AI Tools For Calculus: Enhancing Algorithmic Discovery And Mathematical Understanding

The growing complexity of global sustainability challenges demands advanced analytical and computational approaches. Issues such as climate change, energy transition, urban resilience, and the circular economy require integrated scientific methods for effective solutions. Computational mathematics encompassing mathematical modelling, numerical analysis, optimization theory, uncertainty quantification, and scientific machine learning—plays a crucial role in advancing sustainable engineering innovations. This paper presents a multidisciplinary overview of computational mathematics techniques with machine learning models applied to environmental and climate modelling, circular manufacturing systems, green infrastructure, smart grids, sustainable transportation, and renewable energy technologies. The study critically evaluates several computational approaches, including graph-theoretic models, multi-scale simulations, stochastic systems, partial differential equation (PDE) frameworks, and evolutionary optimization methods in sustainability engineering. It also explores emerging concepts such as digital twins, quantum-inspired optimization, and climate simulations supported by high-performance computing (HPC). Furthermore, the research addresses challenges related to algorithm scalability, model interpretability, uncertainty propagation, and ethical considerations in AI-driven sustainability systems. By integrating mathematics, computer science, and engineering principles, this paper highlights how computational mathematics enables carbon-neutral system design, predictive analytics, and efficient resource management. It also identifies future research directions, including distributed computing architectures, hybrid AI–physics models, and quantum-enhanced optimization to support global Sustainable Development Goals.