Deep learning approaches for $ \beta- $conformable fractional differential equations: A PINN, NRPINN-s, and NRPINN-un based solutions

Sadullah Bulut; Muhammed Yiğider; Sadullah Bulut; Muhammed Yiğider

doi:10.3934/math.2025618

AIMS Mathematics

2025, Volume 10, Issue 6: 13721-13740. doi: 10.3934/math.2025618

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Research article

Deep learning approaches for $ \beta- $conformable fractional differential equations: A PINN, NRPINN-s, and NRPINN-un based solutions

Sadullah Bulut ,
Muhammed Yiğider ^,

Department of Mathematics, Erzurum Technical University, Yakutiye, Erzurum 25050, Turkey

Received: 18 February 2025 Revised: 30 May 2025 Accepted: 04 June 2025 Published: 13 June 2025
MSC : 65M50, 35R11, 68T07

In this paper, we present a numerical approach for solving $ \beta- $conformable fractional differential equations (FDEs) using physics-informed neural networks (PINNs) and their enhanced versions, NRPINN-s and NRPINN-un. The proposed method combines the flexibility of artificial neural networks with the inherent structure of fractional calculus, allowing the solution of complex initial and boundary value problems without domain discretization. The loss function in the model includes initial and boundary conditions and is constructed using modifiable parameters (weights and biases). The efficiency of the proposed methods is demonstrated by numerical experiments on heat and wave equations. The obtained results show that NR-PINNs are superior to classical PINNs in improving convergence, reducing local errors and showing good performance. Visual and tabular comparisons of the solutions obtained from the method with analytical solutions confirm the effectiveness of the approach.
- Physics-Informed Neural Networks (PINNs),
- $ \beta- $conformable fractional differential equations,
- NR-PINN,
- deep learning
Citation: Sadullah Bulut, Muhammed Yiğider. Deep learning approaches for $ \beta- $conformable fractional differential equations: A PINN, NRPINN-s, and NRPINN-un based solutions[J]. AIMS Mathematics, 2025, 10(6): 13721-13740. doi: 10.3934/math.2025618

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Abstract

In this paper, we present a numerical approach for solving $ \beta- $conformable fractional differential equations (FDEs) using physics-informed neural networks (PINNs) and their enhanced versions, NRPINN-s and NRPINN-un. The proposed method combines the flexibility of artificial neural networks with the inherent structure of fractional calculus, allowing the solution of complex initial and boundary value problems without domain discretization. The loss function in the model includes initial and boundary conditions and is constructed using modifiable parameters (weights and biases). The efficiency of the proposed methods is demonstrated by numerical experiments on heat and wave equations. The obtained results show that NR-PINNs are superior to classical PINNs in improving convergence, reducing local errors and showing good performance. Visual and tabular comparisons of the solutions obtained from the method with analytical solutions confirm the effectiveness of the approach.

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