Fractional calculus innovations and machine learning-driven advances in thermal science

This paper explores fractional calculus innovations and machine learning?s role in advancing thermal science, especially in smart textiles. It first introduces three key fractional derivatives (Caputo, Riemann-Liouville, two-scale fractal) for thermal analysis, highlighting their strengths in capturing non-locality, memory effects, and fractal characteristics. Then, it details how the two-scale fractal deriva-tive modifies Caputo and Riemann-Liouville derivatives to better model complex thermal systems in smart textiles, with simplified forms balancing accuracy and computational efficiency. Finally, it discusses machine learning?s synergy with fractional calculus, optimizing model parameters, capturing nonlinearities, and enabling data-driven fractional models, to solve intractable thermal problems in smart textiles, supporting applications like complex textile material heat transfer and electronic thermal management in wearable smart textiles.