Damghan University Press Analytical and Numerical Solutions for Nonlinear Equations 3060-785X 11 2 2026 09 04 A Hybrid Method Based on Rational Legendre Physics Informed Neural Networks and Cuckoo Optimization Algorithm for Solving Nonlinear Inverse Partial Differential Equations 333 351 2139 10.22128/ansne.2026.3307.1213 EN Farhad Fahimi School of Mathematics and Computer Science, Damghan University, Damghan, Iran Reza Pourgholi School of Mathematics and Computer Science, Damghan University, Damghan, Iran 0000-0003-4111-5130 Abdolali Basiri School of Mathematics and Computer Science, Damghan University, Damghan, Iran Journal Article 2026 04 29 We develop an improved physics-informed neural network (PINN) framework for solving nonlinear inverse partial differential equations. Two separate network architectures are explored: one utilizing Legendre polynomials as activation functions and another employing rational Legendre polynomials. In the latter case, there is a crucial hyperparameter, $L$, that significantly impacts both performance and efficiency. To optimize this parameter, we employ the cuckoo optimization algorithm (COA), which fine-tunes $L$ based on the problem's characteristics. The proposed methods are tested on two benchmark problems, the Fisher equation and the Burgers--Fisher equation. The results highlight the effectiveness of both approaches, demonstrating superior accuracy in reconstructing unknown boundary conditions. All experiments were conducted in a Jupyter Notebook environment using PyTorch, confirming the potential of tailored activation functions and optimization techniques in physics-informed deep learning. https://ansne.du.ac.ir/article_2139_4e39992c98f2436f7ce0ff0abdca8ab2.pdf