Low-dimensional Invariant Coordinate Systems for Dynamic Modeling

点击翻译按钮获取中文摘要

9312092 Kirby This proposal presents a new analytic tool for extracting a reduced description of general dynamical systems. It details how systems with dynamics which lie on low-dimensional invariant manifolds or attractors can be reformulated into systems of low dimensionality. In the study of such systems one is typically confronted with complex models wither in the form of partial differential equations, or large systems of ordinary differential or difference equations. This "high- dimensionality" contradicts the conjectured idea that the model of the system should be governable by a small number of equations. This project addresses the issue of how to objectively proceed from an overly detailed and redundant description based on modeling considerations to a reduced model which accurately reflects the true dimensionality of the system. A technique is presented which allows the reformulation of the model to its optimally reduced form and it is shown how this can be used to facilitate the study of dynamical systems with chaotic or turbulent solutions. The basis of the approach is the reduction of the model, wither a partial differential equation or system of ordinary differential equations, to the simplest possible form. This is carried out using a neural network architecture which resembles and bottleneck. The neural network constructs a nonlinear vector function approximation in several variables. The values of the nodes in the bottleneck correspond to the dynamical information optimally compressed onto a manifold of lowest possible dimension. Consequently, the model equations can be formulated in terms of the variables on the bottleneck manifold. This can be viewed as a nonlinear compression of the phase space of the bottleneck manifold. This can be viewed as a nonlinear compression of the phase space of the dynamical system and is mathematically equivalent to a projection of the flow onto a non- orthogonal basis. ***