A Combinatorial and Topological Framework for Deriving Dynamics from Data
Abstract
Extracting dynamics from systems, whether described by mathematical models or by measured data, is of fundamental importance in the applied sciences. Researchers may fit models given by explicit formulae to data, but it is natural to wonder how much of the resulting model dynamics are due to choices made by the modeler in constructing the equations. In addition, standard models are not available for many systems that are measured by data. Here we seek an approach for model construction and analysis that avoids conforming to specified model types. The proposed framework combines tools from nonparametric regression, computational topology, and order theory to extract dynamics directly from data. Recently developed computational methods utilizing set-based techniques and tools from topology have proven remarkably successful in (rigorously) extracting dynamics from discrete-time dynamical systems governed by explicit maps. In addition, the relatively new field of Topological Data Analysis (TDA) has produced very promising techniques for identifying (static) topological structure in point-cloud data and cubical data, even in the presence of noise. Studying dynamics directly from data requires a new approach. The key idea of this proposal is the definition and construction of combinatorial models and combinatorial approximations, structures from which tools from computational dynamics and algebraic topology can extract dynamics. While applicable to a broad class of systems, these combinatorial structures are designed to organize dynamical information from data and measure its robustness. The proposed research combines ideas from computational topology and order theory with regression techniques to produce a relatively flexible method for extracting dynamics from data and high-dimensional systems. Well-developed techniques for non-parametric regression are known to smooth the effects of noise. Topological methods require continuity of the approximation, but are robust under small perturbations and tolerant of bounded noise. Persistent homology can extract structure from spatially explicit data, order theory can describe global dynamics, and the Conley index is able to extract dynamics from combinatorial outer approximations which incorporate bounded error. While all of these methods have proven successful in their own settings, they have not yet been combined into the unified approach we propose here. This unified approach should lead to a significant broadening of the class of systems that may be studied by these techniques and requires fundamental extensions to the mathematical theory as well as careful and systematic development of practical methods for their implementation. Recent advances in data collection and high-dimensional model simulation have led to a dramatic increase in the number of systems for which data is readily available, but for which traditional mathematical analysis fails. Our primary goal in this proposed work is to develop computational techniques that can help to bridge the gap between the large class of mathematical models for which we can now obtain mathematically rigorous results and the still larger class of high-dimensional systems and systems described primarily by data. This work includes developing algorithms, proving foundational results, and studying specific models and data sets.
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Feb 14, 2019
- Source ID
- W911NF1810306
Entities
People
- Sarah Day
Organizations
- Army Contracting Command
- College of William & Mary
- United States Army