SKDMD

Sparsity-promoting Kernel Dynamic Mode Decomposition for our paper on JFM This paper is open access so anyone can download for free.

Table of Contents

• Motivations
• Getting Started
  • Prerequisites
• Usage
• Roadmap
• Contributing
• License
• Contact
• Citation

Motivations

• Why throwing everything in a neural network?

  • neural network approach for finding Koopman operators could suffer from bad minimizers, time consuming (requires a quite large number of GPU training hours), which are all resulted from the non-convex optimization nature

• Why not using classical convex methods?

  • classical nonlinear Koopman analysis method (e.g., EDMD, KDMD) suffers from having hundreds to thousands of approximated Koopman triplets.
  • How to choose an accurate and informative Koopman invariant subspace in Extended/Kernel DMD?

• Traditional linear DMD performs poorly on transient flows

  • due to inherent linear nature, sparsity-promoting DMD (Jovanovic, et al., 2014) cannot provide accurate representation for Koopman mode decomposition in a transient regime.
  • due to lack of truncation on nonlinearly evolving modes, spDMD can end up with spurious unstable Koopman modes for stable flows.

• Not every nonlinear dynamical system requires a neural network to search for Koopman operator

  • we resolve the issue of classical methods by rethinking modes selection in EDMD/KDMD as a multi-task learning problem
  • demonstration on several strongly transient flows shows the effectiveness of the algorithm for providing an accurate reduced-order-model (ROM)

• KDMD can be expensive to computationally evaluate (online)/train (offline)

  • due to cubic complexity in the linear system step (although no one actually performs inverse exactly, but for forward computation it is inevitable)
  • we implement random Fourier features as a way to approximate kernel methods efficiently while enjoying the benefits of EDMD

Getting Started

Prerequisites

• python3
• scipy
• numpy
• scikit-learn
• pyDOE
• mpi4py

Usage

Example: KDMD on 2D cylinder flow past cylinder

Selection of hyperparameter

• go to EXAMPLE/cylinder_re100/hyp
• run parallem hyperparameter selection with 4 processes
  • mpirun -np 4 python3 cv_kdmd_hyp_20d_cyd.py
• then collecting the result and draw the figure by
  • python3 print.py
• the resulting figure result-num.png is a good refernce of choosing hyperparameter

Run SPKDMD

• go to EXAMPLE/cylinder_re100 folder

• run a standard KDMD default parameters

  • python3 example_20d_cyd.py

• perform multi-task learning mode selection

  • note that in run_apo_cyd_ekdmd.py, themax_iter = 1e5 is a very conservative choice for getting accurate result, one can choose max_iter=1e2 to just get a good result of the algorithm
  • python3 run_apo_cyd_ekdmd.py

• postprocessing the result of multi-task learning

  • python3 pps_cyd.py

• Note: that the data is top 20 POD coefficients with mean-subtracted.

Example: Simple toy 2D ODE system

• similarly just go to folder EXAMPLE/lusch

Roadmap

See the open issues for a list of proposed features (and known issues).

Current Limitations

• as it is designed for modal analysis, the error-checking framework is only working on analyzing one single trajectory

  • can be easily generalized to multiple validation trajectories. Just modify the code with several other validation trajectories.

• the standard multi-task solver sklearn.linear_model.MultiTaskElasticNet is okay for trajectory with around 100 to 1000 data points, while can be expensive for higher number of snapshots. However, the number of snapshots for most high-fidelity CFD datasets is no more than 1000 snapshots. It should be further improved by using ideas from variational projection:

  • Askham, Travis, and J. Nathan Kutz. "Variable projection methods for an optimized dynamic mode decomposition." SIAM Journal on Applied Dynamical Systems 17.1 (2018): 380-416.
  • Erichson, N. Benjamin, et al. "Sparse principal component analysis via variable projection." SIAM Journal on Applied Mathematics 80.2 (2020): 977-1002.

Contributing

Contributions are what make the open source community such an amazing place to be learn, inspire, and create. Any contributions you make are greatly appreciated.

1. Fork the Project
2. Create your Feature Branch (git checkout -b feature/AmazingFeature)
3. Commit your Changes (git commit -m 'Add some AmazingFeature')
4. Push to the Branch (git push origin feature/AmazingFeature)
5. Open a Pull Request

License

Distributed under the MIT License. See LICENSE for more information.

Contact

Shaowu Pan - email: shawnpan@umich.edu

Project Link: https://github.com/pswpswpsw/SKDMD

Citation

@article{pan2021sparsity,
title={Sparsity-promoting algorithms for the discovery of informative Koopman-invariant subspaces},
author={Pan, Shaowu and Arnold-Medabalimi, Nicholas and Duraisamy, Karthik},
journal={Journal of Fluid Mechanics},
volume={917},
year={2021},
publisher={Cambridge University Press}
}