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.

翻译：本文提出了一种新颖的运算矩阵方法，用于加速分数阶物理信息神经网络（fPINNs）的训练。该方法通过对分数阶Caputo算子进行非均匀离散化，能够快速计算Caputo型分数阶微分问题中$0<\alpha<1$的分数阶导数。在此方法中，运算矩阵被预先计算，训练阶段则用矩阵-向量乘积替代自动微分。虽然该方案适用于任意网络，但我们重点展示了其在PINNs中的成功实现，特别强调了采用勒让德神经块（LNB）架构时精度的提升。LNB将勒让德多项式融入PINN结构，显著提高了精度。我们通过多种微分方程验证了所提方法的有效性，包括延迟微分方程（DDEs）和微分代数方程组（DAEs）。为展示其通用性，我们将该方法扩展应用于微分方程组领域，重点解决了非线性Pantograph分数阶DDEs/DAEs问题。数值结果的综合分析为上述结论提供了有力支撑。