<p>Accurate parameters identification in nonlinear dynamical systems is crucial for understanding complex phenomena. Fractional Brownian motion is widely used in modeling nonlinear systems due to its long memory, but it also complicates parameter identification. Traditional methods typically rely on linear models or assume Gaussian white noise, which makes them less effective for handling such complex systems. Although data-driven methods have been proposed, most of them utilize simple network architectures and focus primarily on identifying the Hurst exponent or parameters of linear systems. This paper presents a deep learning-based parameter identifier capable of jointly identifying all parameters of a nonlinear dynamical system driven by fractional Brownian motion. Firstly, a new parameter identifier (Transformer-Bidirectional Long Short-Term Memory Neural Network) is constructed by combining the advantages of Transformer in dealing with long-term dependence and Bidirectional Long Short-Term Memory in extracting local features. We then incorporate fractional Brownian motion as the random effect in the nonlinear dynamical system and provide three examples to simulate the system. Finally, the proposed identifier is used to identify parameters in the three systems individually, and its performance is compared to that of the Convolutional Neural Network-Bidirectional Long Short-Term Memory-Attention Neural Network and the Parameter Estimation Neural Network to demonstrate its effectiveness. Taking Example 3 and using the mean absolute error to evaluate identification accuracy. The mean absolute error values for the Transformer-Bidirectional Long Short-Term Memory Neural Network are 0.0171, 0.0699, 0.0328 and 0.0851, which are significantly better than the Convolutional Neural Network-Bidirectional Long Short-Term Memory-Attention Neural Network, with values of 0.0277, 0.1054, 0.0494 and 0.1037, and the Parameter Estimation Neural Network, with values of 0.0419, 0.1265, 0.0708 and 0.1222. In terms of identification speed, the average identification time for the Transformer-Bidirectional Long Short-Term Memory Neural Network is 0.005907 seconds, faster than the Convolutional Neural Network-Bidirectional Long Short-Term Memory-Attention Neural Network at 0.008296 seconds and the Parameter Estimation Neural Network at 0.035498 seconds. These quantitative results demonstrate that the proposed identifier can identify all parameters of the nonlinear dynamical systems more quickly and accurately. Additionally, three key advantages of the proposed identifier are discussed, including the stability of parameter identification results, the rationality of sample size selection, and the advantages of the network construction. The contributions of this paper include: (1) Modeling the complex nonlinear dynamical system driven by fractional Brownian motion; (<InternalRef RefID="Equ2">2</InternalRef>) Proposing a new parameters identifier by integrating Transformer and Bidirectional Long Short-Term Memory; (<InternalRef RefID="Equ3">3</InternalRef>) Employing the orthogonal design table to arrange experiments for optimal parameter combination selection. For a full list of acronyms used in the paper, please refer to Table <InternalRef RefID="Tab1">1</InternalRef>.</p>

Deep learning for parameter identification of nonlinear dynamical system driven by fractional Brownian motion

期刊论文
发表日期：

2025年5月15日

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Wentao Hou,
Shaojuan Ma

摘要

Accurate parameters identification in nonlinear dynamical systems is crucial for understanding complex phenomena. Fractional Brownian motion is widely used in modeling nonlinear systems due to its long memory, but it also complicates parameter identification. Traditional methods typically rely on linear models or assume Gaussian white noise, which makes them less effective for handling such complex systems. Although data-driven methods have been proposed, most of them utilize simple network architectures and focus primarily on identifying the Hurst exponent or parameters of linear systems. This paper presents a deep learning-based parameter identifier capable of jointly identifying all parameters of a nonlinear dynamical system driven by fractional Brownian motion. Firstly, a new parameter identifier (Transformer-Bidirectional Long Short-Term Memory Neural Network) is constructed by combining the advantages of Transformer in dealing with long-term dependence and Bidirectional Long Short-Term Memory in extracting local features. We then incorporate fractional Brownian motion as the random effect in the nonlinear dynamical system and provide three examples to simulate the system. Finally, the proposed identifier is used to identify parameters in the three systems individually, and its performance is compared to that of the Convolutional Neural Network-Bidirectional Long Short-Term Memory-Attention Neural Network and the Parameter Estimation Neural Network to demonstrate its effectiveness. Taking Example 3 and using the mean absolute error to evaluate identification accuracy. The mean absolute error values for the Transformer-Bidirectional Long Short-Term Memory Neural Network are 0.0171, 0.0699, 0.0328 and 0.0851, which are significantly better than the Convolutional Neural Network-Bidirectional Long Short-Term Memory-Attention Neural Network, with values of 0.0277, 0.1054, 0.0494 and 0.1037, and the Parameter Estimation Neural Network, with values of 0.0419, 0.1265, 0.0708 and 0.1222. In terms of identification speed, the average identification time for the Transformer-Bidirectional Long Short-Term Memory Neural Network is 0.005907 seconds, faster than the Convolutional Neural Network-Bidirectional Long Short-Term Memory-Attention Neural Network at 0.008296 seconds and the Parameter Estimation Neural Network at 0.035498 seconds. These quantitative results demonstrate that the proposed identifier can identify all parameters of the nonlinear dynamical systems more quickly and accurately. Additionally, three key advantages of the proposed identifier are discussed, including the stability of parameter identification results, the rationality of sample size selection, and the advantages of the network construction. The contributions of this paper include: (1) Modeling the complex nonlinear dynamical system driven by fractional Brownian motion; (2) Proposing a new parameters identifier by integrating Transformer and Bidirectional Long Short-Term Memory; (3) Employing the orthogonal design table to arrange experiments for optimal parameter combination selection. For a full list of acronyms used in the paper, please refer to Table 1.