Gaussian Process Port-Hamiltonian Systems: Bayesian Learning with Physics Prior

Authors

Thomas Beckers, Jacob Seidman, Paris Perdikaris, George J. Pappas

Abstract

Data-driven approaches achieve remarkable results for the modeling of complex dynamics based on collected data. However, these models often neglect basic physical principles which determine the behavior of any real-world system. This omission is unfavorable in two ways: The models are not as data-efficient as they could be by incorporating physical prior knowledge, and the model itself might not be physically correct. We propose Gaussian Process Port-Hamiltonian systems (GPPHS) as a physics-informed Bayesian learning approach with uncertainty quantification. The Bayesian nature of GP-PHS uses collected data to form a distribution over all possible Hamiltonians instead of a single point estimate. Due to the underlying physics model, a GP-PHS generates passive systems with respect to designated inputs and outputs. Further, the proposed approach preserves the compositional nature of Port-Hamiltonian systems.

Citation

• Journal: 2022 IEEE 61st Conference on Decision and Control (CDC)
• Year: 2022
• Volume:
• Issue:
• Pages: 1447–1453
• Publisher: IEEE
• DOI: 10.1109/cdc51059.2022.9992733

BibTeX

@inproceedings{Beckers_2022,
  title={{Gaussian Process Port-Hamiltonian Systems: Bayesian Learning with Physics Prior}},
  DOI={10.1109/cdc51059.2022.9992733},
  booktitle={{2022 IEEE 61st Conference on Decision and Control (CDC)}},
  publisher={IEEE},
  author={Beckers, Thomas and Seidman, Jacob and Perdikaris, Paris and Pappas, George J.},
  year={2022},
  pages={1447--1453}
}

Download the bib file

References

• Derler, P., Lee, E. A. & Vincentelli, A. S. Modeling Cyber–Physical Systems. Proceedings of the IEEE vol. 100 13–28 (2012) – 10.1109/jproc.2011.2160929
• Close, Modeling and analysis of dynamic systems (2001)
• Andrian, E., Grenier, D. & Rouabhia, M. In Vitro Models of Tissue Penetration and Destruction b Porphyromonas gingivalis. Infection and Immunity vol. 72 4689–4698 (2004) – 10.1128/iai.72.8.4689-4698.2004
• Hou, Z.-S. & Wang, Z. From model-based control to data-driven control: Survey, classification and perspective. Information Sciences vol. 235 3–35 (2013) – 10.1016/j.ins.2012.07.014
• Karniadakis, G. E. et al. Physics-informed machine learning. Nature Reviews Physics vol. 3 422–440 (2021) – 10.1038/s42254-021-00314-5
• Vilasi, G. Hamiltonian Dynamics. (2001) doi:10.1142/3637 – 10.1142/3637
• van der Schaft, A. & Jeltsema, D. Port-Hamiltonian Systems Theory: An Introductory Overview. Foundations and Trends® in Systems and Control vol. 1 173–378 (2014) – 10.1561/2600000002
• van der Schaft, A. L2 - Gain and Passivity Techniques in Nonlinear Control. Communications and Control Engineering (Springer London, 2000). doi:10.1007/978-1-4471-0507-7 – 10.1007/978-1-4471-0507-7
• Greydanus, Hamiltonian neural networks. Advances in Neural Information Processing Systems (2019)
• Bertalan, T., Dietrich, F., Mezić, I. & Kevrekidis, I. G. On learning Hamiltonian systems from data. Chaos: An Interdisciplinary Journal of Nonlinear Science vol. 29 (2019) – 10.1063/1.5128231
• Desai, S. A., Mattheakis, M., Sondak, D., Protopapas, P. & Roberts, S. J. Port-Hamiltonian neural networks for learning explicit time-dependent dynamical systems. Physical Review E vol. 104 (2021) – 10.1103/physreve.104.034312
• Nageshrao, S. P., Lopes, G. A. D., Jeltsema, D. & Babuska, R. Port-Hamiltonian Systems in Adaptive and Learning Control: A Survey. IEEE Transactions on Automatic Control vol. 61 1223–1238 (2016) – 10.1109/tac.2015.2458491
• Rasmussen, C. E. & Williams, C. K. I. Gaussian Processes for Machine Learning. (2005) doi:10.7551/mitpress/3206.001.0001 – 10.7551/mitpress/3206.001.0001
• Rath, K., Albert, C. G., Bischl, B. & von Toussaint, U. Symplectic Gaussian process regression of maps in Hamiltonian systems. Chaos: An Interdisciplinary Journal of Nonlinear Science vol. 31 (2021) – 10.1063/5.0048129
• Ridderbusch, S., Offen, C., Ober-Blobaum, S. & Goulart, P. Learning ODE Models with Qualitative Structure Using Gaussian Processes. 2021 60th IEEE Conference on Decision and Control (CDC) 2896–2896 (2021) doi:10.1109/cdc45484.2021.9683426 – 10.1109/cdc45484.2021.9683426
• Raissi, M., Perdikaris, P. & Karniadakis, G. E. Numerical Gaussian Processes for Time-Dependent and Nonlinear Partial Differential Equations. SIAM Journal on Scientific Computing vol. 40 A172–A198 (2018) – 10.1137/17m1120762
• Bhouri, M. A. & Perdikaris, P. Gaussian processes meet NeuralODEs: a Bayesian framework for learning the dynamics of partially observed systems from scarce and noisy data. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences vol. 380 (2022) – 10.1098/rsta.2021.0201
• Álvarez, M. A., Rosasco, L. & Lawrence, N. D. Kernels for Vector-Valued Functions: A Review. Foundations and Trends® in Machine Learning vol. 4 195–266 (2012) – 10.1561/2200000036
• Maschke, B. M. & van der Schaft, A. J. PORT-CONTROLLED HAMILTONIAN SYSTEMS: MODELLING ORIGINS AND SYSTEMTHEORETIC PROPERTIES. Nonlinear Control Systems Design 1992 359–365 (1993) doi:10.1016/b978-0-08-041901-5.50064-6 – 10.1016/b978-0-08-041901-5.50064-6
• Beckers, T. & Hirche, S. Equilibrium distributions and stability analysis of Gaussian Process State Space Models. 2016 IEEE 55th Conference on Decision and Control (CDC) 6355–6361 (2016) doi:10.1109/cdc.2016.7799247 – 10.1109/cdc.2016.7799247
• Cervera, J., van der Schaft, A. J. & Baños, A. Interconnection of port-Hamiltonian systems and composition of Dirac structures. Automatica vol. 43 212–225 (2007) – 10.1016/j.automatica.2006.08.014
• Ortega, R. & García-Canseco, E. Interconnection and Damping Assignment Passivity-Based Control: A Survey. European Journal of Control vol. 10 432–450 (2004) – 10.3166/ejc.10.432-450
• Adler, R. J. The Geometry of Random Fields. (2010) doi:10.1137/1.9780898718980 – 10.1137/1.9780898718980
• Huber, M. F. Recursive Gaussian process: On-line regression and learning. Pattern Recognition Letters vol. 45 85–91 (2014) – 10.1016/j.patrec.2014.03.004
• de Boor, C. A Practical Guide to Splines. Applied Mathematical Sciences (Springer New York, 1978). doi:10.1007/978-1-4612-6333-3 – 10.1007/978-1-4612-6333-3
• Wilson, Efficiently sampling functions from Gaussian process posteriors. International Conference on Machine Learning
• Beckers, T. & Hirche, S. Prediction With Approximated Gaussian Process Dynamical Models. IEEE Transactions on Automatic Control vol. 67 6460–6473 (2022) – 10.1109/tac.2021.3131988
• Wilson, Kernel interpolation for scalable structured Gaussian processes (KISS-GP). International conference on machine learning