Gaussian Process Assisted Active Learning of Physical Laws
Gaussian Process Assisted Active Learning of Physical Laws
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DOI：
10.1080/00401706.2020.1817790
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发表时间：
2020-10-12
期刊：
影响因子：
2.5
通讯作者：
Lin, Guang
中科院分区：
文献类型：
--
作者：
Chen, Jiuhai;Kang, Lulu;Lin, Guang
在许多科学和工程领域，从有噪声的实验数据中发现控制微分方程是一项重大挑战。这也是理解物理现象以及预测系统未来行为的关键步骤。然而，在很多情况下，收集实验数据既昂贵又耗时。本文提供了一种主动学习方法，能在减少实验数据量的情况下准确估计未知微分方程。我们提出了一种结合D - 最优性和极大极小空间填充准则的自适应设计准则。与其他回归模型的主动学习不同，这里的D - 最优性需要微分方程的未知解以及该解的导数。我们从可用的实验数据中估计高斯过程（GP）回归模型，并将其用作这些未知解函数的替代。推导估计的GP模型的导数，并用于替代解的导数。基于变量选择的回归方法被用于从实验数据中学习微分方程。通过多个案例研究，我们证明了所提出的方法在模型准确性和数据经济性方面优于单独使用D - 最优性和极大极小空间填充设计。
In many areas of science and engineering, discovering the governing differential equations from the noisy experimental data is an essential challenge. It is also a critical step in understanding the physical phenomena and prediction of the future behaviors of the systems. However, in many cases, it is expensive or time-consuming to collect experimental data. This article provides an active learning approach to estimate the unknown differential equations accurately with reduced experimental data size. We propose an adaptive design criterion combining the D-optimality and the maximin space-filling criterion. In contrast to active learning for other regression models, the D-optimality here requires the unknown solution of the differential equations and derivatives of the solution. We estimate the Gaussian process (GP) regression models from the available experimental data and use them as the surrogates of these unknown solution functions. The derivatives of the estimated GP models are derived and used to substitute the derivatives of the solution. Variable selection-based regression methods are used to learn the differential equations from the experimental data. Through multiple case studies, we demonstrate the proposed approach outperforms the D-optimality and the maximin space-filling design alone in terms of model accuracy and data economy.