A Hybrid Method Based on Rational Legendre Physics Informed Neural Networks and Cuckoo Optimization Algorithm for Solving Nonlinear Inverse Partial Differential Equations

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.

کلیدواژه ها:

Physics-informed neural networks ، Nonlinear inverse partial differential equations ، Rational Legendre polynomials ، Cuckoo optimization algorithm ، Fisher equation ، Burgers-Fisher equation