This paper presents a novel operational matrix method to accelerate the training of fractional Physics-Informed Neural Networks (fPINNs). Our approach involves a non-uniform discretization of the fractional Caputo operator, facilitating swift computation of fractional derivatives within Caputo-type fractional differential problems with $0<\alpha<1$. In this methodology, the operational matrix is precomputed, and during the training phase, automatic differentiation is replaced with a matrix-vector product. While our methodology is compatible with any network, we particularly highlight its successful implementation in PINNs, emphasizing the enhanced accuracy achieved when utilizing the Legendre Neural Block (LNB) architecture. LNB incorporates Legendre polynomials into the PINN structure, providing a significant boost in accuracy. The effectiveness of our proposed method is validated across diverse differential equations, including Delay Differential Equations (DDEs) and Systems of Differential Algebraic Equations (DAEs). To demonstrate its versatility, we extend the application of the method to systems of differential equations, specifically addressing nonlinear Pantograph fractional-order DDEs/DAEs. The results are supported by a comprehensive analysis of numerical outcomes.

提出一种新的运算矩阵方法以加速分数型物理信息神经网络(fPINNs)的训练，方法包括对分数型Caputo算子的非均匀离散化，通过矩阵-向量乘积替代自动微分，在各种微分方程中得到验证和数值结果的支持。

使用导数运算矩阵加速分数阶部分积分神经网络