In this work, we demonstrated the use of neural network techniques for solving fractional differential equations (FDEs) governed by the Caputo–Katugampola derivative, specifically standard neural networks and physics-informed neural networks. Our numerical approach was based on the Volterra integral representation of the generalized fractional operator, which provides a natural way to incorporate memory and non-locality effects into the solution procedure. We designed and studied two learning frameworks: (ⅰ) The proposed NFDEs model that approximates the system's dynamics by data-driven learning of the solution trajectory and (ⅱ) the proposed PINFDEs model that informs the learning process by directly embedding the residual form of the governing FDE into the neural network training objective. For numerical implementation, the Caputo–Katugampola derivative was discretized by using a Volterra-type quadrature scheme that accurately and efficiently computed the fractional operator throughout the considered time domain. To demonstrate the performance of the proposed frameworks, four numerical experiments were performed for different fractional orders. Our results emphasized that, while the neural fractional differential equation model can capture the general solution, the proposed PINFDEs model exhibits more accuracy, stability, and generalization for a wide variety of cases, particularly for fractional orders $ \gamma < 1 $, where the nonlocal effects dominate. The improvement in the performance reflected the fact that the explicit incorporation of the physical constraints during the learning stage significantly reduces overfitting and enhances the robustness of the solution. Moreover, the comparative study confirms that embedding physical knowledge into data-driven neural models substantially enhances their predictive capabilities regarding fractional systems.
Citation: Sneha Agarwal, Lakshmi Narayan Mishra. A comparative study of the fractional differential equations using physics-informed neural networks[J]. AIMS Mathematics, 2026, 11(8): 25705-25743. doi: 10.3934/math.20261030
In this work, we demonstrated the use of neural network techniques for solving fractional differential equations (FDEs) governed by the Caputo–Katugampola derivative, specifically standard neural networks and physics-informed neural networks. Our numerical approach was based on the Volterra integral representation of the generalized fractional operator, which provides a natural way to incorporate memory and non-locality effects into the solution procedure. We designed and studied two learning frameworks: (ⅰ) The proposed NFDEs model that approximates the system's dynamics by data-driven learning of the solution trajectory and (ⅱ) the proposed PINFDEs model that informs the learning process by directly embedding the residual form of the governing FDE into the neural network training objective. For numerical implementation, the Caputo–Katugampola derivative was discretized by using a Volterra-type quadrature scheme that accurately and efficiently computed the fractional operator throughout the considered time domain. To demonstrate the performance of the proposed frameworks, four numerical experiments were performed for different fractional orders. Our results emphasized that, while the neural fractional differential equation model can capture the general solution, the proposed PINFDEs model exhibits more accuracy, stability, and generalization for a wide variety of cases, particularly for fractional orders $ \gamma < 1 $, where the nonlocal effects dominate. The improvement in the performance reflected the fact that the explicit incorporation of the physical constraints during the learning stage significantly reduces overfitting and enhances the robustness of the solution. Moreover, the comparative study confirms that embedding physical knowledge into data-driven neural models substantially enhances their predictive capabilities regarding fractional systems.
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Sneha Agarwal, Lakshmi Narayan Mishra. A comparative study of the fractional differential equations using physics-informed neural networks[J]. AIMS Mathematics, 2026, 11(8): 25705-25743. doi: 10.3934/math.20261030
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