论文标题
通过近端梯度方法对非线性动力学系统的拉格朗日稀疏识别
Sparse Identification of Lagrangian for Nonlinear Dynamical Systems via Proximal Gradient Method
论文作者
论文摘要
在许多科学领域，自动提取物理定律引起了极大的兴趣。已经开发了非线性动力学（SINDY）及其变化的稀疏识别，以从观察数据中提取基本的管理方程。但是，当动力学包含有理功能时，信德（Sindy）会面临某些困难。最小动作的原理控制着许多机械系统，在拉格朗日公式中以数学表达。与实际运动的实际方程相比，拉格朗日人更加简洁，尤其是对于复杂的系统，通常不包含机械系统的合理功能。到目前为止，仅提出了一些方法来从测量数据中提取拉格朗日。这样一种方法之一是拉格朗日 - 印度语，可以从数据中提取动力学系统的拉格朗日人的真实形式，但在存在噪声时会受到痛苦。在这项工作中，我们开发了Lagrangian-Sindy（XL-Sindy）的扩展版本，以从嘈杂的测量数据中获取动力学系统的拉格朗日。我们结合了信德岛的概念，并利用近端梯度方法获得拉格朗日的稀疏表达式。我们通过四个非线性动力学证明了XL-Sindy对不同噪声水平的有效性：单个摆，一个推车，pendulum，双子摆和球形摆。此外，我们还验证了Xl-Sindy对Sindy-pi（并行，隐式）的性能，这是Sindy的最新鲁棒变体，可以处理隐式动力学和合理的非线性。我们的实验结果表明，在存在噪声的情况下，XL-Sindy是Sindy-Pi的稳健性的8-20倍。
Distilling physical laws autonomously from data has been of great interest in many scientific areas. The sparse identification of nonlinear dynamics (SINDy) and its variations have been developed to extract the underlying governing equations from observation data. However, SINDy faces certain difficulties when the dynamics contain rational functions. The principle of the least action governs many mechanical systems, mathematically expressed in the Lagrangian formula. Compared to the actual equation of motions, the Lagrangian is much more concise, especially for complex systems, and does not usually contain rational functions for mechanical systems. Only a few methods have been proposed to extract the Lagrangian from measurement data so far. One of such methods, Lagrangian-SINDy, can extract the true form of Lagrangian of dynamical systems from data but suffers when noises are present. In this work, we develop an extended version of Lagrangian-SINDy (xL-SINDy) to obtain the Lagrangian of dynamical systems from noisy measurement data. We incorporate the concept of SINDy and utilize the proximal gradient method to obtain sparse expressions of the Lagrangian. We demonstrated the effectiveness of xL-SINDy against different noise levels with four nonlinear dynamics: a single pendulum, a cart-pendulum, a double pendulum, and a spherical pendulum. Furthermore, we also verified the performance of xL-SINDy against SINDy-PI (parallel, implicit), a recent robust variant of SINDy that can handle implicit dynamics and rational nonlinearities. Our experiment results show that xL-SINDy is 8-20 times more robust than SINDy-PI in the presence of noise.
