Model selection of chaotic systems from data with hidden variables using sparse data assimilation

TL;DR

This paper introduces a novel approach combining variational annealing and sparse optimization to identify models of chaotic systems with hidden variables from limited data, demonstrated on electrical circuit data.

Contribution

It presents a new method for model selection in chaotic systems with unmeasured variables, integrating variational annealing with sparse optimization techniques.

Findings

Successfully recovered circuit equations from experimental data.

Robustness demonstrated against noise and sampling variations.

Applicable to systems with hidden variables and limited measurements.

Abstract

Many natural systems exhibit chaotic behaviour such as the weather, hydrology, neuroscience and population dynamics. Although many chaotic systems can be described by relatively simple dynamical equations, characterizing these systems can be challenging, due to sensitivity to initial conditions and difficulties in differentiating chaotic behavior from noise. Ideally, one wishes to find a parsimonious set of equations that describe a dynamical system. However, model selection is more challenging when only a subset of the variables are experimentally accessible. Manifold learning methods using time-delay embeddings can successfully reconstruct the underlying structure of the system from data with hidden variables, but not the equations. Recent work in sparse-optimization based model selection has enabled model discovery given a library of possible terms, but regression-based methods require measurements of all state variables. We present a method combining variational annealing -- a technique previously used for parameter estimation in chaotic systems with hidden variables -- with sparse optimization methods to perform model identification for chaotic systems with unmeasured variables. We applied the method to experimental data from an electrical circuit with Lorenz-system like behavior to successfully recover the circuit equations with two measured and one hidden variable. We discuss the robustness of our method to varying noise and manifold sampling using ground-truth time-series simulated from the classic Lorenz system.