Synergistic modeling of hydrogel gelation via time-delay dynamics and machine learning algorithms

This paper presents an integrated framework in which delay differential equation (DDE) modeling and machine learning (ML) approaches are coupled to study hydrogel formation kinetics, with emphasis on delayed crosslinker addition. Conventional mechanistic models disclose many physical and kinetic complexities of reacting mixtures; they seldom depict the nonlinear and time-evolving complexities inherent in developing polymer networks. To address this, a mathematical model is developed that examines how the insertion of crosslinkers affects system stability and equilibrium. Analytical and numerical results show that delays nearing critical levels cause bifurcation behavior with substantial implications on gelation kinetics. Sophisticated machine learning systems, including artificial neural networks, support vector regression, and ensemble learning, may accurately estimate gel fractions in complicated parameter spaces with $R^2$ > 0.95. Experimental validation confirms a maximum gel fraction of 0.85$\pm$0.03 with delayed crosslinker addition. By coupling DDE modeling with ML, this framework captures temporal sensitivities, derives generalized kinetic rules, and supports faster optimization of synthetic routes in smart material engineering.