Data-driven Derivation of Partial Differential Equations using Neural Network Model
Data-driven Derivation of Partial Differential Equations using Neural Network Model
复制标题
使用神经网络模型数据驱动偏微分方程的推导
DOI：
10.1142/s1793962321400018
复制
发表时间：
2021
期刊：
影响因子：
--
通讯作者：
Konishi K.
中科院分区：
文献类型：
--
作者：
Koyamada K.;Long Y.;Kawamura T.;Konishi K.
使用偏微分方程(PDE)为流体、大气和宇宙现象等有趣科学现象的大数据开发解释性模型已变得至关重要。大数据通常是在离散的时空点上测量的。在本文中，我们假设一个偏微分方程组，并将多个离散点的偏微分方程组的解数据视为伪测量数据。在数据驱动的PDE推导中，假设PDE是包括部分时间和空间微分项的线性回归模型，其中使用回归分析技术来估计系数，并且PDE推导精度被定义为精确系数和估计系数之间的差(即误差)。需要一个时空模型来计算部分时间和空间微分项，因此，我们采用了一个基于从大数据中获得的神经网络的时空模型。为了开发数据驱动的偏微分方程导出技术，我们使用具有精确解的偏微分方程组，以便我们可以通过在多个离散点上解析地微分来计算微分项。如果我们在NN模型中假设一个激活函数，则可以使用链规则来推导偏微分项。神经网络模型的精度是通过损失函数和精确偏微分项与估计偏微分项之间的误差来衡量的。此外，我们明确了对神经网络结构的要求，该结构可以通过改变神经网络的元参数，即神经网络的层数和神经元的数量，来最大化偏微分方程的派生和神经网络模型的精度。
It has become critical to develop explanatory models using partial differential equations (PDEs) for big data from interesting scientific phenomena, e.g., fluids, atmosphere, and cosmic phenomena. Big data are often measured at discrete spatiotemporal points. In this paper, we assume a PDE and consider a set of the PDE solution data at multiple discrete points as pseudo-measurement data. In a data-driven PDE derivation, it is assumed that the PDE is a linear regression model comprising partial temporal and spatial differential terms, where the coefficients are estimated using regression analysis techniques, and the PDE derivation accuracy is defined as the difference (i.e., error) between the exact and estimated coefficients. A spatiotemporal model is required to calculate the partial temporal and spatial differential terms; thus, we employ a spatiotemporal model based on a neural network (NN) obtained from big data. To develop the data-driven PDE derivation technique, we employ PDEs with exact solutions such that we can calculate the differential terms by differentiating them analytically at multiple discrete points. If we assume an activation function in the NN model, the partial differential term can be derived using a chain rule. The NN model accuracy is measured by the loss function and error between the exact and estimated partial differential terms. In addition, we clarify a requirement for an NN structure that can maximize the PDE derivation and NN model accuracy by varying the NN meta-parameters, i.e., the numbers of NN layers and neurons.