Accelerating Fractional PINNs using Operational Matrices of Derivative

TL;DR

This paper introduces an operational matrix approach to speed up fractional PINNs training by replacing automatic differentiation with matrix-vector products, improving efficiency and accuracy especially with Legendre Neural Blocks.

Contribution

The paper proposes a novel operational matrix method for fractional derivatives, enhancing training speed and accuracy of fractional PINNs, and extends its application to complex differential systems.

Findings

Accelerates fractional PINNs training via operational matrices.

Achieves higher accuracy with Legendre Neural Block architecture.

Successfully applies method to delay and algebraic differential equations.

Abstract

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.