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Accelerating Fractional PINNs using Operational Matrices of Derivative

Tayebeh Taheri, Alireza Afzal Aghaei, Kourosh Parand

TL;DR

The paper addresses accelerating training of fractional PINNs for Caputo-type fractional differential equations with $0<\alpha<1$ by introducing a non-uniform discretization and a precomputed operational matrix $\mathcal{A}$ so that $^{C}D^{\alpha}f \approx \mathcal{A}f$, replacing automatic differentiation with matrix–vector products. It integrates a Legendre Neural Block within PINNs to boost accuracy and demonstrates effectiveness across diverse problems, including delay differential equations, pantograph equations, and fractional differential-algebraic equations, with $L_2$-norm errors often in the $10^{-3}$ to $10^{-4}$ range. The method enables efficient, scalable handling of nonlocal fractional derivatives in PINNs and holds promise for broader fractional systems, though convergence guarantees remain open and extensions to partial and higher-order fractional equations are proposed for future work. The combination of a precomputed operator, non-uniform Caputo discretization, and Legendre-based activations yields improved accuracy and computational efficiency in solving fractional-order dynamical systems.

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 $0<α<1$. 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.

Accelerating Fractional PINNs using Operational Matrices of Derivative

TL;DR

The paper addresses accelerating training of fractional PINNs for Caputo-type fractional differential equations with by introducing a non-uniform discretization and a precomputed operational matrix so that , replacing automatic differentiation with matrix–vector products. It integrates a Legendre Neural Block within PINNs to boost accuracy and demonstrates effectiveness across diverse problems, including delay differential equations, pantograph equations, and fractional differential-algebraic equations, with -norm errors often in the to range. The method enables efficient, scalable handling of nonlocal fractional derivatives in PINNs and holds promise for broader fractional systems, though convergence guarantees remain open and extensions to partial and higher-order fractional equations are proposed for future work. The combination of a precomputed operator, non-uniform Caputo discretization, and Legendre-based activations yields improved accuracy and computational efficiency in solving fractional-order dynamical systems.

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 . 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.
Paper Structure (5 sections, 5 theorems, 37 equations, 11 figures, 9 tables)

This paper contains 5 sections, 5 theorems, 37 equations, 11 figures, 9 tables.

Table of Contents

  1. Introduction
  2. Fractional Calculus
  3. Methodology
  4. Numerical Examples
  5. Conclusion

Key Result

Theorem 1

Let $0 < \alpha < 1$ and the interval $[t_0, t_n]$ is discretized to $n+1$ points, $0 = t_0 < t_1 < \dots < t_n$. Then the following linear combination approximates the Caputo fractional derivative of order $\alpha$: where $f$ is the desired function and $\omega_k$ are real-valued weights.

Figures (11)

  • Figure 1: Legendre Neural Block.
  • Figure 2: physics-informed neural networks.
  • Figure 3: (a) Comparison of Results (b) Absolute Error of Ex.\ref{['ex1_4']}.
  • Figure 4: (a) Comparison of Results (b) Absolute Error of Ex.\ref{['ex2_4']}.
  • Figure 5: (a) Comparison of Results (b) Absolute Error of Ex.\ref{['ex3_4']}.
  • ...and 6 more figures

Theorems & Definitions (18)

  • Definition 1
  • Definition 2
  • Theorem 1
  • proof
  • Corollary 1
  • proof
  • Corollary 2
  • Theorem 2
  • proof
  • Theorem 3
  • ...and 8 more