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Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved November 21, 2020 (received for review October 12, 2020)
Significance
Multiscale structure is all around us: in biological tissues, active matter, oceans, networks, and images. Identifying the multiscale features of these systems is crucial to our understanding and control of them. We introduce a method that rationally extracts localized multiscale features from data, which may be thought of as the building blocks of the underlying phenomena.
Abstract
Many materials, processes, and structures in science and engineering have important features at multiple scales of time and/or space; examples include biological tissues, active matter, oceans, networks, and images. Explicitly extracting, describing, and defining such features are difficult tasks, at least in part because each system has a unique set of features. Here, we introduce an analysis method that, given a set of observations, discovers an energetic hierarchy of structures localized in scale and space. We call the resulting basis vectors a “data-driven wavelet decomposition.” We show that this decomposition reflects the inherent structure of the dataset it acts on, whether it has no structure, structure dominated by a single scale, or structure on a hierarchy of scales. In particular, when applied to turbulence—a high-dimensional, nonlinear, multiscale process—the method reveals self-similar structure over a wide range of spatial scales, providing direct, model-free evidence for a century-old phenomenological picture of turbulence. This approach is a starting point for the characterization of localized hierarchical structures in multiscale systems, which we may think of as the building blocks of these systems.
Footnotes
- ↵1To whom correspondence may be addressed. Email: mdgraham{at}wisc.edu.
Author contributions: D.F. and M.D.G. designed research, performed research, analyzed data, and wrote the paper.
The authors declare no competing interest.
This article is a PNAS Direct Submission.
This article contains supporting information online at https://www.pnas.org/lookup/suppl/doi:10.1073/pnas.2021299118/-/DCSupplemental.
Data Availability.
Simulation data and code have been deposited in GitHub, available at https://github.com/dfloryan/DDWD.
Published under the PNAS license.
References
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Discovering multiscale and self-similar structure with data-driven wavelets
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- Physical Sciences
- Applied Physical Sciences
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