Improved modeling of crystallization processes by Universal Differential Equations

Abstract

Crystallization is a crucial process in the pharmaceutical industry, usually modeled by Population Balance Models (PBM). This study introduces a novel approach, combining PBM with machine learning techniques, specifically the Universal Differential Equation (UDE), to describe a batch cooling crystallization process. Unlike conventional methods, UDE eliminates the need for defining a supersaturation term and was applied to model the nucleation, growth, and dissolution of potassium sulfate in water. In this study, three UDE models for supersaturated condition and one for undersaturated condition were developed, aiming to predict particle count, length, surface area, volume, and concentration. Initially, these models were trained using all available experimental data from a previous study for the potassium sulfate batch crystallization. They were validated with simulated data generated by the PBM. In a second scenario, the models were trained with a subset of the experimental data and tested with the remainder. The UDE models performed efficiently, exhibiting similar mean squared error and mean absolute error compared to conventional PBMs. Notably, the UDE approach demonstrated an advantage in requiring a smaller training dataset. This innovative coupling of UDE and PBM offers a less complex yet effective model to capture essential aspects of crystallization kinetics and variables’ dynamics. Particularly, UDE models excel in predicting nucleation kinetics, making them a valuable alternative to conventional representations. Furthermore, this approach can be easily adapted to various crystallization processes involving different phenomena, enhancing its versatility and potential impact.