Data-driven Dynamical Systems AnalysisTraditional dynamical systems analysis is restricted to systems for which the dynamics are given in a mathematically tractable set of differential equations in some a-priori known coordinates (which is a prerequisite to traditional methods). We develop the ability to construct ab initio representations for data observations originating from unknown dynamical systems, without the need to explicitly write (or analyze) the model equations. Such an ability facilitates the analysis of currently inaccessible problems involving nonlinear time-varying systems. Our effort includes both validation of our techniques on linear mechanical systems with well-established models and analysis, as well as the investigation of nonlinear time-varying mechanical systems, which pose a challenge for current tools and methods.
In dynamical systems, diffusion operators have been recently used to empirically build the Koopman composition operator directly from data. The Koopman operator enables the representation of data through the space of scalar functions defined on them and encompasses some important properties. First, the Koopman operator is linear even when describing a highly nonlinear propagation. Second, the Koopman operator has a discrete spectrum. Third, it was shown that even when representing a nonlinear dynamical system, the temporal evolution of the eigenfunctions of the Koopman operator is linear as well. Combining these theoretical properties with the ability to approximate the Koopman operator from data using diffusion operators equips us with substantial power.
By relying on Koopman operator theory and results, our recent work has contributed to the growing literature on nonparametric time-series filtering in a novel and promising direction, enabling us, without rigid model assumptions, to unveil their model and to filter highly nonstationary time-series. Typically, temporal information is overlooked when existing geometric methods are applied to analyze time-series, since time-series are regarded as data sets of samples, ignoring their dynamics and temporal order. Recently, we and others have attempted to incorporate the time dependency of consecutive samples into the manifold learning framework.
We recently showed that using diffusion operators, in a data-driven manner, with minimal prior knowledge and rigid model assumptions, merely from observations from an unknown nonlinear dynamical system, we are able not only to discover the system’s hidden state, dynamics, and observation function, but to attain a compact linear description of the full system.
In our recent paper published in PNAS, we developed a geometric/analytic learning algorithm capable of creating minimal descriptions of parametrically dependent unknown nonlinear dynamical systems. This was accomplished by the data-driven discovery of useful intrinsic-state variables and parameters in terms of which one can empirically model the underlying dynamics. We presented an informed observation geometry that enables us to formulate models without first principles as well as without closed-form equations, and solely from observations, to accurately represent the system’s underlying dynamics.
These results facilitated the development of a new method for analyzing neuronal activity acquired from awake behaving animals by two-photon imaging. We showed that the new method enables us to extract hidden biological features and accurate indications of pathological dysfunction. In addition, we demonstrated its capability to identify, solely from observations, patterns of neuronal activity, variability related to external triggers, and behavioral events (e.g., a sequence of motor actions), at different time scales, and among specific neuronal sub-groups.
A notable limitation of existing manifold learning techniques is that the manifold is assumed to remain fixed in time, which is very restrictive, especially for nonstationary time-series spanning long periods of time. Pushing manifold and geometry learning frontiers, we study time-varying manifolds. A particularly exciting direction we currently pursue is to devise a multiresolution manifold analysis framework. Such multiresolution analysis framework can be seen as analogous to the classical wavelet analysis under the manifold setting. More concretely, we develop new multiscale diffusion operators that infer the propagation law between consecutive manifolds in a sequence of time-varying manifolds. These operators are constructed first at a fine temporal resolution, and then, in a recursive manner, at a coarser and coarser time scales from the operators at the finer time scales, giving rise to a dyadic tree of operators discovering different time scales of temporal variations. This allows us to construct nested function spaces and multi-resolution functional diffusion operators, which represent the underlying state of the dynamical system at different time scales. In addition, the new diffusion operators between frames have a strong analogy to low- and high-pass (manifold) filters, giving rise to the discovery of the slowly and rapidly changing sub-manifold components. As a result, we are able to decompose the manifold structure into sub-manifolds in different scales of temporal variations. Such a decomposition consists of data-driven Koopman operators defined at multiple time scales, describing the data in multiple levels of temporal coherence. Naturally, such a formulation inherits the powerful properties of Koopman operators, namely, linearity in functional space as well as linear dynamics.
Traditional dynamical systems analysis is restricted to systems for which the dynamics are given in a mathematically tractable set of differential equations in some a-priori known coordinates (which is a prerequisite to traditional methods). The ability to construct ab initio representations for data observations originating from unknown dynamical systems, without the need to explicitly write (or analyze) the model equations, broadens this scope, facilitating the analysis of currently inaccessible problems involving nonlinear time-varying systems. We have already validated our techniques on linear mechanical systems with well-established models and analysis, and we now investigate nonlinear time-varying mechanical systems, which pose a challenge for current tools and methods. To demonstrate the broad scope of our methodology, we now address key applications in physics, involving the identification of phase transitions. Since our methods do not depend on order parameters, knowledge of the topological content of the phases, or any other specifics of the transition at hand, they pave the way to the development of a generic tool for identifying unexplored phase transitions. This direction involving nonlinear dynamical systems is ambitious; yet, in our recent work, we have already shown it is feasible.
