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This toolbox contains methods for the approximation of transfer operators and their eigenfunctions as well as methods for learning the governing equations from data:

• DMD, TICA, AMUSE
• Ulam's method
• EDMD, kernel EDMD, generator EDMD
• SINDy
• kernel PCA, kernel CCA
• CMD
• SEBA

The algorithms are implemented based on the following publications:

• F. Bach and M. Jordan. Kernel Independent Component Analysis.
• S. Brunton, J. Proctor, and J. Kutz. Discovering governing equations from data by sparse identification of nonlinear dynamical systems.
• G. Froyland, C. Rock, and K. Sakellariou. Sparse eigenbasis approximation: Multiple feature extraction across spatiotemporal scales with application to coherent set identification.
• F. Noé and F. Nüske. A variational approach to modeling slow processes in stochastic dynamical systems.
• B. Schölkopf, A. Smola, and K.-R. Müller. Nonlinear component analysis as a kernel eigenvalue problem.
• C. Schwantes and V. Pande. Modeling Molecular Kinetics with tICA and the Kernel Trick.
• L. Tong, V. Soon, Y. Huang, and R. Liu. AMUSE: a new blind identification algorithm.
• J. Tu, C. Rowley, D. Luchtenburg, S. Brunton, and J. Kutz. On dynamic mode decomposition: Theory and applications.
• M. Williams, I. Kevrekidis, and C. Rowley. A data-driven approximation of the Koopman operator: Extending dynamic mode decomposition.
• M. Williams, C. Rowley, and I. Kevrekidis. A kernel-based method for data-driven Koopman spectral analysis.
• S. Klus, P. Koltai, and C. Schütte. On the numerical approximation of the Perron-Frobenius and Koopman operator.
• S. Klus, F. Nüske, P. Koltai, H. Wu, I. Kevrekidis, C. Schütte, and F. Noé. Data-driven model reduction and transfer operator approximation.
• S. Klus, I. Schuster, and K. Muandet. Eigendecompositions of transfer operators in reproducing kernel Hilbert spaces.
• S. Klus, B. E. Husic, and M. Mollenhauer: Kernel canonical correlation analysis approximates operators for the detection of coherent structures in dynamical data.

The ODE/SDE solvers required by some examples to generate trajectory data are implemented in C++. In order to create Python bindings, first install pybind11, then go to the cpp directory and compile the code by executing the following command:

Linux: g++ -O3 -Wall -shared -std=c++11 -fPIC `python3 -m pybind11 --includes` systems.cpp -o ../d3s/systems`python3-config --extension-suffix`
MAC:   c++ -O3 -Wall -shared -std=c++11 -undefined dynamic_lookup `python3 -m pybind11 --includes` systems.cpp -o ../d3s/systems`python3-config --extension-suffix`