Modeling Batch Crystallization under Uncertainty Using Physics-informed Machine Learning

The development of robust and reliable modeling approaches for crystallization processes is often challenging because of non-idealities in real data arising from various sources of uncertainty. This study investigated the effectiveness of physics-informed recurrent neural networks (PIRNNs) that integrate the mechanistic population balance model with recurrent neural networks under the presence of systematic and model uncertainties. Such uncertainties are represented by using synthetic data containing controlled noise, solubility shift, and limited sampling. The research demonstrates that PIRNNs achieve strong generalization and physical consistency, maintain stable learning behavior, and accurately recover kinetic parameters despite significant stochastic variations in the training data. In the case of systematic errors in the solubility model, the inclusion of physics regularization improved the test performance by more than an order of magnitude compared to purely data-driven models, whereas excessive weighting of physics increased error arising due to the model mismatch. The results also show that PIRNNs are able to recover model parameters and replicate crystallization dynamics even in the limit of very low sampling resolution. These findings validate the robustness of physics-informed machine learning in handling data imperfections and incomplete domain knowledge, providing a potential pathway toward reliable and practical hybrid modeling of crystallization dynamics and industrial process monitoring and control.

💡 Research Summary

This paper presents a comprehensive study on the use of physics‑informed recurrent neural networks (PIRNNs) for modeling batch cooling crystallization under various sources of uncertainty. The authors begin by outlining the challenges inherent in crystallization processes—namely, the need for accurate prediction of particle size distribution, purity, and polymorphism, and the difficulty of parameter estimation for mechanistic population balance models (PBMs) when experimental data are noisy, sparse, or biased. To address these issues, they propose a hybrid modeling framework that embeds the PBM’s governing ordinary differential equations directly into the loss function of a recurrent neural network, thereby enforcing physical consistency during training.

Synthetic data are generated from a well‑established PBM for paracetamol batch cooling, using the method of moments to reduce the PDE to a set of ODEs for the zeroth through third moments and solute concentration. The authors deliberately introduce three classes of uncertainty: (1) aleatoric measurement noise (10 %, 30 %, and 100 % of the signal standard deviation), (2) epistemic model mismatch by shifting the solubility polynomial by 10 % across the temperature range, and (3) epistemic sampling scarcity, where only 2, 3, 5, or 9 time points per run are retained (total training points ranging from 20 to 90). This systematic manipulation creates a realistic testbed for evaluating model robustness.

The PIRNN architecture consists of an LSTM (or vanilla RNN) core that predicts the time evolution of the state variables. The loss function combines a data‑fit term (mean‑squared error between predictions and noisy observations) with a physics‑regularization term that penalizes violations of the PBM equations. A weighting factor λ controls the trade‑off between data fidelity and physical adherence. Extensive hyper‑parameter sweeps reveal that moderate λ values (≈10–100) yield the best performance; overly large λ forces the network to obey an inaccurate physics model (e.g., the shifted solubility), leading to increased prediction error, while λ≈0 reduces the model to a purely data‑driven RNN that overfits noise.

Results show that PIRNNs achieve markedly lower test errors than both pure data‑driven RNNs and the standalone PBM. In high‑noise scenarios, the mean absolute error of PIRNN predictions remains below 0.045 g/g, compared with >0.12 g/g for the RNN and >0.20 g/g for the PBM. Moreover, the hybrid model accurately recovers kinetic parameters (secondary nucleation constant k_b2, growth pre‑exponential factor k_g, and exponentials α, β, γ) within 5 % of their true values, even when training data are severely down‑sampled. Notably, with only two measurement points per run (initial and final), PIRNN still converges to reasonable parameter estimates, demonstrating the strong regularizing effect of the embedded physics.

The study also highlights the sensitivity of performance to the physics‑regularization weight. A “U‑shaped” error curve is observed as λ varies: error decreases as λ grows from zero, reaches a minimum at moderate λ, then rises sharply when λ becomes too large. This underscores the necessity of balancing model fidelity against model mismatch, especially when the underlying thermodynamic correlations (e.g., solubility) are imperfect.

In the discussion, the authors acknowledge that real experimental data would require conversion of indirect measurements (FBRM counts, Raman spectra) into moments, a step they bypass by assuming a perfect sensor. They suggest that integrating a shallow neural network for this conversion, as demonstrated in prior work, would make the approach directly applicable to laboratory data. Future extensions include Bayesian treatment of λ, uncertainty quantification of predictions, and application to multi‑polymorphic systems.

In conclusion, the paper provides strong evidence that physics‑informed machine learning can robustly handle aleatoric and epistemic uncertainties in crystallization modeling. By embedding mechanistic knowledge into a flexible recurrent network, PIRNNs retain interpretability, improve extrapolation, and enable reliable parameter estimation even with limited, noisy data. This hybrid strategy offers a promising pathway for real‑time monitoring, control, and scale‑up of industrial crystallization processes.