Mathematical modelling using generative artificial intelligence

Traditional mathematical modelling relies heavily on human intuition, trial-and-error parameter tuning, and domain expertise. This paper presents a comprehensive framework for leveraging Generative Artificial Intelligence (AI) and Large Language Models (LLMs) to automate the formulation, analysis, and optimization of mathematical models. We investigate the integration of Physics-Informed Neural Networks (PINNs), symbolic regression, and LLM-driven code generation to bridge the gap between empirical data and analytical equations. Our results demonstrate that AI-driven frameworks can discover governing differential equations up to 10 times faster than manual derivation while maintaining physical consistency. This study establishes a paradigm shift in computational mathematics, enabling rapid, automated scientific discovery across fluid dynamics, epidemiology, and financial systems.