Hao Zhihang, Yang Chunhua, Huang Keke
School of Automation, Central South University, Changsha 410083, China.
Chaos. 2023 Aug 1;33(8). doi: 10.1063/5.0164484.
In the field of science and engineering, identifying the nonlinear dynamics of systems from data is a significant yet challenging task. In practice, the collected data are often contaminated by noise, which often severely reduce the accuracy of the identification results. To address the issue of inaccurate identification induced by non-stationary noise in data, this paper proposes a method called weighted ℓ1-regularized and insensitive loss function-based sparse identification of dynamics. Specifically, the robust identification problem is formulated using a sparse identification mathematical model that takes into account the presence of non-stationary noise in a quantitative manner. Then, a novel weighted ℓ1-regularized and insensitive loss function is proposed to account for the nature of non-stationary noise. Compared to traditional loss functions like least squares and least absolute deviation, the proposed method can mitigate the adverse effects of non-stationary noise and better promote the sparsity of results, thereby enhancing the accuracy of identification. Third, to overcome the non-smooth nature of the objective function induced by the inclusion of loss and regularization terms, a smooth approximation of the non-smooth objective function is presented, and the alternating direction multiplier method is utilized to develop an efficient optimization algorithm. Finally, the robustness of the proposed method is verified by extensive experiments under different types of nonlinear dynamical systems. Compared to some state-of-the-art methods, the proposed method achieves better identification accuracy.
在科学与工程领域，从数据中识别系统的非线性动力学是一项重要但具有挑战性的任务。在实际应用中，收集到的数据常常受到噪声污染，这往往会严重降低识别结果的准确性。为了解决数据中非平稳噪声导致的识别不准确问题，本文提出了一种基于加权ℓ1正则化和不敏感损失函数的动力学稀疏识别方法。具体而言，利用一个稀疏识别数学模型来定量考虑非平稳噪声的存在，从而构建鲁棒识别问题。然后，提出了一种新颖的加权ℓ1正则化和不敏感损失函数，以适应非平稳噪声的特性。与最小二乘法和最小绝对偏差等传统损失函数相比，该方法能够减轻非平稳噪声的不利影响，更好地促进结果的稀疏性，进而提高识别的准确性。第三，为了克服由于包含损失项和正则化项而导致的目标函数的非光滑性，提出了非光滑目标函数的平滑近似，并利用交替方向乘子法开发了一种高效的优化算法。最后，通过在不同类型非线性动力系统下的大量实验验证了该方法的鲁棒性。与一些现有方法相比，该方法取得了更好的识别精度。
