To overcome these limitations, this study introduces a novel two-timescale-based hybrid modeling methodology that integrates FPMs for capturing rapid dynamics with deep neural networks (DNNs) as surrogate models for slower, computationally demanding dynamics [3,4]. The proposed hybrid approach systematically decomposes a complex system into two subsystems aligned with their intrinsic timescales. The fast subsystem dynamics, typically governed by well-established physical mechanisms and characterized by rapid responses to operational changes, are accurately and efficiently represented using first-principles equations. Conversely, the slow subsystem, often involving stiff and nonlinear processes, is represented through DNN or other data-driven models [5]. These neural network surrogates are trained using extensive, high-fidelity data generated from detailed simulations of complex slow dynamics.
This two-timescale-based hybrid modeling strategy significantly mitigates the computational stiffness and nonlinear complexities traditionally associated with fully mechanistic models, enabling efficient and robust predictions suitable for real-time implementation within advanced control frameworks. To illustrate the efficacy of the proposed methodology, the hybrid modeling approach is applied to pharmaceutical crystallization processes, specifically targeting the challenging problem of transforming bimodal or multimodal crystal size distributions (CSDs) into unimodal distributions [6,7]. Here, the crystallization mechanism with fast dynamics like nucleation and growth is represented using method-of-moments (MoM) equations, and the complex slow dynamics of particle aggregation are described by Smoluchowski population balance equations. Within a model predictive control (MPC) framework, the hybrid model facilitates dynamic optimization of critical operational parameters, including temperature and supersaturation. By accurately capturing and predicting the interactions between fast nucleation/growth dynamics and slower aggregation phenomena, the MPC strategically regulates these parameters to minimize secondary nucleation and reduce aggregation-driven broadening. As a result, the control system effectively guides the crystallization process toward desired unimodal distributions, substantially enhancing product uniformity and downstream processing efficiency.
Preliminary simulation results underscore the advantages of the two-timescale-based hybrid modeling approach, highlighting significant computational speedups compared to conventional fully mechanistic models without compromising predictive accuracy. This methodological advancement represents a major step forward in the capability to handle real-time control scenarios for stiff, nonlinear, multiscale processes. The proposed hybrid modeling and control approach, while demonstrated through crystallization systems exhibiting bimodal-to-unimodal distribution transformation, presents broad potential applicability across various chemical engineering domains facing similar multiscale dynamic challenges.
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