Robust data-driven discovery of governing physical laws with error bars
Robust data-driven discovery of governing physical laws with error bars
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DOI：
10.1098/rspa.2018.0305
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发表时间：
2018-09
期刊：
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通讯作者：
Sheng Zhang;Guang Lin
中科院分区：
文献类型：
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作者：
Sheng Zhang;Guang Lin
在许多科学和工程研究领域，从嘈杂的数据中发现支配性的物理定律是一个巨大的挑战。我们提出了一种新的方法来数据驱动的发现常微分方程（ode）和偏微分方程（PDEs），在显式或隐式形式。我们在广泛的问题上展示了我们的方法，包括浅水方程和Navier-Stokes方程。关键思想是使用量纲分析为基础方程选择候选项，并使用阈值稀疏贝叶斯回归近似带有误差条的项的权重。该算法采用贝叶斯推理对超参数进行自动调优。我们的方法是有效的，稳健的，并且能够通过为每个发现的候选方程提供误差条来量化不确定性。通过经典ode和偏微分方程的集合验证了算法的有效性。数值实验证明了我们的算法对噪声数据的鲁棒性，以及它发现各种候选方程的能力，这些方程带有表示量化不确定性的误差条。通过与序列阈值最小二乘算法和lasso算法在噪声时间序列测量中的比较，表明该方法具有更好的鲁棒性和准确性。此外，使用发现的控制物理定律的带有误差条的数据驱动的动力学预测比经典的多项式回归更准确和健壮。
Discovering governing physical laws from noisy data is a grand challenge in many science and engineering research areas. We present a new approach to data-driven discovery of ordinary differential equations (ODEs) and partial differential equations (PDEs), in explicit or implicit form. We demonstrate our approach on a wide range of problems, including shallow water equations and Navier–Stokes equations. The key idea is to select candidate terms for the underlying equations using dimensional analysis, and to approximate the weights of the terms with error bars using our threshold sparse Bayesian regression. This new algorithm employs Bayesian inference to tune the hyperparameters automatically. Our approach is effective, robust and able to quantify uncertainties by providing an error bar for each discovered candidate equation. The effectiveness of our algorithm is demonstrated through a collection of classical ODEs and PDEs. Numerical experiments demonstrate the robustness of our algorithm with respect to noisy data and its ability to discover various candidate equations with error bars that represent the quantified uncertainties. Detailed comparisons with the sequential threshold least-squares algorithm and the lasso algorithm are studied from noisy time-series measurements and indicate that the proposed method provides more robust and accurate results. In addition, the data-driven prediction of dynamics with error bars using discovered governing physical laws is more accurate and robust than classical polynomial regressions.