Data-driven Reconstruction of Nonlinear Dynamics from Sparse Observation

Abstract

We present a data-driven model to reconstruct nonlinear dynamics from a very sparse times series data, which relies on the strength of the echo state network (ESN) in learning nonlinear representation of data. With an assumption of the universal function approximation capability of ESN, it is shown that the reconstruction problem can be formulated as a fixed-point problem, in which the trajectory of the dynamical system is a fixed point of the ESN. An under-relaxed fixed-point iteration is proposed to reconstruct the nonlinear dynamics from a sparse observation. The proposed fixed-point ESN is tested against both univariate and multivariate chaotic dynamical systems by randomly removing up to 95% of the data. It is shown that the fixed-point ESN is able to reconstruct the complex dynamics from only 5 ~ 10% of the data. For a relatively simple non-chaotic dynamical system, the numerical experiments on a forced van der Pol oscillator show that it is possible to reconstruct the nonlinear dynamics from only 1~2% of the data.

References in corpus (6)

• Deep Learning in Neural Networks: An Overview
• Attractor Reconstruction by Machine Learning
• BRITS: Bidirectional Recurrent Imputation for Time Series
• Deep learning algorithm for data-driven simulation of noisy dynamical system
• Using a reservoir computer to learn chaotic attractors, with applications to chaos synchronisation and cryptography
• Short note on the behavior of recurrent neural network for noisy dynamical system

Cited by in corpus (6)

• Embedding and Approximation Theorems for Echo State Networks
• Echo State Networks trained by Tikhonov least squares are L2(μ) approximators of ergodic dynamical systems
• Reservoir Computing Benchmarks: a tutorial review and critique
• Semi Conditional Variational Auto-Encoder for Flow Reconstruction and Uncertainty Quantification from Limited Observations
• Stability of a Parametrically Driven, Coupled Oscillator System: An Auxillary Function Method Approach
• (Thesis) Reservoir Computing With Dynamical Systems