求解时间分数阶相场微分方程的自适应分数阶物理信息网络

Adaptive Fractional Physical Information Neural Network for Solving Time Fractional Phase Field Partial Differential Equations

摘要: 本文提出具有自适应权重的分数阶物理信息神经网络(adaptive-fPINN-PQI)求解时间分数阶偏微分方程。首先，利用Hadamard有限部分积分意义上的分段二次插值(PQI)对时间分数阶导数进行离散。其次，为降低自动微分引入的误差，本文采用中心差分法代替自动微分求导，计算空间各阶偏导数，提高了预测解精度。此外，本文构建的自适应权重残差网络，基于残差网络架构有效防止梯度消失。并通过建立自适应权重来自动调整不同损失项的权重，显著平衡其梯度，进一步提升预测解精度。最后，将adaptive-fPINN-PQI用于求解时间分数阶相场偏微分方程，证明了该网络的高精度和高效率。

Abstract: In this paper, an accurate fractional physical information neural network with an adaptive learning rate (adaptive-fPINN-PQI) is proposed for solving time fractional partial differential equations. Firstly, the time-fractional derivative in the Caputo sense is discretized by piecewise quadratic interpolation (PQI) in the sense of the Hadamard finite-part integral. Next, the central difference method is used instead of automatic differentiation to reduce the error of automatic differentiation. Besides, the present adaptive-fPINN-PQI is based on the ResNet architecture to effectively overcome the issue of gradient vanishing. The adaptive learning rate is constructed to automatically adjust the weights of different loss terms, significantly balancing their gradients and improving the accuracy of the predicted solutions. Finally, time fractional phase field equations have been solved using the proposed adaptive-fPINN-PQI to demonstrate its high precision and efficiency.