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Koopman operator based identification of nonlinear networks

Ramachandran Anantharaman, Alexandre Mauroy

TL;DR

The paper addresses the challenge of identifying nonlinear networked dynamics with heterogeneous node behavior and inputs from data. It introduces a two-step Koopman operator–based framework: first, identify the network topology (neighbors and inputs) via a dual lifting method; then, perform local dynamics identification using a lifted linear model. The approach scales to large, sparse networks and yields a modular, linear lifted representation of the whole network, enabling analysis and control design. Numerical examples on Erdos-Renyi graphs, non-polynomial vector fields, and Hindmarsh-Rose neuron networks demonstrate accurate topology reconstruction and parameter estimation, outperforming prior methods in data efficiency and scalability.

Abstract

In this work, we develop a method to identify continuous-time nonlinear networked dynamics via the Koopman operator framework. The proposed technique consists of two steps: the first step identifies the neighbors of each node, and the second step identifies the local dynamics at each node from a predefined set of dictionary functions. The technique can be used to either identify the Boolean network of interactions (first step) or to solve the complete network identification problem that amounts to estimating the local node dynamics and the nature of the node interactions (first and second steps). Under a sparsity assumption, the data required to identify the complete network dynamics is significantly less than the total number of dictionary functions describing the dynamics. This makes the proposed approach attractive for identifying large dimensional networks with sparse interconnections. The accuracy and performance of the proposed identification technique are demonstrated with several examples.

Koopman operator based identification of nonlinear networks

TL;DR

The paper addresses the challenge of identifying nonlinear networked dynamics with heterogeneous node behavior and inputs from data. It introduces a two-step Koopman operator–based framework: first, identify the network topology (neighbors and inputs) via a dual lifting method; then, perform local dynamics identification using a lifted linear model. The approach scales to large, sparse networks and yields a modular, linear lifted representation of the whole network, enabling analysis and control design. Numerical examples on Erdos-Renyi graphs, non-polynomial vector fields, and Hindmarsh-Rose neuron networks demonstrate accurate topology reconstruction and parameter estimation, outperforming prior methods in data efficiency and scalability.

Abstract

In this work, we develop a method to identify continuous-time nonlinear networked dynamics via the Koopman operator framework. The proposed technique consists of two steps: the first step identifies the neighbors of each node, and the second step identifies the local dynamics at each node from a predefined set of dictionary functions. The technique can be used to either identify the Boolean network of interactions (first step) or to solve the complete network identification problem that amounts to estimating the local node dynamics and the nature of the node interactions (first and second steps). Under a sparsity assumption, the data required to identify the complete network dynamics is significantly less than the total number of dictionary functions describing the dynamics. This makes the proposed approach attractive for identifying large dimensional networks with sparse interconnections. The accuracy and performance of the proposed identification technique are demonstrated with several examples.

Paper Structure

This paper contains 24 sections, 3 theorems, 99 equations, 9 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Koopman operator framework for dynamical systems
  3. The Koopman operator and its infinitesimal generator
  4. Data-driven approximation of the Koopman operator and lifted dynamics
  5. System identification using the Koopman operator framework
  6. Dual lifting method for systems with inputs
  7. Lifting and construction of the Koopman Operator
  8. Test functions for computing the Koopman Operator
  9. Example.
  10. Network Identification using Koopman Operator
  11. Problem formulation
  12. First step - Identification of the topology
  13. Choice of node functions
  14. Estimating the neighbors of a node
  15. Second step - local identification of the dynamics
  16. ...and 9 more sections

Key Result

proposition thmcounterproposition

Given the sets of test functions $G_i$ and the associated errors eq:e_i-eq:te_i, with $i=1,\dots,r$,

Figures (9)

  • Figure 1: According to Proposition \ref{['lem:Test-fun']}, the errors $e$ and $\tilde{e}$ reach their minimum at the same value of the parameter $\gamma$.
  • Figure 2: Effect of varying the amount of data on a network with $N = 400$, probability of an edge $\rho = 0.005$, noise variance $\sigma = 0.01$, and threshold $\delta = 0.1$.
  • Figure 3: Effect of varying the size of a network with data points $K = N$, probability of an edge $\rho = 0.005$, noise variance $\sigma = 0.01$, and threshold $\delta = 0.1$.
  • Figure 4: Effect of varying the noise variance on a network with $N = 400$, probability of an edge $\rho = 0.005$, number of data points $K = 400$, and threshold $\delta = 0.1$.
  • Figure 5: Effect of varying the edge probability $\rho$ with $N = 50$, noise variance $\sigma = 0.01$, and number of data points $K = 600$.
  • ...and 4 more figures

Theorems & Definitions (6)

  • proposition thmcounterproposition
  • proof
  • lemma thmcounterlemma
  • proof
  • theorem 1
  • proof