Two-timescale-based hybrid modeling framework for crystallization processes

Many real-world chemical processes exhibit dynamics that span widely separated timescales, creating challenges in modeling, simulation, and control. Accurately resolving fast transients alongside slow evolutions is computationally demanding, as stiff systems often require fine temporal discretization for numerical stability, particularly when using detailed first-principles models. Crystallization processes exemplify such two-timescale systems, where rapid nucleation and crystal growth interact with slower aggregation dynamics to shape the evolving crystal size distribution (CSD). Traditional population balance models, such as the Smoluchowski framework, become computationally intractable due to the nonlinear and stiff nature of aggregation terms. To overcome these challenges, we propose a two-timescale-based hybrid modeling framework that integrates method-of-moments (MoM) equations to capture fast dynamics from nucleation and growth, while a deep neural network (DNN) surrogate replaces the computationally intensive aggregation terms. This separation allows fast and slow subsystems to be handled using tailored modeling strategies, improving numerical stability and simulation speed without compromising accuracy. We embed this hybrid model within a model predictive control (MPC) architecture to manipulate temperature trajectories, enabling independent control of nucleation, growth, and aggregation rates. Our results demonstrate that this framework enables precise regulation of the CSD and shows strong potential for enhancing product uniformity and quality in pharmaceutical crystallization processes.