(DURIP) ACCURATE COMPUTATIONS FOR IMPRECISE NONLINEAR DYNAMICS
Abstract
Data driven and multiscale systems, which are ubiquitous in current challenges in science and engineering, typically involve many unknown, unmeasured parameters and measurements of the key variables are often imprecise. Furthermore, accepted models, e.g. differential equations, for which solutions are guaranteed to provide quantitative or qualitative understanding of the phenomenon of interest are often lacking. Thus understanding the time evolution of these systems requires a computationally efficient framework that accurately and robustly captures coarse behavior over large ranges of parameters without an explicit analytic representation of the system. To address these issues this proposal asks for the computational facilities to develop, test, and apply a novel mathematically rigorous, systematic, and computationally efficient approach to nonlinear dynamics. The framework for this approach is based on four distinct computational efforts. (i) Combinatorial dynamics that are organized using ideas from order theory. (ii) Homological algebra that is used to translate the finite combinatorial information into a continuous setting. (iii) Conley theory that provides a bridge from homological invariants to dynamical objects of interest such as fixed points, periodic orbits, and chaotic dynamics. (iv) Validated numerics that provides high precision mathematically rigorous guarantees that the numerical computations have produced correct results. Each of these steps is computationally intensive, but many are easily parallelized. Thus the request for a modest cluster that can easily be accessed by collaborators, research scientists, postdoctoral fellows, graduate students, and undergraduate students associated with PI Mischaikow’ s research efforts. The current applications of these techniques are to problem of immediate interest to the Department of Defense including understanding the dynamics of deep learning, automated control of robotic systems, and analysis and design of biological circuits. Longer term impacts of this work will include analysis of complex high dimensional spatiotemporal systems such as those arising in fluids and dense granular systems, trophic systems in mathematical ecology, and evolutionary dynamics at the interface of systems biology and ecology.
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Mar 07, 2023
- Source ID
- FA95502210102
Entities
People
- Konstantin Mischaikow
Organizations
- Air Force Office of Scientific Research
- Rutgers University
- United States Air Force