Interpretable and uncertainty-aware machine learning for trustworthy prediction in batch crystallization

Abstract

Symbolic regression was applied to model the nucleation and crystal growth in potassium sulfate batch crystallization. A methodology combining multivariable analysis, symbolic regression, meta-heuristic optimization, and statistical analysis was developed and compared with an alternative neural network-based approach. Experimental data were used to identify models via symbolic regression for predicting the first four moments of the crystal size distribution (CSD) and to train neural networks for the same purpose. Both methods efficiently modeled the crystallization process, achieving similar mean squared error (MSE) values. Additionally, solute concentration predictions were successful. The models were tested on simulated batches with supersaturation conditions not present in the experimental dataset. Both approaches performed comparably to the population balance model (PBM). However, symbolic regression required fewer parameters than neural networks. Parameter estimation for symbolic regression was done using particle swarm optimization (PSO), and confidence regions were constructed via Fisher’s test, revealing ellipsoidal shapes for most variables. For the zero-order moment, reparameterization achieved well-defined confidence regions. Symbolic regression provided interpretable and generalizable models that describe key phenomena, such as nucleation and crystal growth, aligning with population balance theory. These models offer advantages for modeling, controlling, and optimizing crystallization processes over traditional ODE systems.