Attractor Modeling and Empirical Nonlinear Model Reduction of dissipative Dynamical Systems

In a broad sense, model reduction means producing a low-dimensional dynamical system that replicates either approximately, or more strictly, exactly and topologically, the output of a dynamical system. Model reduction has an important role in the study of dynamical systems and also with engineering problems. In many cases, there exists a good low-dimensional model for even very high-dimensional systems, even infinite dimensional systems in the case of a PDE with a low-dimensional attractor. The theory of global attractors approaches these issues analytically, and focuses on finding (depending on the question at hand), a slow-manifold, inertial manifold, or center manifold, on which a restricted dynamical system represents the interesting behavior of the dynamical system; the main issue depends on defining a stable invariant manifold in which the dynamical system is invariant. These approaches are analytical in nature, however, and are therefore not always appropriate for dynamical systems known only empirically through a dataset. Empirically, the collection of tools available are much more restricted, and are essentially linear in nature. Usually variants of Galerkin’s method, project the dynamical system onto a function linear subspace spanned by modes of some chosen spanning set. Even the popular Karhunen–Loeve decomposition, or POD, method is exactly such a method. As such, it is forced to either make severe errors in the case that the invariant space is intrinsically a highly nonlinear manifold, or bypass low-dimensionality by retaining many modes in order to capture the manifold. In this work, we present a method of modeling a low-dimensional nonlinear manifold known only through the dataset. The manifold is modeled as a discrete graph structure. Intrinsic manifold coordinates will be found specifically through the ISOMAP algorithm recently developed in the Machine Learning community originally for purposes of image recognition.