Development of Mamadu–Njoseh Polynomial Neural Networks for Physics-Informed Learning

Objective This present work seeks to address the challenges in solving differential equations through the development of an improved Physics-informed neural network approach aimed at ensuring efficient convergence and precise solutions. Material and Methods A novel approach, termed the Mamadu-Njoseh polynomial physics-informed neural Network (MNP-PINN), is introduced. This approach utilises the orthogonal Mamadu-Njoseh polynomials (MNP) as activation functions in the network architecture. Physical constraints arising from the governing differential equation are embedded into the loss function to guide the learning process. The performance validation for the new approach is carried out by testing it with the one-dimensional Poisson equation. All computations are implemented in Python 3.11, with consistent procedures for network initialisation, collocation point generation, and training to ensure reproducibility. Results The results show excellent convergence of MNP-PINN solutions with the analytical solution of the Poisson equation. The solution has spectral accuracy with errors consistently small throughout the solution domain. In addition, there is stability in the optimisation process of the training algorithm, showing improved efficiency in the machine learning process compared with the analytical PINN solution. Conclusion The MNP-PINN has provided a robust and highly accurate framework for the solution of the differential equations. By utilizing the concepts of orthogonality and approximation provided by the MNP, improvement has been made in convergence acceleration and accuracy to that provided in PINNs. It appears from the results that the MNP-PINN framework has tremendous potential for further generalisation and expansion to more complex models.