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Deep-learning based discovery of partial differential equations in integral form from sparse and noisy data
arXiv - CS - Numerical Analysis Pub Date : 2020-11-24 , DOI: arxiv-2011.11981
Hao Xu, Dongxiao Zhang, Nanzhe Wang

Data-driven discovery of partial differential equations (PDEs) has attracted increasing attention in recent years. Although significant progress has been made, certain unresolved issues remain. For example, for PDEs with high-order derivatives, the performance of existing methods is unsatisfactory, especially when the data are sparse and noisy. It is also difficult to discover heterogeneous parametric PDEs where heterogeneous parameters are embedded in the partial differential operators. In this work, a new framework combining deep-learning and integral form is proposed to handle the above-mentioned problems simultaneously, and improve the accuracy and stability of PDE discovery. In the framework, a deep neural network is firstly trained with observation data to generate meta-data and calculate derivatives. Then, a unified integral form is defined, and the genetic algorithm is employed to discover the best structure. Finally, the value of parameters is calculated, and whether the parameters are constants or variables is identified. Numerical experiments proved that our proposed algorithm is more robust to noise and more accurate compared with existing methods due to the utilization of integral form. Our proposed algorithm is also able to discover PDEs with high-order derivatives or heterogeneous parameters accurately with sparse and noisy data.

中文翻译:

基于稀疏和嘈杂数据的积分形式偏微分方程的深度学习发现

数据驱动的偏微分方程(PDE)的发现近年来引起了越来越多的关注。尽管已取得重大进展,但仍有一些未解决的问题。例如,对于具有高阶导数的PDE,现有方法的性能不能令人满意,尤其是当数据稀疏且嘈杂时。很难发现异构参数嵌入偏微分算子中的异构参数PDE。在这项工作中,提出了一种将深度学习和整体形式相结合的新框架,以同时解决上述问题,并提高PDE发现的准确性和稳定性。在该框架中,首先使用观察数据训练深度神经网络,以生成元数据并计算导数。然后,定义一个统一的积分形式,并采用遗传算法发现最佳结构。最后,计算参数的值,并确定参数是常量还是变量。数值实验证明,由于利用了积分形式,因此与现有方法相比,本文提出的算法具有更强的抗噪性和准确性。我们提出的算法还能够利用稀疏和嘈杂的数据准确地发现具有高阶导数或异构参数的PDE。数值实验证明,由于利用了积分形式,因此与现有方法相比,本文提出的算法具有更强的抗噪性和准确性。我们提出的算法还能够利用稀疏和嘈杂的数据准确地发现具有高阶导数或异构参数的PDE。数值实验证明,由于利用了积分形式,因此与现有方法相比,本文提出的算法具有更强的抗噪性和准确性。我们提出的算法还能够利用稀疏和嘈杂的数据准确地发现具有高阶导数或异构参数的PDE。
更新日期:2020-11-25
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