WEAK SINDy: GALERKIN-BASED DATA-DRIVEN MODEL SELECTION

We present a novel weak formulation and discretization for discovering governing equations from noisy measurement data. This method of learning differential equations from data fits into a new class of algorithms that replace pointwise derivative approximations with linear transformations and variance reduction techniques. Compared to the standard SINDy algorithm presented in [S. L. Brunton, J. L. Proctor, and J. N. Kutz, Proc. Natl. Acad. Sci. USA, 113 (2016), pp. 3932-3937], our so-called weak SINDy (WSINDy) algorithm allows for reliable model identification from data with large noise (often with ratios greater than 0.1) and reduces the error in the recovered coefficients to enable accurate prediction. Moreover, the co-efficient error scales linearly with the noise level, leading to high-accuracy recovery in the low-noise regime. Altogether, WSINDy combines the simplicity and efficiency of the SINDy algorithm with the natural noise reduction of integration, as demonstrated in [H. Schaeffer and S. G. McCalla, Phys. Rev. E, 96 (2017), 023302], to arrive at a robust and accurate method of sparse recovery.

我们提出了一种新的弱形式和离散化方法，用于从含噪测量数据中发现控制方程。这种从数据中学习微分方程的方法属于一类新的算法，该算法用线性变换和方差降低技术取代了逐点导数近似。与[S. L. 布伦顿、J. L. 普罗克特和J. N. 库茨，《美国国家科学院院刊》，113（2016），第3932 - 3937页]中提出的标准SINDy算法相比，我们所谓的弱SINDy（WSINDy）算法能够从具有大噪声（通常噪声比大于0.1）的数据中可靠地识别模型，并降低恢复系数的误差以实现准确预测。此外，系数误差与噪声水平呈线性比例关系，从而在低噪声情况下能够高精度恢复。总之，WSINDy将SINDy算法的简单性和高效性与积分的自然降噪特性相结合，如[H. 谢弗和S. G. 麦卡拉，《物理评论E》，96（2017），023302]中所展示的那样，形成了一种稳健且准确的稀疏恢复方法。