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Discovering governing equations from data by sparse identification of nonlinear dynamical systems
- S. Brunton, J. Proctor, J. Kutz
- Mathematics, MedicineProceedings of the National Academy of Sciences
- 11 September 2015
This work develops a novel framework to discover governing equations underlying a dynamical system simply from data measurements, leveraging advances in sparsity techniques and machine learning and using sparse regression to determine the fewest terms in the dynamic governing equations required to accurately represent the data.
On dynamic mode decomposition: Theory and applications
- Jonathan H. Tu, C. Rowley, D. Luchtenburg, S. Brunton, J. Kutz
- Computer Science, Mathematics
- 29 November 2013
A theoretical framework in which dynamic mode decomposition is defined as the eigendecomposition of an approximating linear operator, which generalizes DMD to a larger class of datasets, including nonsequential time series, and shows that under certain conditions, DMD is equivalent to LIM.
Dynamic mode decomposition - data-driven modeling of complex systems
This first book to address the DMD algorithm presents a pedagogical and comprehensive approach to all aspects of DMD currently developed or under development, and blends theoretical development, example codes, and applications to showcase the theory and its many innovations and uses.
Data-driven discovery of partial differential equations
- S. Rudy, S. Brunton, J. Proctor, J. Kutz
- Medicine, Computer ScienceScience Advances
- 21 September 2016
The sparse regression method is computationally efficient, robust, and demonstrated to work on a variety of canonical problems spanning a number of scientific domains including Navier-Stokes, the quantum harmonic oscillator, and the diffusion equation.
Deep learning for universal linear embeddings of nonlinear dynamics
It is often advantageous to transform a strongly nonlinear system into a linear one in order to simplify its analysis for prediction and control, so the authors combine dynamical systems with deep learning to identify these hard-to-find transformations.
Dynamic Mode Decomposition with Control
- J. Proctor, S. Brunton, J. Kutz
- Computer Science, MathematicsSIAM J. Appl. Dyn. Syst.
- 22 September 2014
This work develops a new method which extends dynamic mode decomposition (DMD) to incorporate the effect of control to extract low-order models from high-dimensional, complex systems and provides the additional innovation of being able to disambiguate between the underlying dynamics and the effects of actuation, resulting in accurate input-output models.
Chaos as an intermittently forced linear system
- S. Brunton, B. Brunton, J. Proctor, E. Kaiser, J. Kutz
- Mathematics, MedicineNature Communications
- 18 August 2016
A universal, data-driven decomposition of chaos as an intermittently forced linear system is presented, combining delay embedding and Koopman theory to decompose chaotic dynamics into a linear model in the leading delay coordinates with forcing by low-energy delay coordinates.
Data-driven discovery of coordinates and governing equations
- Kathleen P. Champion, Bethany Lusch, J. Kutz, S. Brunton
- Medicine, Computer ScienceProceedings of the National Academy of Sciences
- 29 March 2019
A custom deep autoencoder network is designed to discover a coordinate transformation into a reduced space where the dynamics may be sparsely represented, and the governing equations and the associated coordinate system are simultaneously learned.
Data-Driven Sparse Sensor Placement for Reconstruction: Demonstrating the Benefits of Exploiting Known Patterns
- K. Manohar, B. Brunton, J. Kutz, S. Brunton
- Computer Science, MathematicsIEEE Control Systems
- 26 January 2017
This article explores how to design optimal sensor locations for signal reconstruction in a framework that scales to arbitrarily large problems, leveraging modern techniques in machine learning and sparse sampling.