(147b) Data-Driven Modeling and Analysis of Complex Flows

Paul Steen’s mentorship and friendship meant a great deal to me and had a long and positive impact on my life and career. In this presentation I describe some of my group’s recent research on themes that Paul introduced to me when I was a graduate student. I had been looking forward to our next meeting to discuss these topics with him.

Many complex flow phenomena are characterized by chaotic dynamics, a large number of degrees of freedom, and hierarchical, multiscale structure in space and time. In two vignettes, we describe some recent work aimed at developing and applying machine learning and data science tools for systems displaying these characteristics.

The first vignette builds on the idea that while partial differential equations are formally infinite-dimensional, the presence of energy dissipation drives the long-time dynamics onto a finite-dimensional invariant manifold sometimes called an inertial manifold (IM). We describe a data-driven framework to represent chaotic dynamics on this manifold and illustrate it with data from simulations of the Kuramoto-Sivashinsky equation, a toy model for spatiotemporally chaotic systems like turbulent flows. A hybrid method combining linear and nonlinear (neural-network) dimension reduction transforms between coordinates in the full state space and on the IM. Additional neural networks predict time evolution on the IM, yielding good trajectory-wise predictions of short-time dynamics as well as properly capturing long-time statistics.

The second vignette addresses how to represent fields with multiscale structure in space or time. We describe a method, inspired by wavelet analysis, that adaptively decomposes a dataset into an hierarchy of structures (specifically orthogonal basis vectors) localized in scale and space: a “data-driven wavelet decomposition”. This decomposition reflects the inherent structure of the dataset it acts on. In particular, when applied to turbulent flow data, it reveals spatially localized, self-similar, hierarchical structures. It is important emphasize that self-similarity is not built into the analysis, rather, it emerges from the data. This approach is a starting point for the characterization of localized hierarchical structures, and we expect that it will find application to other systems, such as atmospheres, oceans, biological tissues, active matter and many others, that display multiscale spatiotemporal structure.