Data-Driven Discovery of Population Balance Equations for the Particulate Sciences

Understanding the behavior of particles in a dispersed phase system via population balances holds fundamental importance in studies of particulate sciences across various fields. Particle behavior, however, is sophisticated as a single particle can undergo internal property changes (e.g., size, cell age, and energy content) through various mechanisms. When confronted with an unknown distributed particulate system, discovering the underlying population balance equation (PBE) entails firstly learning the underlying particulate phenomena followed by the associated phenomenological laws that govern the kinetics and mechanisms of particle transformations in their local conditions. Conventional inverse problem approaches reveal the shape of phenomenological functions for predetermined forms of PBE (e.g., pure breakage/aggregation PBE, etc.). However, these methods can be limited in their ability to uncover the mechanisms which govern uncharacterized particulate systems from data. Leveraging the increasing abundance of data, we devise a data-driven framework based on sparse regression to learn PBEs as linear combinations of an extensive pool of candidate terms. Thus, this approach enables effective and accurate functional identification of PBEs without assuming the structure a priori, hence mitigating any potential loss of details, while minimizing model overfitting and providing a more interpretable representation of particulate systems. We showcase the proficiency of our approach across a wide spectrum of particulate systems, ranging from simple canonical pure breakage and pure aggregation systems to complex systems with multiple particulate processes. Our approach holds the potential to generalize the discovery of PBEs along with their phenomenological laws from data, thus facilitating wider adoption of population balances.