Machine learning in parameter estimation of nonlinear systems

Abstract 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- $$1.2\%$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>1.2</mml:mn> <mml:mo>%</mml:mo> </mml:mrow> </mml:math> relative errors for key parameters in the Lorenz system, significantly outperforming NLS, which diverges with errors exceeding $$12\%$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>12</mml:mn> <mml:mo>%</mml:mo> </mml:mrow> </mml:math> under identical noise conditions. The use of SiLU activation improves convergence, yielding statistically significant reductions in estimation errors ( $$p &lt; 0.01$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mn>0.01</mml:mn> </mml:mrow> </mml:math> ). Furthermore, the framework operates up to $$8\times $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>8</mml:mn> <mml:mo>×</mml:mo> </mml:mrow> </mml:math> faster than conventional optimization techniques while reducing root-mean-square error by over $$99.9\%$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>99.9</mml:mn> <mml:mo>%</mml:mo> </mml:mrow> </mml:math> 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. Graphic Abstract