This project is being conducted jointly with Prof. Michael Kirby of the Department of Mathematics, CSU.
In general, determining solutions of realistic model equations governing complicated nonlinear phenomena presents an enormous challenge. Consider, for example, the Navier-Stokes (N-S) equations governing fluid motion. Mathematically, the model consists of a system of 5 coupled nonlinear partial differential equations with a set of auxiliary constraints due to boundary conditions. These equations are generally difficult or impossible to solve analytically, even in the simplest cases. As a result researchers must resort to determining approximate solutions which hopefully reflect, at least qualitatively, the true solutions of the original model equations.
It is important that the physical laws, e.g. conservation laws in the case of the Navier-Stokes equations, used to derive models of nonlinear phenomena are generally coordinate free. It is the use of approximate numerical techniques which require the introduction of a coordinate system. The choice of coordinatization plays a central role in the resulting approximate model. It is our thesis that the majority of approaches to this problem, e.g., the Galerkin projection, use inadequate coordinate systems resulting in overly large and complicated models. In addition, we contend that the model itself plays a key role in the utility of the solutions for understanding the complicated nonlinear phenomena in question.
Our research outlines an approach for the simplification of complex mathematical models of physical phenomena. We are developing a procedure for empirically extracting an optimal nonlinear-nonorthogonal coordinate system for any given model. Our approach uses a neural network as a tool for learning the nonlinear geometry of phase space. The network is capable of generating mappings which provide the appropriate nonlinear transformation of coordinates to the model's natural manifold. The emphasis of our study differs from general time series analysis in that we assume we possess a set of model equations governing the data production. It is our goal to simplify the model and view this as a form of preprocessing of the raw data, effectively rendering it more amenable to further analysis. We assert that the understanding, prediction, control or manipulation of physical models will be greatly facilitated by recasting the models into a form which incorporates the topology of the solution curves. In what follows we outline our technique with examples of the methodology applied to simple systems.