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Data-driven discovery methods are having a profound impact across the physical, engineering and biological sciences. The goal of these emerging methods is to mine big data content for low-dimensional patterns of activity and/or clustering of behaviors. Such pattern recognition algorithms, typically falling under the aegis of machine learning methods, are then capitalized on for control, optimization or characterization of the system under consideration. The success of these methods, such as self-driving automobiles, have brought society to the brink of a technology revolution. As we have recently shown, optical systems are far less complex than self-driving algorithms, thus suggesting that self-tuning methods should be easily implementable in the much more limited parameter space afforded by, for example, mode-locked fiber lasers and/or metamaterial antennas.

Sparsity for Classification of Dynamical Modes

Machine learning is based upon algorithms. These algorithms aim to learn and clusters patterns typically in a low-dimensional spaces. The algorithmic procedure used for dimensionally reducing, classifying and reconstruction relies first on building a library of modes acquired, in an offline manner, by low-rank projections (SVD) of data. The

machine learning portion of the algorithm thus relies on (a) data sampling of the full, highly-resolved complex system which is then (b) dimensionally reduced through SVD/POD/PCA and (c) stored in a libraries (matrices) matrix. Once constructed, (d) sparse sensing at a limited number of locations can be used to (e) classify the dynamical regime by solving an L1 convex optimization problem. The L1 norm promotes sparsity and allows the classification to identify the current relevant dynamic modes from the learned libraries. Further, the a reconstruction of the data can be then made from limited measurements and the correct library modes. Such a framework is ideal for high-dimensional dynamical systems where low-dimensional patterns of activity dominate many meaningful behaviors.

machine learning portion of the algorithm thus relies on (a) data sampling of the full, highly-resolved complex system which is then (b) dimensionally reduced through SVD/POD/PCA and (c) stored in a libraries (matrices) matrix. Once constructed, (d) sparse sensing at a limited number of locations can be used to (e) classify the dynamical regime by solving an L1 convex optimization problem. The L1 norm promotes sparsity and allows the classification to identify the current relevant dynamic modes from the learned libraries. Further, the a reconstruction of the data can be then made from limited measurements and the correct library modes. Such a framework is ideal for high-dimensional dynamical systems where low-dimensional patterns of activity dominate many meaningful behaviors.

Selected Recent Publications

- S. Brunton, J. Proctor and J. N. Kutz, Discovering governing equations from data: sparse identification of nonlinear dynamical systems arXiv:1509.03580.
- J. N. Kutz, X. Fu, S. Brunton and J. Grosek, Dynamic Mode Decomposition for Robust PCA with Applications to Foreground/Background Subtraction in Video Streams and Multi-Resolution Analysis, in CRC Handbook on Robust Low-Rank and Sparse Matrix Decomposition: Applications in Image and Video Processing, T. Bouwmans Ed. (CRC Press, 2015).
- I. Bright, G. Lin and J. N. Kutz, Classification of Spatio-Temporal Data via Asynchronous Sparse Sampling: Application to Flow Around a Cylinder, arXiv:1506.00661.
- I. Bright, G. Lin and J. N. Kutz, Compressive sensing based machine learning strategy for characterizing the flow around a cylinder with limited pressure measurements, Physics of Fluids 25 (2013) 127102.
- S. Brunton, J. Tu, I. Bright and J. N. Kutz, Compressive sensing and low-rank libraries for classification of bifurcation regimes in nonlinear dynamical systems, SIAM Journal of Applied Dynamical Systesm 13 (2014) 1716-1732.