Koopman-based Deep Learning for Nonlinear System Estimation (2405.00627v2)
Abstract: Nonlinear differential equations are encountered as models of fluid flow, spiking neurons, and many other systems of interest in the real world. Common features of these systems are that their behaviors are difficult to describe exactly and invariably unmodeled dynamics present challenges in making precise predictions. In this paper, we present a novel data-driven linear estimator based on Koopman operator theory to extract meaningful finite-dimensional representations of complex non-linear systems. The Koopman model is used together with deep reinforcement networks that learn the optimal stepwise actions to predict future states of nonlinear systems. Our estimator is also adaptive to a diffeomorphic transformation of the estimated nonlinear system, which enables it to compute optimal state estimates without re-learning.
- S. Ostojic and N. Brunel, “From spiking neuron models to linear-nonlinear models,” PLoS computational biology, vol. 7, no. 1, p. e1001056, 2011.
- J. H. Tu, “Dynamic mode decomposition: Theory and applications,” Ph.D. dissertation, Princeton University, 2013.
- P. J. Schmid, “Dynamic mode decomposition of numerical and experimental data,” Journal of fluid mechanics, vol. 656, pp. 5–28, 2010.
- M. O. Williams, I. G. Kevrekidis, and C. W. Rowley, “A data–driven approximation of the koopman operator: Extending dynamic mode decomposition,” Journal of Nonlinear Science, vol. 25, pp. 1307–1346, 2015.
- S. L. Brunton, “Notes on koopman operator theory,” Universität von Washington, Department of Mechanical Engineering, Zugriff, vol. 30, 2019.
- E. Yeung, S. Kundu, and N. Hodas, “Learning deep neural network representations for koopman operators of nonlinear dynamical systems,” in 2019 American Control Conference (ACC). IEEE, 2019, pp. 4832–4839.
- P. Bevanda, J. Kirmayr, S. Sosnowski, and S. Hirche, “Learning the koopman eigendecomposition: A diffeomorphic approach,” in 2022 American Control Conference (ACC). IEEE, 2022, pp. 2736–2741.
- P. Bevanda, M. Beier, S. Kerz, A. Lederer, S. Sosnowski, and S. Hirche, “Diffeomorphically learning stable koopman operators,” IEEE Control Systems Letters, vol. 6, pp. 3427–3432, 2022.
- S. Mowlavi, M. Benosman, and S. Nabi, “Reinforcement learning-based estimation for partial differential equations,” arXiv preprint arXiv:2302.01189, 2023.
- S. L. Brunton and B. R. Noack, “Closed-loop turbulence control: Progress and challenges,” Applied Mechanics Reviews, vol. 67, no. 5, p. 050801, 2015.
- S. Klus, P. Koltai, and C. Schütte, “On the numerical approximation of the perron-frobenius and koopman operator,” arXiv preprint arXiv:1512.05997, 2015.
- M. Korda and I. Mezić, “On convergence of extended dynamic mode decomposition to the koopman operator,” Journal of Nonlinear Science, vol. 28, pp. 687–710, 2018.
- T. P. Lillicrap, J. J. Hunt, A. Pritzel, N. Heess, T. Erez, Y. Tassa, D. Silver, and D. Wierstra, “Continuous control with deep reinforcement learning,” arXiv preprint arXiv:1509.02971, 2015.
- Z. Sun and J. Baillieul, “Emulation learning for neuromimetic systems,” 2023 62nd IEEE Conference on Decision and Control (CDC) (and also arXiv preprint arXiv:2305.03196), pp. 8292–8299, 2023.
- Y. Lan and I. Mezić, “Linearization in the large of nonlinear systems and koopman operator spectrum,” Physica D: Nonlinear Phenomena, vol. 242, no. 1, pp. 42–53, 2013.