Data-driven approaches to dynamical networks: Integrating equation-free methods, machine learning and sparsity J. Nathan Kutz Department of Applied Mathematics.

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Presentation transcript:

Data-driven approaches to dynamical networks: Integrating equation-free methods, machine learning and sparsity J. Nathan Kutz Department of Applied Mathematics University of Washington Seattle, WA 98195-3925 Email: kutz@uw.edu

Mathematical Foundations Dimensionality Reductions + Machine Learning - Goal is production of 2D light bullet Phase velocity in z direction Field confined to waveguides with Bragg Grating Contact layer pumps 0th waveguide

Dynamical Systems + PDEs i. Generic nonlinear , time-dependent system ii. Measurements (assimilation) - Goal is production of 2D light bullet Phase velocity in z direction Field confined to waveguides with Bragg Grating Contact layer pumps 0th waveguide iii. My commitment – low-dimensional subspaces x

Encoding Dynamics

Sparsity + Sensors - Goal is production of 2D light bullet Phase velocity in z direction Field confined to waveguides with Bragg Grating Contact layer pumps 0th waveguide

Compressive Sensing: A Cartoon - Goal is production of 2D light bullet Phase velocity in z direction Field confined to waveguides with Bragg Grating Contact layer pumps 0th waveguide

Compressive Sensing for Fluids - Goal is production of 2D light bullet Phase velocity in z direction Field confined to waveguides with Bragg Grating Contact layer pumps 0th waveguide

Flow Around a Cylinder

Compressive Sensing Reconstruction 1000 Re What about sensor placement?

Dynamic Mode Decomposition Equation-Free Dynamic Mode Decomposition & Koopman Operators - Goal is production of 2D light bullet Phase velocity in z direction Field confined to waveguides with Bragg Grating Contact layer pumps 0th waveguide

Dynamic Mode Decomposition - Goal is production of 2D light bullet Phase velocity in z direction Field confined to waveguides with Bragg Grating Contact layer pumps 0th waveguide

Regression to Dynamical Systems Linear dynamics (equation-free) Eigenfunction expansion - Goal is production of 2D light bullet Phase velocity in z direction Field confined to waveguides with Bragg Grating Contact layer pumps 0th waveguide Least-square fit

Multi-Scale Physics - Goal is production of 2D light bullet Phase velocity in z direction Field confined to waveguides with Bragg Grating Contact layer pumps 0th waveguide

El Nino data (1990s-2010+) - Goal is production of 2D light bullet Phase velocity in z direction Field confined to waveguides with Bragg Grating Contact layer pumps 0th waveguide

Compressive DMD & Control - Goal is production of 2D light bullet Phase velocity in z direction Field confined to waveguides with Bragg Grating Contact layer pumps 0th waveguide

With Yu Hu, Eric Shea-Brown, Steve Brunton, Nick Cain & Stefan Mihalas Dynamic Networks & Functionality With Yu Hu, Eric Shea-Brown, Steve Brunton, Nick Cain & Stefan Mihalas - Goal is production of 2D light bullet Phase velocity in z direction Field confined to waveguides with Bragg Grating Contact layer pumps 0th waveguide

Network Motifs - Goal is production of 2D light bullet Phase velocity in z direction Field confined to waveguides with Bragg Grating Contact layer pumps 0th waveguide

Engineering Second Order Motifs - Goal is production of 2D light bullet Phase velocity in z direction Field confined to waveguides with Bragg Grating Contact layer pumps 0th waveguide Yu Hu, Brunton …. Kutz, Shea-Brown, arxiv 2015 (soon)

Bringing It All Together - Goal is production of 2D light bullet Phase velocity in z direction Field confined to waveguides with Bragg Grating Contact layer pumps 0th waveguide