  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.
