带一致导数的时间分数阶扩散方程正、反问题的深度神经网络方法
Deep neural network methods for solving forward and inverse problems of time fractional diffusion equations with conformable derivative
摘要：
物理信息神经网络（PINNs）在求解偏微分方程方面显示出巨大的优势。本文首次提出用PINNs研究保形时间分数阶扩散方程。通过解决监督学习任务，我们设计了一个新的时空函数逼近器，具有较高的数据效率。采用L-BFGS算法对损失函数进行优化，采用反向传播算法对参数进行更新，得到数值解。对于正问题，我们可以用IC/BC作为数据，用PINN求解相应的偏微分方程。三个数值例子证明了我们的方法的有效性。特别地，当保形分数阶导数$α$的阶数趋于$1$时，引入一类加权PINN来克服解的奇异性所引起的精度下降.对于反问题，我们利用所获得的数据对神经网络进行训练，并详细阐述了方程中参数$λ$的估计。同样，我们给出了三个数值例子，表明即使训练数据中含有1%的不相关噪声，我们的方法也能准确地识别参数。
英文摘要：
Physics-informed neural networks (PINNs) show great advantages in solving partial differential equations. In this paper, we for the first time propose to study conformable time fractional diffusion equations by using PINNs. By solving the supervise learning task, we design a new spatio-temporal function approximator with high data efficiency. L-BFGS algorithm is used to optimize our loss function, and back propagation algorithm is used to update our parameters to give our numerical solutions. For the forward problem, we can take IC/BCs as the data, and use PINN to solve the corresponding partial differential equation. Three numerical examples are are carried out to demonstrate the effectiveness of our methods. In particular, when the order of the conformable fractional derivative $α$ tends to $1$, a class of weighted PINNs is introduced to overcome the accuracy degradation caused by the singularity of solutions. For the inverse problem, we use the data obtained to train the neural network, and the estimation of parameter $λ$ in the equation is elaborated. Similarly, we give three numerical examples to show that our method can accurately identify the parameters, even if the training data is corrupted with 1\% uncorrelated noise.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
