Geometrical Methods for Feature Extraction and Prediction in Autonomous and Non-Autonomous Dynamical Systems: Applicatio
Abstract
Short Work StatementThe PI will develop novel geometrical methods for data analysis and prediction in dynamical systems.ObjectiveThe goal of this project is to develop geometrical methods in dynamical systems, taking into account time-dependent exogenous factors and spatiotemporal data relationships. In order to meet these objectives, the PI proposes to develop new techniques that take advantage of the foliated geometry of data generated by non-autonomous dynamical systems to extract intrinsic components due to natural (internal) variability and exogenous trends with no prior knowledge of the equations of motion.ApproachThe general framework of the proposed research, encountered in many science and engineering disciplines, is dynamical systems operating in high-dimensional phase spaces, but generating data with low-dimensional, nonlinear geometric structures. When the dynamics are influenced by time-dependent exogenous factors, the data acquires a foliated geometry where each leaf corresponds to the dynamical states at a fixed value of the external forcing. This geometry naturally leads to an intrinsic notion of internal dynamics and trend based on the components of the fulldynamical vector field tangent and orthogonal to the leaves, respectively. The PI s approach is to identify these components and their associated spatiotemporal patterns by analyzing ensemble experiments through kernel methods. Kernel methods also form a natural mathematical framework to construct function spaces on nonlinear geometric datasets with a well-defined notion of smoothness, which we propose to use for feature extraction, operator approximation, and prediction of spatiotemporal data. Specifically, the PI proposes a novel nonparametric forecasting approach which uses kernels on the product of the system~s phase space and the spatial domain to assign an initial probability distribution on spatiotemporal training data, and forecast function-valued observables (e.g., scalar fields) viaan analog approach (i.e., by following the evolution of that density on the training data). The PI proposes to use the same kernels to extract spatiotemporal features through their eigenfunctions, and represent the Koopman operators governing the evolution of function-valued observables. New approaches for nonparametric analog forecasting are proposed that are based on combining ideas from Takens delay-coordinate maps and operator-valued kernels to capture spatiotemporal relationships in high-dimensional data. These kernels will also be employed in feature extraction of time-evolving data with both temporal and spatialintermittency, and for Galerkin approximation of Koopman operators governing the time evolution of function-valued (spatially extended) observables. Applications are proposed in three high-impact areas in atmosphere ocean science, namely analysis of large-ensemble climate simulations with trends, reconstruction and forecasting of Arctic and Antarctic sea ice, and extraction and prediction of multiscale convective waves in the tropical atmosphere.Overall Merits and ONR RelevanceThe PI proposes the development of a new suite of analytical and computational methods to address the problem of finding the low-dimensional manifolds on which critical information in very high-dimensional datasets reside. The PI s effort, if successful will enhance the Navy and the DoD s capabilities in uncovering meaning embedded in very high-dimensional datasets that critical DoD applications routinely generate, especially in those applications that arise in oceanic and atmospheric processes,
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Aug 12, 2016
- Source ID
- N000141612649
Entities
People
- Dimitrios Giannakis
Organizations
- New York University
- Office of Naval Research
- United States Navy