Messenger Daniel A, Bortz David M
Department of Applied Mathematics, University of Colorado, Boulder, CO 80309-0526 USA.
Multiscale Model Simul. 2021;19(3):1474-1497. doi: 10.1137/20m1343166. Epub 2021 Sep 7.
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, , 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 coefficient 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, , 96 (2017), 023302], to arrive at a robust and accurate method of sparse recovery.
我们提出了一种用于从噪声测量数据中发现控制方程的新的弱形式和离散化方法。这种从数据中学习微分方程的方法属于一类新算法，这类算法用线性变换和方差缩减技术取代了逐点导数近似。与文献[S. L. 布伦顿、J. L. 普罗克特和J. N. 库茨，，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）的数据中进行可靠的模型识别，并减少恢复系数中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实现高精度恢复。总体而言，WSINDy将SINDy算法的简单性和效率与积分固有的噪声降低特性相结合，正如文献[H. 舍费尔和S. G. 麦卡拉，，96 (2017)，023302]所展示的那样，从而得到一种用于稀疏恢复的稳健且准确的方法。
