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Data-driven identification of 2D Partial Differential Equations using extracted physical features
Computer Methods in Applied Mechanics and Engineering ( IF 7.2 ) Pub Date : 2021-04-21 , DOI: 10.1016/j.cma.2021.113831
Kazem Meidani , Amir Barati Farimani

Many scientific phenomena are modeled by Partial Differential Equations (PDEs). The development of data gathering tools along with the advances in machine learning (ML) techniques have raised opportunities for data-driven identification of governing equations from experimentally observed data. We propose an ML method to discover the terms involved in the equation from two-dimensional spatiotemporal data. Robust and useful physical features are extracted from data samples to represent the specific behaviors imposed by each mathematical term in the equation. Compared to the previous models, this idea provides us with the ability to discover 2D equations with time derivatives of different orders, and also to identify new underlying physics on which the model has not been trained. Moreover, the model can work with small sets of low-resolution data while avoiding instability caused by numerical differentiations. The results indicate robustness of the features extracted based on prior knowledge in comparison to automatically detected features by a Three-dimensional Convolutional Neural Network (3D CNN) given the same amounts of data. Although particular PDEs are studied in this work, the idea of the proposed approach could be extended for reliable identification of various PDEs.



中文翻译:

使用提取的物理特征进行数据驱动的2D偏微分方程的识别

许多科学现象都是通过偏微分方程(PDE)建模的。数据收集工具的发展以及机器学习(ML)技术的进步为从实验观察到的数据进行数据驱动的控制方程式识别提供了机会。我们提出了一种ML方法来从二维时空数据中发现方程中涉及的项。从数据样本中提取出稳健而有用的物理特征,以表示每个数学术语施加的特定行为在等式中。与以前的模型相比,此思想使我们能够发现具有不同阶次时间导数的2D方程,并能够确定尚未训练模型的新基础物理学。此外,该模型可以处理少量的低分辨率数据,同时避免了由数值微分引起的不稳定性。结果表明,与给定相同数据量的三维卷积神经网络(3D CNN)自动检测到的特征相比,基于先验知识提取的特征具有较强的鲁棒性。尽管在这项工作中对特定的PDE进行了研究,但是可以将提出的方法的思想扩展到各种PDE的可靠识别中。

更新日期:2021-04-21
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