Finding nonlinear system equations and complex network structures from data: a sparse optimization approach

Abstract

In applications of nonlinear and complex dynamical systems, a common situation is that the system can be measured but its structure and the detailed rules of dynamical evolution are unknown. The inverse problem is to determine the system equations and structure based solely on measured time series. Recently, methods based on sparse optimization have been developed. For example, the principle of exploiting sparse optimization such as compressive sensing to find the equations of nonlinear dynamical systems from data was articulated in 2011 by the Nonlinear Dynamics Group at Arizona State University. This article presents a brief review of the recent progress in this area. The basic idea is to expand the equations governing the dynamical evolution of the system into a power series or a Fourier series of a finite number of terms and then to determine the vector of the expansion coefficients based solely on data through sparse optimization. Examples discussed here include discovering the equations of stationary or nonstationary chaotic systems to enable prediction of dynamical events such as critical transition and system collapse, inferring the full topology of complex networks of dynamical oscillators and social networks hosting evolutionary game dynamics, and identifying partial differential equations for spatiotemporal dynamical systems. Situations where sparse optimization is effective and those in which the method fails are discussed. Comparisons with the traditional method of delay coordinate embedding in nonlinear time series analysis are given and the recent development of model-free, data driven prediction framework based on machine learning is briefly introduced.

arXiv comment23 pages, 2 figures. arXiv admin note: text overlap with arXiv:1704.08764

References in corpus (30)

• Scikit-learn: Machine Learning in Python
• Epidemic spreading in scale-free networks
• Catastrophic cascade of failures in interdependent networks
• Statistical physics of social dynamics
• Epidemic processes in complex networks
• Evolutionary games on graphs
• Evolutionary prisoner's dilemma game on a square lattice
• Practical implementation of nonlinear time series methods: The TISEAN package
• Dynamical interplay between awareness and epidemic spreading in multiplex networks
• Voter Model on Heterogeneous Graphs
• Using Machine Learning to Replicate Chaotic Attractors and Calculate Lyapunov Exponents from Data
• Attractor Reconstruction by Machine Learning
• Predicting catastrophes in nonlinear dynamical systems by compressive sensing
• The extreme vulnerability of interdependent spatially embedded networks
• Hybrid Forecasting of Chaotic Processes: Using Machine Learning in Conjunction with a Knowledge-Based Model
• Data Based Identification and Prediction of Nonlinear and Complex Dynamical Systems
• Phase transitions in contagion processes mediated by recurrent mobility patterns
• Reconstructing propagation networks with natural diversity and identifying hidden sources
• Binary-state dynamics on complex networks: pair approximation and beyond
• Robust Reconstruction of Complex Networks from Sparse Data
• Forecasting Chaotic Systems with Very Low Connectivity Reservoir Computers
• A Unified Approach to Attractor Reconstruction
• Reservoir computing with a single time-delay autonomous Boolean node
• Using Noisy or Incomplete Data to Discover Models of Spatiotemporal Dynamics
• Using reservoir computers to distinguish chaotic signals
• Machine-learning inference of fluid variables from data using reservoir computing
• Robust and optimal sparse regression for nonlinear PDE models
• How Entropic Regression Beats the Outliers Problem in Nonlinear System Identification
• A universal data based method for reconstructing complex networks with binary-state dynamics
• Data-driven discovery of partial differential equation models with latent variables

Cited by in corpus (6)

• Next Generation Reservoir Computing
• Learning Spatiotemporal Chaos Using Next-Generation Reservoir Computing
• Dynamical System Identification, Model Selection and Model Uncertainty Quantification by Bayesian Inference
• Symbolic Regression via Neural Networks
• Observing network dynamics through sentinel nodes
• Dynamical and statistical properties of estimated high-dimensional ODE models: The case of the Lorenz'05 type II model