 metric (W(1)) to RRI spectral data.

On the architecture of cell regulation networks.

Background: With the rapid development of high-throughput experiments, detecting functional modules has become increasingly important in analyzing biological networks. However, the growing size and complexity of these networks preclude structural breaking in terms of simplest units. We propose a novel graph theoretic decomposition scheme combined with dynamics consideration for probing the architecture of complex biological networks.

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.

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 Mezić (Ph.D. thesis, Caltech, 1994) and Mezić and Wiggins [Chaos 9, 213 (1999)].

A new mixing diagnostic and Gulf oil spill movement.

Chaotic advection has served as the paradigm for mixing in fluid flows with simple time dependence. Its skeletal structure is based on analysis of invariant attracting and repelling manifolds in fluid flows. Here we develop a finite-time theory for two-dimensional incompressible fluid flows with arbitrary time dependence and introduce a new mixing diagnostic based on it.