Reconstruction of dynamic systems using genetic algorithms with dynamic search limits

TL;DR

This work tackles the problem of reconstructing governing dynamic equations from time-series data by introducing a polynomial-ansatz representation of the dynamics and estimating the coefficients with a genetic algorithm. A key innovation is the dynamic search-limits mechanism, which adaptively tightens or relaxes coefficient bounds and prunes negligible terms to promote sparsity and avoid local optima. Across linear, nonlinear, and Lorenz systems, the approach with dynamic limits achieves near-perfect fits ($R^2$ ~ 0.9999) and very low $ISE$ values, while a fixed-limit GA struggles to recover the Lorenz dynamics. The method yields parsimonious, interpretable models and demonstrates robustness in data-driven dynamic model discovery, albeit with higher computational cost and without a universal hyperparameter-tuning recipe.

Abstract

Mathematical modeling is a powerful tool for describing, predicting, and understanding complex phenomena exhibited by real-world systems. However, identifying the equations that govern a system's dynamics from experimental data remains a significant challenge without a definitive solution. In this study, evolutionary computing techniques are presented to estimate the governing equations of a dynamical system using time-series data. The main approach is to propose polynomial equations with unknown coefficients, and subsequently perform a parametric estimation using genetic algorithms. Some of the main contributions of the present study are an adequate modification of the genetic algorithm to remove terms with minimal contributions, and a mechanism to escape local optima during the search. To evaluate the proposed method, we applied it to three dynamical systems: a linear model, a nonlinear model, and the Lorenz system. Our results demonstrate a reconstruction with an Integral Square Error below 0.22 and a coefficient of determination R-squared of 0.99 for all systems, indicating successful reconstruction of the governing dynamic equations.

Figures (12)

• Figure 1: General scheme of the proposed strategy for implementing dynamic limits for the genetic algorithms.
• Figure 2: Performance of the GA for the reconstruction of the linear a) and nonlinear dynamical system\ref{['oscCub']}.
• Figure 3: Comparison between the real values, $\dot{x}$ and $\dot{y}$, and those estimated by the GA, $\dot{x}_{est}$ and $\dot{y}_{est}$. a) and c) for the linear system \ref{['oscLin']}, and b) and d) for the nonlinear system \ref{['oscCub']}.
• Figure 4: Comparisons of real and estimated signals for a) the linear system, and b) the nonlinear system.
• Figure 5: Performance of the genetic algorithm in the reconstruction of the Lorenz system.