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 \(\mathcal {O}(n)\) and not \(\mathcal {O}(n^2)\), as suggested by the standard parametrization of these systems.
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References
Gonzalez, O.: Time integration and discrete Hamiltonian systems. In: Mechanics: From Theory to Computation. Springer, New York, NY (2000). https://doi.org/10.1007/978-1-4612-1246-1_10
Grigoryeva, L., Ortega, J.P.: Dimension reduction in recurrent networks by canonicalization. J. Geom. Mech. 13(4), 647–677 (2021). https://doi.org/10.3934/jgm.2021028
Kalman, R.E.: Mathematical description of linear dynamical systems. J. Soc. Indus. Appl. Math. Ser. A Control. 1(2), 152–192 (1963). https://doi.org/10.1137/0301010
Karniadakis, G.E., Kevrekidis, I.G., Lu, L., Perdikaris, P., Wang, S., Yang, L.: Physics-informed machine learning. Nat. Rev. Phys. 3(6), 422–440 (2021)
Leimkuhler, B., Reich, S.: Simulating Hamiltonian Dynamics. Cambridge University Press, Cambridge (2004)
Marsden, J.E., West, M.: Discrete mechanics and variational integrators. Acta Numer. 10, 357–514 (2001). https://doi.org/10.1017/S096249290100006X
McLachlan, R.I., Quispel, G.R.W.: Geometric integrators for ODEs. J. Phys. A Math. Gener. 39(19), 5251 (2006)
Medianu, S., Lefevre, L., Stefanoiu, D.: Identifiability of linear lossless Port-controlled Hamiltonian systems. In: 2nd International Conference on Systems and Computer Science, pp. 56–61 (2013). https://doi.org/10.1109/IcConSCS.2013.6632023
Ortega, J.P., Yin, D.: Learnability of linear port-Hamiltonian systems. arXiv preprint arXiv:2303.15779 (2023)
Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics informed deep learning (part i): Data-driven solutions of nonlinear partial differential equations. arXiv preprint arXiv:1711.10561 (2017)
van der Schaft, A., Jeltsema, D.: Port-Hamiltonian systems theory: an introductory overview, vol. 1 (2014).https://doi.org/10.1561/2600000002
Sontag, E.: Mathematical Control Theory: Deterministic Finite Dimensional Systems. Springer-Verlag, Heidelberg (1998). https://doi.org/10.1007/978-3-540-69532-5_16
Williamson, J.: On the algebraic problem concerning the normal forms of linear dynamical systems. Am. J. Math. 58(1), 141–163 (1936)
Wu, J.L., Xiao, H., Paterson, E.: Physics-informed machine learning approach for augmenting turbulence models: a comprehensive framework. Phys. Rev. Fluids 3(7), 74602 (2018)
Acknowledgements
The authors thank Lyudmila Grigoryeva for helpful discussions and remarks and acknowledge partial financial support from the Swiss National Science Foundation (grant number 175801/1) and the School of Physical and Mathematical Sciences of the Nanyang Technological University. DY is funded by the Nanyang President’s Graduate Scholarship of Nanyang Technological University.
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Ortega, JP., Yin, D. (2023). Expressiveness and Structure Preservation in Learning Port-Hamiltonian Systems. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2023. Lecture Notes in Computer Science, vol 14072. Springer, Cham. https://doi.org/10.1007/978-3-031-38299-4_33
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