A comprehensive review of an artificial intelligence-based framework for classifying and solving modified heat partial differential equations

Abstract

Partial differential equations (PDEs) fundamentally describe the dynamic relationships among physical quantities and therefore play an essential role in modelling physical processes such as heat transfer, fluid flow, population dynamics, and material behavior, unlike purely algebraic relations. This study presents a critical narrative review of analytical, numerical, and artificial-intelligence-based approaches relevant to modified heat equations, with emphasis on their assumptions, limitations, and suitability for different problem classes. Although some existing hybrid methods integrate governing physics with machine learning to achieve physical informativeness and computational efficiency, such as physics-informed neural networks (PINNs) and neural-operator methods, there remains an unfulfilled need to develop a consistent framework for systematically categorizing modified PDE forms and to identify an appropriate analytical, numerical, and AI-based solution strategy. It provides a critical review of traditional methods of analysis, such as separation of variables, similarity transformations, and perturbation methods, as well as numerical methods, including finite-difference and finite-element methods, alongside modern machine-learning techniques, particularly deep learning networks, and their relative advantages, weaknesses, and integration possibilities. A classification and integration framework of modified heat equations, based on their physical and mathematical properties, is introduced, connecting suitable analytical, numerical, and data-driven solution methods coherently to the complex heat equations. This integrative perspective is intended to support physically interpretable method selection, computational efficiency, and scalable modelling of modified heat equations across scientific and engineering applications.