分野別セミナー

Data-driven analysis of dynamical systems is fundamental in analyses of dynamic processes around us, where machine learning techniques often play important roles to develop estimation algorithms. Recently, operator-theoretic methods have attracted attention for this purpose, where the behavior of nonlinear dynamical systems is analyzed through the representations with transfer operators. In particular, Koopman operator plays an important role in the analysis because the spectral analysis of the operator directly gives physical interpretations of evolution equations. And, dynamic mode decomposition (DMD), which is a popular estimation method for the spectral analysis, has been applied to various scientific and engineering fields in a decade.


In this talk, I first overview the basic idea behind operator-theoretic analysis of dynamical systems, focusing on the one with transfer operators including Koopman operator, and then give brief reviews on DMD and its variants. Then, I describe some of our related works such as DMD using machine learning techniques, such as reproducing kernels, and the application of DMD-type methods to analyses of collective motions.


[1] Y. Kawahara, "Dynamic mode decomposition with reproducing kernels for Koopman spectral analysis," Advances in Neural Information Processing Systems 29 (Proc. of NIPS’16), pp.911-919, 2016.
[2] K. Fujii, Y. Inaba, and Y. Kawahara, "Koopman spectral kernels for comparing complex dynamics: Application to multiagent sport plays,” in Proc. of the 2017 European Conf. on Machine Learning and Principles and Practice of Knowledge Discovery in Databases (ECML-PKDD’17), pp.127–139, 2017.
[3] I. Ishikawa, K. Fujii, M. Ikeda, Y. Hashimoto, and Y. Kawahara, "Metric for nonlinear dynamical systems with Koopman operators," Advances in Neural Information Processing Systems 31 (Proc. of NeurIPS’18), pp.2858-2868, 2018.
[4] K. Fujii, T. Kawasaki, Y. Inaba, and Y. Kawahara, "Prediction and classification in equation-free collective motion dynamics," PLoS Computational Biology, 14(11): e1006545, 2018.
[5] K. Fujii, and Y. Kawahara, “Dynamic mode decomposition in vector-valued repro- ducing kernel Hilbert spaces for extracting dynamical structure among observables,” Neural Networks, 117: 94-103, 2019.