A MACHINE LEARNING-INTEGRATED NUMERICAL FRAMEWORK FOR SOLVING NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS IN CLIMATE MODELING OF PAKISTAN

Abstract

This study developed a machine learning–integrated numerical framework for solving nonlinear fractional differential equations (NFDEs) in climate modeling applications in Pakistan. The primary objective was to address the computational limitations of conventional numerical methods in capturing nonlinear, multiscale, and memory-dependent climatic dynamics. The proposed framework integrated scientific machine learning techniques, including physics-informed neural networks and neural operator approximations, with fractional calculus-based numerical methods to enhance predictive accuracy, computational efficiency, and numerical stability. A quantitative and computational research design was employed using secondary climate datasets representing key meteorological variables of Pakistan, including temperature, precipitation, and atmospheric variability indicators. The performance of the proposed framework was evaluated and compared with traditional numerical approaches using standard metrics such as RMSE, MAE, execution time, convergence behavior, and stability indices. The results demonstrated that the proposed framework significantly outperformed conventional methods, reducing computational cost and prediction errors while improving stability and forecasting accuracy. Furthermore, the framework effectively captured nonlinear interactions and long-term memory effects inherent in climatic processes. The findings confirmed that integrating machine learning with fractional differential equation solvers offers a robust and scalable approach for climate modeling in highly complex and uncertain environments. The study contributes to computational mathematics, scientific machine learning, and climate science by introducing an advanced hybrid modeling paradigm suitable for climate-vulnerable regions such as Pakistan.