【作者】 王蕾蕾；

【导师】 杜宁；

【作者基本信息】 山东大学 ， 计算数学， 2023， 硕士

【摘要】 基于物理的神经网络(PINN)是求解整数阶偏微分方程(PDE)的有效方法。PINN采用的是标准前馈神经网络(NN),PDE通过自动微分技术嵌入到NN中,再通过最小化损失函数训练网络,进而预测解。在本文中,利用PINN的扩展分数阶PINN,即fPINNs来求解变阶Caputo时间分数阶扩散方程。fPINNs的核心思想是引入了一种混合方法,它使用整数阶算子的自动微分和分数阶算子的数值离散来构造损失函数中的残差,这种方法绕过了自动微分不适用于分数阶算子的困难,因为整数演算中的标准链式法则在分数演算中是无效的。在本文中我们采用的是Caputo分数阶导数的L1离散格式,对于时间项和拉普拉斯项均使用的是自动微分技术,我们用均方误差来构造损失函数。通过实验调整了神经网络中的参数,用几个例子验证了 fPINNs方法对于变阶Caputo时间分数阶扩散方程的适用性,并得到了较好的结果。通过与传统方法比较,我们发现fPINNs的一个突出优点就是可以通过很少的训练点,算出精确度较高的数值解,这也意味着用较少的时间可以算出与传统算法一样精度的结果,大大提高了运算效率。并且由于fPINNs本质上是由数据驱动的,它的运算不依赖于网格剖分,所以在处理高维问题和不规则区域模型时具有很大的优势。更多还原

【Abstract】 The physics-based neural network(PINN)is an effective method for solving integral order partial differential equations(PDE).PINN uses standard feedforward neural network(NN),PDE is embedded into NN through automatic differentiation technology,and then the network is trained by minimizing the constructed loss function to predict the solution.In this paper,the extended fractional PINN of PINN,namely fPINNs,is used to solve the variable order Caputo time fractional diffusion equation.The core idea of fPINNs is to introduce a hybrid method,which uses the automatic differentiation of integer order operators and the numerical discretization of fractional order operators to construct the residual in the loss function.This method bypasses the difficulty that automatic differentiation is not applicable to fractional order operators,because the standard chain rule in integer calculus is invalid in fractional calculus.In this paper,we use the L1 discretization scheme of Caputo fractional derivative.For the time term and Laplace term,we apply the automatic differential technique.We use the mean square error to construct the loss function.The appropriate parameters of the neural network are adjusted through experiments.The applicability of the fPINNs method to the variable-order Caputo time-fractional diffusion equation is verified by several examples,and good results are obtained.By comparing with traditional methods,we find that one outstanding advantage of fPINNs is that it can calculate numerical solutions with high accuracy with few training points,which also means that it takes less time to calculate results with the same accuracy as the traditional algorithm,which greatly improves the operation efficiency.Because fPINNs is essentially data-driven,its operations do not depend on mesh partitioning,it has certain advantages in dealing with higher-dimensional problems and irregular region models.更多还原