(614g) Integrating Timescales in Process Modeling: A Hybrid Modeling Approach Combining First Principles and Neural Networks, with Applications in Crystallization Mechanisms

In the multifaceted domain of process modeling, epsilon (ε)-based timescale modeling using singular perturbation theory has proven to be instrumental, particularly for processes containing multiple simultaneous phenomena at differing timescales [1,2]. These phenomena, ranging from rapid nucleation in crystallization to slower thermal effects in chemical reactors, are underpinned by rate constants or any meaningful variable expressed as small parameters in the form of ε, which encapsulate the crux of the system's dynamic response [3]. Capturing and accurately estimating the behavior of ε and its impact on system dynamics poses significant challenges due to the inherent complexity and potential for system instability, that occurs due to the difference in timescales of the governing equations. This complexity is further increased by nonlinearities and the non-explicit separation of timescales in many chemical processes, such as in continuous stirred tank reactors (CSTRs), multi-phase and catalytic reactions, chemical reaction networks, etc [4]. The primary challenge is that in many cases, these ε parameters are not constant and vary with operating conditions like temperature, and flow rates, which further affects the output states.

In this work, a hybrid modeling approach is presented, which utilizes neural networks to estimate time-varying ε values that are then integrated into a first-principles model. This novel approach attempts to address the limitations of previous two-timescale studies by using a machine learning (ML) component that allows for the system to adapt to process changes affecting this timescale separation, thus providing a more accurate and robust estimation of ε [5]. To this end, a hybrid modeling framework is developed where a neural network is used to estimate the timescale separation parameters. The network is trained through backpropagation using experimental data that reflects the underlying physical phenomena. Once estimated, the ε values are used in a transformed coordinate system, converting the original ODEs into a standard singularly perturbed form where fast and slow kinetics are distinct so that a rigorous analysis of the resulting subsystems can be conducted. Similar to the concepts detailed in the literature on singular perturbation modeling of nonlinear processes [6], this transformation is essential for accurate timescale integration. The neural network predictions are iteratively refined by comparing the ODE simulation outputs against actual process data, adjusting the network's weights and biases to home in on the true ϵ dynamics. Further, this splitting of the subsystems allows us to develop and implement efficient control strategies for either of the slow/fast subsystems separately.

This approach is applied to a case study focusing on the crystallization process to illustrate the efficacy of this approach. Distinct ε values for nucleation and aggregation rates are estimated from the neural network, with the ODEs simulated across various temperatures and solubility conditions to generate a rich dataset that predicts the number of crystals, crystal size distribution (CSD), and solute concentration. This data drives the iterative learning process, fine-tuning the neural network to more precisely capture the ε dynamics under changing conditions. Once these ε parameters are obtained, they are used to segregate the fast nucleation subsystem and the slower aggregation subsystem, for further separate controller development on these subsystems. This work thus demonstrates a robust hybrid model capable of learning complex time-varying ε dynamics, improving the prediction and control of crystallization processes. The implications for industrial applications are profound, as this approach significantly enhances the analysis and control of processes with timescale multiplicity. This framework can also be used to guide the optimization and design of industrial processes, setting a precedent for marrying the use of hybrid models with the established ε-based singular perturbation models.

References:

1. Kumar, Aditya, Panagiotis D. Christofides, and Prodromos Daoutidis. "Singular perturbation modeling of nonlinear processes with nonexplicit time-scale multiplicity." Chemical Engineering Science 53.8 (1998): 1491-1504.
2. Van Breusegem, Vincent, and George Bastin. "Reduced order dynamical modelling of reaction systems: a singular perturbation approach." Proceedings of the 30th IEEE Conference on Decision and Control. Vol. 2. 1991.
3. Christofides, Panagiotis D., and Prodromos Daoutidis. "Feedback control of two-time-scale nonlinear systems." International Journal of Control 63.5 (1996): 965-994.
4. Abdullah, Fahim, Zhe Wu, and Panagiotis D. Christofides. "Data-based reduced-order modeling of nonlinear two-time-scale processes." Chemical Engineering Research and Design 166 (2021): 1-9.
5. Shah, P., Sheriff, M. Z., Bangi, M. S. F., Kravaris, C., Kwon, J. S. I., Botre, C., & Hirota, J. (2022). Deep neural network-based hybrid modeling and experimental validation for an industry-scale fermentation process: Identification of time-varying dependencies among parameters. Chemical Engineering Journal, 135643.
6. Kokotović, Petar, Hassan K. Khalil, and John O'reilly. Singular perturbation methods in control: analysis and design. Society for Industrial and Applied Mathematics, 1999.