References

The method implemented here

  • Khan2023 Nigel Bruce Khan, Sparse Identification of Symplectic Hamiltonian Dynamics for Predictive Modeling and Analysis. Master's Thesis, Technische Universität München, 30 November 2023. Examiner: Univ.-Prof. Dr. Eric Sonnendrücker; scientific advisor: Dr. Michael Kraus. mediaTUM 1747893

    The source of the Hamiltonian-SINDy method and of this package. Introduces both the Hamiltonian-SINDy algorithm and an Auto-Encoder-Hamiltonian-SINDy variant that discovers canonical conjugate coordinates jointly with the dynamics; only the former is implemented here.

    Statements taken from this thesis are checked in scripts/verify_thesis_examples.jl rather than transcribed. Three of its equations do not hold as printed — see Nonlinear Oscillator and When It Fails.

Foundational SINDy

  • BruntonProctorKutz2016 Steven L. Brunton, Joshua L. Proctor, J. Nathan Kutz, Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proceedings of the National Academy of Sciences 113(15), 3932–3937, 2016. doi:10.1073/pnas.1517384113 · arXiv:1509.03580

    The original method: the library formulation $\dot X = \Theta(X)\Xi$, sequentially thresholded least squares, and the Lorenz and damped-oscillator examples reproduced in this documentation.

  • ZhangSchaeffer2019 Linan Zhang, Hayden Schaeffer, On the Convergence of the SINDy Algorithm. Multiscale Modeling & Simulation 17(3), 948–972, 2019. doi:10.1137/18M1189828 · arXiv:1805.06445

    The convergence analysis: STLSQ as a fixed-point iteration on $\lVert Ax-b\rVert_2^2 + \lambda^2\lVert x\rVert_0$, termination in at most $p$ steps, strict decrease, and convergence to a local minimiser. Also the ridge variant (STRidge) and the reason $\lambda$ and $\gamma$ compose rather than trade off.

Extensions referenced in the text

  • MessengerBortz2021 Daniel A. Messenger, David M. Bortz, Weak SINDy: Galerkin-Based Data-Driven Model Selection. Multiscale Modeling & Simulation 19(3), 1474–1497, 2021. doi:10.1137/20M1343166 · arXiv:2005.04339

    The weak formulation, which integrates against compactly supported test functions and so never differentiates noisy data. The principal answer to the derivative-estimation problem discussed in Sparse Identification. Not yet implemented here.

  • Champion2019 Kathleen Champion, Bethany Lusch, J. Nathan Kutz, Steven L. Brunton, Data-driven discovery of coordinates and governing equations. PNAS 116(45), 22445–22451, 2019. doi:10.1073/pnas.1906995116 · arXiv:1904.02107

    The autoencoder–SINDy coupling that the thesis's Auto-Encoder-Hamiltonian-SINDy builds on.

  • Fasel2022 Urban Fasel, J. Nathan Kutz, Bingni W. Brunton, Steven L. Brunton, Ensemble-SINDy: Robust sparse model discovery in the low-data, high-noise limit, with active learning and control. Proceedings of the Royal Society A 478(2260), 20210904, 2022. doi:10.1098/rspa.2021.0904 · arXiv:2111.10992

  • Kaptanoglu2022 Alan A. Kaptanoglu et al., PySINDy: A comprehensive Python package for robust sparse system identification. Journal of Open Source Software 7(69), 3994, 2022. doi:10.21105/joss.03994

    The reference implementation in Python, and a useful checklist of optimizers and feature libraries.

Structure-preserving identification

  • LeeTraskStinis2021 Kookjin Lee, Nathaniel Trask, Panos Stinis, Structure-preserving Sparse Identification of Nonlinear Dynamics for Data-driven Modeling. Proceedings of Mathematical and Scientific Machine Learning, PMLR 190, 65–80, 2022. arXiv:2109.05364

    A more general bracket-based framework covering both Poisson (conservative) and metric (dissipative) structure.

  • Greydanus2019 Samuel Greydanus, Misko Dzamba, Jason Yosinski, Hamiltonian Neural Networks. NeurIPS 2019. arXiv:1906.01563

    The neural counterpart: parametrise $H$ by a network rather than a sparse basis. Accurate but not interpretable, which is the trade-off sparse identification exists to avoid.

Software