Koopman Operator Theory for Nonautonomous and Stochastic Systems

Sepsis, a manifestation of the body’s inflammatory response to injury and infection, has a mortality rate of between 28%-50% and affects approximately 1 million patients annually in the United States. Currently, there are no therapies targeting the cellular/molecular processes driving sepsis that have demonstrated the ability to control this disease process in the clinical setting. We propose that this is in great part due to the considerable heterogeneity of the clinical trajectories that constitute clinical “sepsis,” and that determining how this system can be controlled back into a state of health requires the application of concepts drawn from the field of dynamical systems. In this work, we consider the human immune system to be a random dynamical system, and investigate its potential controllability using an agent-based model of the innate immune response (the Innate Immune Response ABM or IIRABM) as a surrogate, proxy system. Simulation experiments with the IIRABM provide an explanation as to why single/limited cytokine perturbations at a single, or small number of, time points is unlikely to significantly improve the mortality rate of sepsis. We then use genetic algorithms (GA) to explore and characterize multi-targeted control strategies for the random dynamical immune system that guide it from a persistent, non-recovering inflammatory state (functionally equivalent to the clinical states of systemic inflammatory response syndrome (SIRS) or sepsis) to a state of health. We train the GA on a single parameter set with multiple stochastic replicates, and show that while the calculated results show good generalizability, more advanced strategies are needed to achieve the goal of adaptive personalized medicine. This work evaluating the extent of interventions needed to control a simplified surrogate model of sepsis provides insight into the scope of the clinical challenge, and can serve as a guide on the path towards true “precision control” of sepsis.

Random dynamical systems can be generated by stochastic differential equations (sde) on the one hand, and by random differential equations (rde), i.e. randomly parametrized ordinary differential equations on the other hand. Due to conflicting concepts in stochastic calculus and ergodic theory, asymptotic problems for systems associated with sde are harder to treat. We show that both objects are basically identical, modulo a stationary coordinate change (cohomology) on the state space. This observation opens completely new opportunities for the treatment of asymptotic problems for systems related to sde: just study them for the conjugate rde, which is often possible by simple path-by-path classical arguments. This is exemplified for the problem of local linearization of random dynamical systems, the classical analogue of which leads to the Hartman–Grobman theorem.

In this study, we propose a new method that is useful for estimating unknown parameter values of stochastic differential equation (SDE) models, based on probability density function (PDF) data measured from random dynamical systems. As our method does not require explicit description of PDF, it can be applied to the SDE models even when their PDFs are hardly derived in explicit forms due to multiplicative-noise terms, nonlinear terms, and so on. Therefore, our method is expected to provide a versatile tool to dynamically parameterize measured PDF data. In our proposed method, it is assumed that a measured PDF is obtained from a random dynamical system whose structure is described by a known SDE model with unknown parameter values. With the help of It\^o calculus, the Fokker-Planck equation (FPE) is derived from the SDE model. The measured PDF and a candidate of parameter values are substituted into the FPE to calculate a FPE residual. Our method is applied to two random vibration systems. Their FPE residuals tend to zero as the parameter values tend to exact values, showing that our proposed FPE residual can be utilized for unknown parameter estimation of SDE models.

We review the recent notion of a nonautonomous dynamical system (NDS), which has been introduced as an abstraction of both random dynamical systems and continuous skew product flows. Our focus is on fundamental analogies and discrepancies brought about by these two principal classes of NDS. We discuss base dynamics mainly through almost periodicity and almost automorphy, and we emphasize the importance of these concepts for NDS which are generated by differential and difference equations. Nonautonomous dynamics is presented by means of selected examples. We also mention several natural yet unresolved questions.

The theory and applications of random dynamical systems (RDS) are at the cutting edge of research in mathematics and economics, particularly in modeling the long-run evolution of economic systems subject to exogenous random shocks. Despite this interest, there are no books available that solely focus on RDS in finance and economics. Exploring this emerging area, Random Dynamical Systems in Finance shows how to model RDS in financial applications. Through numerous examples, the book explains how the theory of RDS can describe the asymptotic and qualitative behavior of systems of random and stochastic differential/difference equations in terms of stability, invariant manifolds, and attractors. The authors present many models of RDS and develop techniques for implementing RDS as approximations to financial models and option pricing formulas. For example, they approximate geometric Markov renewal processes in ergodic, merged, double-averaged, diffusion, normal deviation, and Poisson cases and apply the obtained results to option pricing formulas. With references at the end of each chapter, this book provides a variety of RDS for approximating financial models, presents numerous option pricing formulas for these models, and studies the stability and optimal control of RDS. The book is useful for researchers, academics, and graduate students in RDS and mathematical finance as well as practitioners working in the financial industry.

Conceptual Analysis and Random Attractor for Dissipative Random Dynamical Systems

We consider attractors for certain types of random dynamical systems. These are skew–product systems whose base transformations preserve an ergodic invariant measure. We discuss definitions of invariant sets, attractors and invariant measures for deterministic and random dynamical systems. Under assumptions that include, for example, iterated function systems, but that exclude stochastic differential equations, we demonstrate how random attractors can be seen as examples of Milnor attractors for a skew–product system. We discuss the minimality of these attractors and invariant measures supported by them. As a further connection between random dynamical systems and deterministic dynamical systems, we show how dynamical or D–bifurcations of random attractors with multiplicative noise can be seen as blowout bifurcations, and we relate the issue of branching at such D–bifurcations to branching at blowout bifurcations.

The study of qualitative properties of random and stochastic differential equations is now one of the most active fields in the modern theory of dynamical systems. In the deterministic case, the properties of flattening and squeezing in infinite-dimensional autonomous dynamical systems require the existence of a bounded absorbing set and imply the existence of a global attractor. The flattening property involves the behaviour of individual trajectories while the squeezing property involves the difference of trajectories. It is shown here that the flattening property is implied by the squeezing property and is in fact weaker, since the attractor in a system with the flattening property can be infinite-dimensional, whereas it is always finite-dimensional in a system with the squeezing property. The flattening property is then generalized to random dynamical systems, for which it is called the pullback flattening property. It is shown to be weaker than the random squeezing property, but equivalent to pullback asymptotic compactness and pullback limit-set compactness, and thus implies the existence of a random attractor. The results are also valid for deterministic non-autonomous dynamical systems formulated as skew-product flows.

We present a new algorithm for analysis of neuronal data from single-channel time-series. Using a novel view of stochastic geometry to obtain a realistic model of nonlinear oscillatory systems, we extract traces of multiple oscillations from single-electrode extracellular recordings obtained from a representative neuron. Empiric data from neuronal recording has supported the observation that oscillations are prevalent in dynamics of the brain [1]. The literature on mathematical and computational modeling of neuronal networks of coupled oscillators is rich and includes numerous examples of successful explanations of experimental data. A number of mathematical approaches to model oscillatory networks of neurons take a deterministic approach to modeling the dynamics of such sophisticated complex biological systems where is added, if at all, as a test of stability of the system and its behavior under small perturbations. There are possible situations where the dynamics of the system depends fundamentally on the actual non-stationary statistics and the transient nature of noise in the system. The most natural mathematical theories in this context involve nonlinear dynamics and probability theory, which has been the subject of research in describing networks as well as individual neurons. The key theory that enables us to achieve a mathematically rigorous synergy between an ideal geometric theory and the biological reality is a recent merging of stochastic analysis and ergodic theory, known as the theory of Random Dynamical Systems (RDS) whose systematic foundations are laid out in the seminal work of L. Arnold [2]. The far-reaching ideas of RDS require a demanding technical mastery of stochastic analysis and ergodic. Our geometric and topological view will illustrate some of the remarkable theoretical and numerical achievements that RDS offers for biologically realistic modeling of nonlinear dynamics, while at the same time, it illuminates the ideas behind its algorithmic development.

A new result on the fractal dimension estimates of random attractor for non-autonomous random 2D stochastic dynamical type systems

In this dissertation we focus on the application of thermodynamic formalism to non-autonomous and random dynamical systems. Specifically we use the thermodynamic formalism to investigate the dimension of various fractal constructions via the, now standard, technique of Bowen which he developed in his 1979 paper on quasi-Fuchsian groups. Bowen showed, roughly speaking, that the dimension of a fractal is equal to the zero of the relevant topological pressure function. We generalize the results of Rempe-Gillen and Urbanski on non-autonomous iterated function systems to the setting of non-autonomous graph directed Markov systems and then show that the Hausdorff dimension of the fractal limit set is equal to the zero of the associated pressure function provided the size of the alphabets at each time step do not grow too quickly. In trying to remove these growth restrictions, we present several other systems for which Bowen's formula holds, most notably ascending systems. We then use these various constructions to investigate the Hausdorff dimension of various subsets of the Julia set for different large classes of transcendental meromorphic functions of finite order which have been perturbed non-autonomously. In particular we find lower and upper bounds for the dimension of the subset of the Julia set whose points escape to infinity, and in many cases we find the exact dimension. While the upper bound was known previously in the autonomous case, the lower bound was not known in this setting, and all of these results are new in the non-autonomous setting. We also use transfer operator techniques to prove an almost sure invariance principle for random dynamical systems for which the thermodynamical formalism has been well established. In particular, we see that if a system exhibits a fiberwise spectral gap property and the base dynamical system is sufficiently well behaved, i.e. it exhibits an exponential decay of correlations, then the almost sure invariance principle holds. We then apply these results to uniformly expanding random systems like those studied by Mayer, Skorulski, and Urbanski and Denker and Gordin.

Stochastic physics-informed neural ordinary differential equations

Conley index for random dynamical systems

Steven L. Brunton, Marko Budišić, Eurika Kaiser, and J. Nathan Kutz are the authors of the Survey and Review paper in this issue, “Modern Koopman Theory for Dynamical Systems.” Koopman theory is a valuable formalism for the study of dynamical systems; it has gained popularity in recent years in connection with data-driven analysis, control theory, and other areas. The basic idea is simple. If $(d/dt) {x} = {f}({x})$ is a dynamical system in ${\mathbb R}^n$, its solution flow is the one-parameter family of maps ${F}^t: {\mathbb R}^n\rightarrow {\mathbb R}^n$ such that $t\mapsto {F}^t({x}_0)$ is the solution that at time $t=0$ takes the value ${x}_0$. In an alternative description, rather than looking at points in ${\mathbb R}^n$ being moved by the flow, one may consider real-valued functions $g({x})$ being transformed by the dynamical system as $g \mapsto {\cal K}^t(g)$, where the Koopman (or composition) operator ${\cal K}^t$ is defined by ${\cal K}(g)({x}) = g({F}^t({x}))$. Thus the Koopman operator acts on an infinite-dimensional space of functions $g$ (bad news), but it is linear, even if the original dynamical system is not (good news). The idea behind the operator ${\cal K}^t$ is particularly useful when the dynamical system is being studied by big data$/$machine learning techniques in cases where $f$ and ${F}^t$ are unknown but measurements $g_i({x}(t_j))$ along a solution ${x}(t)$ are available. While the use of composition operators is very much older, B. O. Koopman noted in a seminal 1931 paper that, in the particular case of conservative dynamics, the operator ${\cal K}^t$ will be unitary in a suitable $L^2$ space, an observation that allowed him to apply to classical mechanics the theory of Hilbert space operators being developed at the time to formulate mathematically quantum mechanics. Koopman's results have been extended in many directions to dissipative or conservative systems, in continuous or discrete time. The survey in this issue does not assume a previous knowledge of Koopman theory and reviews recent developments on Koopman operators, with particular emphasis on the dynamic mode decomposition (DMD) algorithm and its variants used in many real-life applications. The paper contains almost five hundred references, shows applications to fluid mechanics, epidemiology, neuroscience, plasma physics, finance, robotics, the power grid, and other fields, and discusses connections with many areas, including numerical linear algebra, control theory, statistics, model reduction, and uncertainty qualification. I believe it will be of interest to a wide range of readers.

We study the motion of an inertial particle in a fractional Gaussian random field. The motion of the particle is described by Newton's second law, where the force is proportional to the difference between the background fluid velocity and the particle velocity. The fluid velocity satisfies a linear stochastic partial differential equation driven by an infinite-dimensional fractional Brownian motion with an arbitrary Hurst parameter H ∈ (0, 1). The usefulness of such random velocity fields in simulations is that we can create random velocity fields with desired statistical properties, thus generating artificial images of realistic turbulent flows. This model also captures the clustering phenomenon of preferential concentration, observed in real world and numerical experiments, i.e. particles cluster in regions of low-vorticity and high-strain rate. We prove almost sure existence and uniqueness of particle paths and give sufficient conditions to rewrite this system as a random dynamical system with a global random pullback attractor. Finally, we visualize the random attractor through a numerical experiment.