Model adaptive phase space reconstruction

Author:

Dhadphale Jayesh M.1ORCID,Hauke Kraemer K.2ORCID,Gelbrecht Maximilian23ORCID,Kurths Jürgen2ORCID,Marwan Norbert24ORCID,Sujith R. I.1ORCID

Affiliation:

1. Department of Aerospace Engineering, Indian Institute of Technology Madras 1 , Chennai, Tamil Nadu 600036, India

2. Potsdam Institute for Climate Impact Research, Member of the Leibniz Association 2 , 14473 Potsdam, Germany

3. School of Engineering & Design, Technical University of Munich 3 , 80333 Munich, Germany

4. Institute of Physics and Astronomy, University of Potsdam 4 , 14476 Potsdam, Germany

Abstract

Phase space reconstruction (PSR) methods allow for the analysis of low-dimensional data with methods from dynamical systems theory, but their application to prediction models, such as those from machine learning (ML), is limited. Therefore, we here present a model adaptive phase space reconstruction (MAPSR) method that unifies the process of PSR with the modeling of the dynamical system. MAPSR is a differentiable PSR based on time-delay embedding and enables ML methods for modeling. The quality of the reconstruction is evaluated by the prediction loss. The discrete-time signal is converted into a continuous-time signal to achieve a loss function, which is differentiable with respect to the embedding delays. The delay vector, which stores all potential embedding delays, is updated along with the trainable parameters of the model to minimize prediction loss. Thus, MAPSR does not rely on any threshold or statistical criterion for determining the dimension and the set of delay values for the embedding process. We apply the MAPSR method to uni- and multivariate time series stemming from chaotic dynamical systems and a turbulent combustor. We find that for the Lorenz system, the model trained with the MAPSR method is able to predict chaotic time series for nearly seven to eight Lyapunov time scales, which is found to be much better compared to other PSR methods [AMI-FNN (average mutual information-false nearest neighbor) and PECUZAL (Pecora-Uzal) methods]. For the univariate time series from the turbulent combustor, the long-term cumulative prediction error of the MAPSR method for the regime of chaos stays between other methods, and for the regime of intermittency, MAPSR outperforms other PSR methods.

Funder

Ministry of Education, India

Office of Global Engagement, IIT Madras

Volkswagen Foundation

Publisher

AIP Publishing

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