Pseudo-Hamiltonian neural networks with state-dependent external forces
Authors
Sølve Eidnes, Alexander J. Stasik, Camilla Sterud, Eivind Bøhn, Signe Riemer-Sørensen
Abstract
Hybrid machine learning based on Hamiltonian formulations has recently been successfully demonstrated for simple mechanical systems, both energy conserving and not energy conserving. We introduce a pseudo-Hamiltonian formulation that is a generalization of the Hamiltonian formulation via the port-Hamiltonian formulation, and show that pseudo-Hamiltonian neural network models can be used to learn external forces acting on a system. We argue that this property is particularly useful when the external forces are state dependent, in which case it is the pseudo-Hamiltonian structure that facilitates the separation of internal and external forces. Numerical results are provided for a forced and damped mass–spring system and a tank system of higher complexity, and a symmetric fourth-order integration scheme is introduced for improved training on sparse and noisy data.
Keywords
Pseudo-Hamiltonian neural networks; Physics-informed machine learning; Hybrid machine learning
Citation
- Journal: Physica D: Nonlinear Phenomena
- Year: 2023
- Volume: 446
- Issue:
- Pages: 133673
- Publisher: Elsevier BV
- DOI: 10.1016/j.physd.2023.133673
BibTeX
@article{Eidnes_2023,
title={{Pseudo-Hamiltonian neural networks with state-dependent external forces}},
volume={446},
ISSN={0167-2789},
DOI={10.1016/j.physd.2023.133673},
journal={Physica D: Nonlinear Phenomena},
publisher={Elsevier BV},
author={Eidnes, Sølve and Stasik, Alexander J. and Sterud, Camilla and Bøhn, Eivind and Riemer-Sørensen, Signe},
year={2023},
pages={133673}
}
References
- Hamilton, On a general method in dynamics. Philos. Trans. R. Soc. (1834)
- 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
- Rashad, R., Califano, F., van der Schaft, A. J. & Stramigioli, S. Twenty years of distributed port-Hamiltonian systems: a literature review. IMA Journal of Mathematical Control and Information vol. 37 1400–1422 (2020) – 10.1093/imamci/dnaa018
- Greydanus, Hamiltonian neural networks. (2019)
- 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
- Duong, Hamiltonian-based neural ODE networks on the SE(3) manifold for dynamics learning and control. (2021)
- Duong, (2021)
- Jin, Learning Poisson systems and trajectories of autonomous systems via Poisson neural networks. IEEE Trans. Neural Netw. Learn. Syst. (2022)
- Chen, Neural symplectic form: Learning Hamiltonian equations on general coordinate systems. Adv. Neural Inf. Process. Syst. (2021)
- Finzi, Simplifying Hamiltonian and Lagrangian neural networks via explicit constraints. Adv. Neural Inf. Process. Syst. (2020)
- Celledoni, E., Leone, A., Murari, D. & Owren, B. Learning Hamiltonians of constrained mechanical systems. Journal of Computational and Applied Mathematics vol. 417 114608 (2023) – 10.1016/j.cam.2022.114608
- McLachlan, R. I., Quispel, G. R. W. & Robidoux, N. Geometric integration using discrete gradients. Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences vol. 357 1021–1045 (1999) – 10.1098/rsta.1999.0363
- Van Der Schaft, Port-Hamiltonian systems: an introductory survey. (2006)
- Beattie, C. A., Mehrmann, V. & Van Dooren, P. Robust port-Hamiltonian representations of passive systems. Automatica vol. 100 182–186 (2019) – 10.1016/j.automatica.2018.11.013
- Cherifi, K. An overview on recent machine learning techniques for Port Hamiltonian systems. Physica D: Nonlinear Phenomena vol. 411 132620 (2020) – 10.1016/j.physd.2020.132620
- Benner, P., Goyal, P. & Van Dooren, P. Identification of port-Hamiltonian systems from frequency response data. Systems & Control Letters vol. 143 104741 (2020) – 10.1016/j.sysconle.2020.104741
- Cherifi, K., Goyal, P. & Benner, P. A non-intrusive method to inferring linear port-Hamiltonian realizations using time-domain data. ETNA - Electronic Transactions on Numerical Analysis vol. 56 102–116 (2022) – 10.1553/etna_vol56s102
- Morandin, (2022)
- Grmela, M. & Öttinger, H. C. Dynamics and thermodynamics of complex fluids. I. Development of a general formalism. Physical Review E vol. 56 6620–6632 (1997) – 10.1103/physreve.56.6620
- Öttinger, H. C. & Grmela, M. Dynamics and thermodynamics of complex fluids. II. Illustrations of a general formalism. Physical Review E vol. 56 6633–6655 (1997) – 10.1103/physreve.56.6633
- Zhang, GFINNs: GENERIC formalism informed neural networks for deterministic and stochastic dynamical systems. Philos. Trans. Roy. Soc. A (2022)
- Matsubara, Deep energy-based modeling of discrete-time physics. (2020)
- Kingma, (2014)
- Jin, P., Zhang, Z., Zhu, A., Tang, Y. & Karniadakis, G. E. SympNets: Intrinsic structure-preserving symplectic networks for identifying Hamiltonian systems. Neural Networks vol. 132 166–179 (2020) – 10.1016/j.neunet.2020.08.017
- David, (2021)
- van der Schaft, A. J. & Maschke, B. M. Port-Hamiltonian Systems on Graphs. SIAM Journal on Control and Optimization vol. 51 906–937 (2013) – 10.1137/110840091
- De Persis, C. & Kallesoe, C. S. Pressure Regulation in Nonlinear Hydraulic Networks by Positive and Quantized Controls. IEEE Transactions on Control Systems Technology vol. 19 1371–1383 (2011) – 10.1109/tcst.2010.2094619
- Hairer, (2006)
- van Bokhoven, W. M. G. Efficient higher order implicit one-step methods for integration of stiff differential equations. BIT Numerical Mathematics vol. 20 34–43 (1980) – 10.1007/bf01933583
- CASH, J. R. & SINGHAL, A. Mono-implicit Runge—Kutta Formulae for the Numerical Integration of Stiff Differential Systems. IMA Journal of Numerical Analysis vol. 2 211–227 (1982) – 10.1093/imanum/2.2.211
- DiPietro, Sparse symplectically integrated neural networks. Adv. Neural Inf. Process. Syst. (2020)
- Desai, S. A., Mattheakis, M. & Roberts, S. J. Variational integrator graph networks for learning energy-conserving dynamical systems. Physical Review E vol. 104 (2021) – 10.1103/physreve.104.035310
- Lee, Structure-preserving sparse identification of nonlinear dynamics for data-driven modeling. (2022)
- Cardoso-Ribeiro, Port-Hamiltonian modeling, discretization and feedback control of a circular water tank. (2019)
- Brugnoli, A., Haine, G., Serhani, A. & Vasseur, X. Numerical Approximation of Port-Hamiltonian Systems for Hyperbolic or Parabolic PDEs with Boundary Control. Journal of Applied Mathematics and Physics vol. 09 1278–1321 (2021) – 10.4236/jamp.2021.96088