Mathematically Justified Computational Platform for Nonlinear Dynamics
Abstract
The goal of this proposal is to develop novel mathematical, computationally efficient approaches to nonlinear dynamics, focusing on applicability to scientific and engineering challenges. In the context of data-driven science, often there is no close form system that is a true model, and hence it is essential that characterizations of dynamics obtained from different data sets be comparable. Our approach and the associated algorithms are based on coarse, but robust representations of dynamics that will enable both qualitative and quantitative comparisons. Our framework is combinatorial and algebraic, utilizing order theory and algebraic topology. We propose to develop algorithms and a suite of computer code to- define surrogate models from data, construct polygonal decompositions of phase space based on neural networks, use these decompositions to build combinatorial models of dynamics, extract combinatorial representations with guaranteed uncertainty bounds of the dynamics, and use algebraic topological methods to identify the dynamics in the underlying continuous models. We will provide tutorials on how to use this code. As there are minimal constraints on the theoretical mathematical foundations that underlies the development of the proposed computational infrastructure, the potential impact of this research is quite general. These mathematical ideas and the current computational tools have been tested on simple problems coming from machine learning, robotic control, ecology, systems biology, and synthetic biology. We will continue to use more complex and realistic examples from these domains to guide the development of the proposed advances of mathematical techniques, algorithms, and code.
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Feb 22, 2024
- Source ID
- FA95502310011
Entities
People
- Konstantin Mischaikow
Organizations
- Air Force Office of Scientific Research
- Rutgers University
- United States Air Force