Many complex nonequilibrium systems, including turbulent flows, 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 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; this can be done in either the discrete-time (difference equation) or continuous-time (ordinary differential equation) setting. The formalism accounts for translation invariance and energy conservation, and substantially outperforms linear dimension reduction, reproducing very well key dynamic and statistical features of the attractor.

The second vignette addresses how to represent flow or other fields with multiscale structure. 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 turbulent structures that we may think of as the building blocks of turbulence. It will also find application to other systems, such as atmospheres, oceans, biological tissues, active matter and many others, that display multiscale spatiotemporal structure.