https://doi.org/10.1140/epjb/s10051-025-00904-7
Regular Article - Statistical and Nonlinear Physics
Machine learning in parameter estimation of nonlinear systems
Institute for Mathematics, Heidelberg University, Im Neuenheimer Feld 205, 69120, Heidelberg, Baden-Württemberg, Germany
a
kaushal.kumar@uni-heidelberg.de
Received:
19
December
2024
Accepted:
18
March
2025
Published online:
9
April
2025
Accurate parameter estimation in nonlinear dynamical systems remains a fundamental challenge due to noise, limited data, and model complexity. Traditional methods, such as gradient-based optimization and nonlinear least squares (NLS), often struggle under real-world multiplicative noise, exhibiting sensitivity to outliers and high computational demands. This study introduces a neural network framework integrating the Huber loss function to achieve robust and efficient parameter estimation. Applied to canonical dynamical systems, including damped oscillators, van der Pol oscillators, Lotka–Volterra models, and chaotic Lorenz dynamics, the proposed method demonstrates superior accuracy and resilience to noise. Notably, it maintains sub-
relative errors for key parameters in the Lorenz system, significantly outperforming NLS, which diverges with errors exceeding 
under identical noise conditions. The use of SiLU activation improves convergence, yielding statistically significant reductions in estimation errors (
). Furthermore, the framework operates up to 
faster than conventional optimization techniques while reducing root-mean-square error by over 
in high-noise regimes. These results establish a robust, data-driven approach for parameter estimation in complex dynamical systems, bridging machine learning with nonlinear physics and enabling real-time applications in noisy environments.
© The Author(s) 2025
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