Expressiveness and Structure Preservation in Learning Port-Hamiltonian Systems

Authors

Abstract

A well-specified parametrization for single-input/single-output (SISO) linear port-Hamiltonian systems amenable to structure-preserving supervised learning is provided. The construction is based on controllable and observable normal form Hamiltonian representations for those systems, which reveal fundamental relationships between classical notions in control theory and crucial properties in the machine learning context, like structure-preservation and expressive power. The results in the paper suggest parametrizations of the estimation problem associated with these systems that amount, at least in the canonical case, to unique identification and prove that the parameter complexity necessary for the replication of the dynamics is only \( \)\mathcal {O}(n)\( \) O ( n ) and not \( \)\mathcal {O}(n^2)\( \) O ( n 2 ) , as suggested by the standard parametrization of these systems.

Keywords

Linear port-Hamiltonian system; machine learning; structure-preserving algorithm; systems theory; physics-informed machine learning

Citation

ISBN: 9783031382987
Publisher: Springer Nature Switzerland
DOI: 10.1007/978-3-031-38299-4_33
Note: International Conference on Geometric Science of Information

BibTeX

@inbook{Ortega_2023,
  title={{Expressiveness and Structure Preservation in Learning Port-Hamiltonian Systems}},
  ISBN={9783031382994},
  ISSN={1611-3349},
  DOI={10.1007/978-3-031-38299-4_33},
  booktitle={{Geometric Science of Information}},
  publisher={Springer Nature Switzerland},
  author={Ortega, Juan-Pablo and Yin, Daiying},
  year={2023},
  pages={313--322}
}

Download the bib file

References

Gonzalez, O. Time Integration and Discrete Hamiltonian Systems. Mechanics: From Theory to Computation 257–275 (2000) doi:10.1007/978-1-4612-1246-1_10 – 10.1007/978-1-4612-1246-1_10
Grigoryeva, L. & Ortega, J.-P. Dimension reduction in recurrent networks by canonicalization. JGM 13, 647 (2021) – 10.3934/jgm.2021028
Kalman, R. E. Mathematical Description of Linear Dynamical Systems. Journal of the Society for Industrial and Applied Mathematics Series A Control 1, 152–192 (1963) – 10.1137/0301010
Karniadakis, G. E. et al. Physics-informed machine learning. Nat Rev Phys 3, 422–440 (2021) – 10.1038/s42254-021-00314-5
Leimkuhler, B. & Reich, S. Simulating Hamiltonian Dynamics. (2005) doi:10.1017/cbo9780511614118 – 10.1017/cbo9780511614118
Marsden, J. E. & West, M. Discrete mechanics and variational integrators. Acta Numerica 10, 357–514 (2001) – 10.1017/s096249290100006x
McLachlan, R. I. & Quispel, G. R. W. Geometric integrators for ODEs. J. Phys. A: Math. Gen. 39, 5251–5285 (2006) – 10.1088/0305-4470/39/19/s01
Medianu, S., Lefevre, L. & Stefanoiu, D. Identifiability of linear lossless Port-controlled Hamiltonian systems. 2nd International Conference on Systems and Computer Science 56–61 (2013) doi:10.1109/icconscs.2013.6632023 – 10.1109/icconscs.2013.6632023
van der Schaft, A. & Jeltsema, D. Port-Hamiltonian Systems Theory: An Introductory Overview. FnT in Systems and Control 1, 173–378 (2014) – 10.1561/2600000002
Menoncin, F. An Approximate Solution for Optimal Portfolio in Incomplete Markets. Mathematical Control Theory and Finance 293–310 (2008) doi:10.1007/978-3-540-69532-5_16 – 10.1007/978-3-540-69532-5_16
Williamson, J. On the Algebraic Problem Concerning the Normal Forms of Linear Dynamical Systems. American Journal of Mathematics 58, 141 (1936) – 10.2307/2371062
Wu, J.-L., Xiao, H. & Paterson, E. Physics-informed machine learning approach for augmenting turbulence models: A comprehensive framework. Phys. Rev. Fluids 3, (2018) – 10.1103/physrevfluids.3.074602