A physics-informed neural network approach for the one-dimensional wave equation: data-driven solution with Fibonacci polynomial activation functions

Ali Algan; Faruk Düşünceli; Ali Algan; Faruk Düşünceli

doi:10.3934/math.20261223

AIMS Mathematics

2026, Volume 11, Issue 9: 30886-30903. doi: 10.3934/math.20261223

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A physics-informed neural network approach for the one-dimensional wave equation: data-driven solution with Fibonacci polynomial activation functions

Ali Algan ¹,
Faruk Düşünceli ^{2
,
,}

1.
Department of Mathematics, Institute of Graduate Studies, Mardin Artuklu University, Türkiye
2.
Department of Economics, Mardin Artuklu University, Türkiye

Received: 22 July 2026 Revised: 11 September 2026 Accepted: 16 September 2026 Published: 21 September 2026
MSC : 65M70, 68T07, 35L05, 11B39

In this study, a data-driven computational framework based on Physics-Informed Neural Networks (PINNs) was proposed for the numerical solution of the one-dimensional wave equation. Conventional numerical methodologies, including finite difference and finite element schemes, necessitated intricate discretization processes and became computationally onerous for high-resolution simulations. The proposed PINN framework reinterprets the problem as a supervised learning task that learns the underlying physical behavior directly from data in a mesh-free manner, using training sets generated from the governing equation together with the initial and boundary conditions. The central contribution of this study is the integration of Fibonacci polynomials as activation functions within the network architecture. The continuous and structurally rich mathematical properties of Fibonacci polynomials enhance the representational capacity of the network relative to conventional activation functions such as Tanh, improving learning dynamics, training stability, and accuracy in capturing the oscillatory nature of wave phenomena. The performance of the model was evaluated through error analysis, utilizing standard metrics such as Mean Squared Error (MSE) and Relative-L2 error, in comparison to known reference solutions. The proposed Fibonacci-PINN was directly compared with a standard Tanh-based PINN under identical training conditions. The findings demonstrated that the Fibonacci-PINN consistently attains reduced prediction error in comparison to the standard Tanh-PINN across all three numerical illustrations examined, thereby evidencing enhanced approximation precision with diminished computational demands. In contradistinction to conventional discretization-based methodologies, the model engenders smooth, continuous functional outputs, thereby facilitating efficacious visualization of wave propagation. In this study, we demonstrate the efficiency and flexibility of combining artificial neural networks with structurally rich mathematical activation functions as a viable alternative.
- Data-driven modeling,
- partial differential equations,
- activation function design,
- Levenberg-Marquardt optimization
Citation: Ali Algan, Faruk Düşünceli. A physics-informed neural network approach for the one-dimensional wave equation: data-driven solution with Fibonacci polynomial activation functions[J]. AIMS Mathematics, 2026, 11(9): 30886-30903. doi: 10.3934/math.20261223

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Abstract

In this study, a data-driven computational framework based on Physics-Informed Neural Networks (PINNs) was proposed for the numerical solution of the one-dimensional wave equation. Conventional numerical methodologies, including finite difference and finite element schemes, necessitated intricate discretization processes and became computationally onerous for high-resolution simulations. The proposed PINN framework reinterprets the problem as a supervised learning task that learns the underlying physical behavior directly from data in a mesh-free manner, using training sets generated from the governing equation together with the initial and boundary conditions. The central contribution of this study is the integration of Fibonacci polynomials as activation functions within the network architecture. The continuous and structurally rich mathematical properties of Fibonacci polynomials enhance the representational capacity of the network relative to conventional activation functions such as Tanh, improving learning dynamics, training stability, and accuracy in capturing the oscillatory nature of wave phenomena. The performance of the model was evaluated through error analysis, utilizing standard metrics such as Mean Squared Error (MSE) and Relative-L2 error, in comparison to known reference solutions. The proposed Fibonacci-PINN was directly compared with a standard Tanh-based PINN under identical training conditions. The findings demonstrated that the Fibonacci-PINN consistently attains reduced prediction error in comparison to the standard Tanh-PINN across all three numerical illustrations examined, thereby evidencing enhanced approximation precision with diminished computational demands. In contradistinction to conventional discretization-based methodologies, the model engenders smooth, continuous functional outputs, thereby facilitating efficacious visualization of wave propagation. In this study, we demonstrate the efficiency and flexibility of combining artificial neural networks with structurally rich mathematical activation functions as a viable alternative.

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