Conference Agenda

In today’s chemical industry, the pursuit of more profitable, sustainable and safer processes is of paramount importance, yet remaining a challenging task given the complex nature of chemical processes. Such systems are defined by governing equations—mass, energy and momentum balances—which are often described by Ordinary Differential Equations (ODEs), Partial Differential Equations (PDEs) or Differential Algebraic Equations (DAEs).

Numerical methods have been traditionally used to replace differential constraints with algebraic equations, enabling the use of state-of-the-art optimization solvers. However, these methods are computationally expensive, limiting their applicability to real-world problems. In this study, we explore the capabilities of Physics-Informed Neural Networks (PINNs) (Raissi et al., 2019) to optimize the design and operation of chemical reactors.

PINNs have emerged as a powerful resource to model complex systems by incorporating physical knowledge into the learning process. This technique outperforms purely data driven strategies in terms of predictive performance and reduces the dependency on large datasets (Ghalambaz et al., 2024)—a key advantage in chemical reactor engineering, where data availability is often scarce. This work expands the applications of PINNs beyond their traditional role as surrogate models, introducing them as an alternative optimization method. Unlike previous studies that have applied a sequential approach (i.e., decoupling training and optimization) (Patel et al., 2023; Ryu et al., 2023), we propose a unified framework where PINNs simultaneously describe the behavior of a reactive system and provide the optimal solutions to a given task (Seo, 2024).

To showcase our methodology, we present two case studies: one focusing on the optimal design, and the other on the optimal control of a reactor. In the former, reactor dynamics and economic goals are integrated into the network architecture to identify the design parameters that minimize capital cost while maintaining performance. The latter case aims at identifying the optimal temperature profile within the reactor. These examples illustrate the potential of PINNs in chemical reactor optimization, arising as an alternative to addressing optimization problems that traditional methods often struggle to solve.

References:

• Ghalambaz, M., Sheremet, M.A., Khan, M.A., Raizah, Z., Shafi, J., 2024. Physics-informed neural networks (PINNs): application categories, trends and impact. Int J Numer Methods Heat Fluid Flow. https://doi.org/10.1108/HFF-09-2023-0568
• Patel, R., Bhartiya, S., Gudi, R., 2023. Optimal temperature trajectory for tubular reactor using physics informed neural networks. J Process Control 128, 103003. https://doi.org/10.1016/J.JPROCONT.2023.103003
• Raissi, M., Perdikaris, P., Karniadakis, G.E., 2019. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. J Comput Phys 378, 686–707. https://doi.org/10.1016/J.JCP.2018.10.045
• Ryu, Y., Shin, S., Liu, J.J., Lee, W., Na, J., 2023. Physics-informed neural networks for optimization of polymer reactor design, in: Kokossis, A.C., Georgiadis, M.C., Pistikopoulos, E. (Eds.), 33rd European Symposium on Computer Aided Process Engineering, Computer Aided Chemical Engineering. Elsevier, pp. 493–498. https://doi.org/https://doi.org/10.1016/B978-0-443-15274-0.50079-2
• Seo, J., 2024. Solving real-world optimization tasks using physics-informed neural computing. Sci Rep 14, 202. https://doi.org/10.1038/s41598-023-49977-3

Artificial neural networks (ANNs) are widely used as surrogate models to represent complex underlying models in process systems engineering [6]. However, it is well-known that ANNs do not guarantee physically consistent predictions thereby preventing its adoption in various real-world scenarios [1]. To mitigate this limitation, significant research has been carried out to enforce known mechanistic relationships between inputs and predictions in neural networks [4,7]. However, current approaches (1) are limited to specific problems governed by specialized mathematical formulations and (2) rely on external solvers that increase the computational cost to train and evaluate the ANN. Addressing the latter issue, Chen et al. proposed KKT-hPINN: a computationally efficient projection method based on Karush-Kuhn-Tucker (KKT) conditions to enforce linear constraints in ANNs [3]. Yet, the method is limited to linear constraints.

We enforce physical laws that are nonlinear in nature by extending the KKT-hPINN approach. For enforcing nonlinear constraints, we propose two approaches: (1) based on the Picard iteration method to enforce multiplicatively separable constraints by sequentially fixing one of the participating variables, and (2) approximate the solution of nonlinear constraints by linearizing them via Taylor expansion with minimum deviation. We test both approaches to train ANNs for two case studies from the literature: (1) catalytic packed bed reactors for methanol synthesis [5], and (2) Gibbs reactor in an autothermal reforming process [2]. We observe that the proposed approaches can be used to efficiently enforce enthalpy balances expressed via nonlinear constraints ensuring physically consistent predictions while retaining inexpensive training and inference. Additionally, atomic conservation laws expressed via linear constraints are imposed. Enforcing conservation laws ensures ANNs improve accuracy even in data-scarce conditions and when using smaller architectures compared to vanilla ANNs. We expect our method to promote wider adoption of ANNs in real-world applications, especially for scenarios such as large-scale simulations and optimization where the observance of fundamental laws is paramount.

References
[1] P. Bedué, A. Fritzsche, Apr. 2021. Can we trust AI? An empirical investigation of trust requirements and guide to successful AI adoption. Journal of Enterprise Information Management 35 (2), 530–549.

[2] S. I. Bugosen, C. D. Laird, R. B. Parker, Jul. 2024. Process Flowsheet Optimization with Surrogate and Implicit Formulations of a Gibbs Reactor 3, 113–120.

[3] H. Chen, G. E. C. Flores, C. Li, Oct. 2024. Physics-Informed Neural Networks with Hard Linear Equality Constraints. Computers & Chemical Engineering 189, 108764.

[4] P. L. Donti, D. Rolnick, J. Z. Kolter, 2021. DC3: A learning method for optimization with hard constraints.

[5] G. Lastrucci, M. F. Theisen, A. M. Schweidtmann, 2024. Physics-informed neural networks and time-series transformer for modeling of chemical reactors, 571–576.

[6] K. McBride, K. Sundmacher, Jan. 2019. Overview of Surrogate Modeling in Chemical Process Engineering. Chemie Ingenieur Technik 91 (3), 228–239.

[7] A. Mukherjee, D. Bhattacharyya, Aug. 2024. On the development of steady-state and dynamic mass-constrained neural networks using noisy transient data. Computers & Chemical Engineering 187, 108722.

Electrochemical Separation (ECS) technologies, including Electrodialysis (ED), Electrodeionization (EDI), and Capacitive Deionization (CDI), are vital for efficient water treatment and desalination processes. However, optimizing the operational conditions of these systems to achieve higher separation efficiency remains a complex challenge due to their nonlinear and dynamic nature. In this paper, we propose a Reinforcement Learning (RL)-based control framework to address this challenge. By applying various RL algorithms, such as model-free and actor-critic methods, we develop an intelligent control strategy that adapts to different system configurations and conditions. This approach autonomously learns the optimal operational parameters, significantly improving ion removal efficiency. The proposed RL-based control system enhances the performance of ECS processes, providing a versatile and adaptive solution for optimizing separation across multiple electrochemical technologies. This work demonstrates the potential of RL in advancing the design and control of sustainable water purification systems.

Robust modeling of process systems is a complex and lengthy procedure. These spatially distributed dynamical systems, usually modeled as systems of partial differential equations, are dependant on both time and space discretization and exhibit highly non-linear behaviour. Analytical solution of the resulting model is often not possible and numerical methods are employed, which are computationally expensive and might encounter instabilities, especially in inverse problems like optimization and state identification or parameter estimation.

Data-driven methods typically have less computational costs and their development requires of less process insights resulting in more efficient surrogate models. Deep learning methods have recently been promising in modeling spatially distributed process systems. Graph Neural Networks (GNNs) are a form of deep learning methods where the system is represented as a graph with a set of nodes and their edge relationships [1]. However, these models are often poor at extrapolation and explainability as they do not respect process boundaries/conditions.

The integration of mechanistic and surrogate models balances the strengths and weaknesses of both modeling principles [2]. The resultant hybrid model, once properly tuned, has better prediction accuracy, increased interpretability and tends to respect the governing physical laws better.

This study aims to develop a Physics-Informed Graph Neural Network (PI-GNN) framework tailored for a catalytic CO₂ methanation reactor in a Power-to-Methane process. The approach integrates a mechanistic model for a single-tube fixed bed methanation reactor [3] and a grid-based graph structure utilizing Graph Attention Networks (GATs) [4]. The inclusion of process insights is anticipated to enhance the model’s predictive capabilities and the explainability of its predictions. The performance of the GNN model, Derivative-Informed GNN, and PI-GNN will be compared to demonstrate these improvements. The hybrid nature of the framework is expected to allow for the use of a less representative mechanistic model in PI-GNN, potentially reducing the time and effort required for model development and accelerating the model development to online deployment pipeline.

References-

[1] Gori M., Monfardini G., and Scarselli F., ”A new model for learning in graph domains”, in Proceedings 2005 IEEE International Joint Conference on Neural Networks, vol. 2. IEEE (2005), pp. 729–734, doi: 10.1109/IJCNN.2005.1555942.

[2] von Stosch M., Oliveira R., Peres J., de Azevedo S.F., ”Hybrid semi-parametric modeling in process systems engineering: Past, present and future.”, Computers & Chemical Engineering, 60 (2014), pp. 86-101, doi: 10.1016/j.compchemeng.2013.08.008.

[3] Zimmermann, R. T., Bremer J., and Sundmacher K., ”Load-flexible fixed-bed reactors by multi-period design optimization.” Chemical Engineering Journal 428 (2022): 130771, doi: 10.1016/j.cej.2021.130771.

[4] Peterson, L., Forootani, A., Sanchez Medina, E.I., Gosea, I.V., Sundmacher, K., Benner, P.: Comparison of data-driven approaches for simulating the dynamic operation of a CO₂ methanation reactor. Submitted to IEEE Transaction on Automation. Science and Engineering (2024).

Digitalization in industrial biotechnology is general slow as most processes still rely on experience and regression-based models, which struggle to address increasing complexity. Digital Twins (DT) are gaining interest for industrial applications, due to their potential to enhance process efficiency and resource utilization. DT is still a fast-evolving concept, aiming at a technology that is modular, generic, and scalable. By leveraging Physics-Informed Neural Networks (PINNs) on the development of DT for bioreactors, we address challenges related to limited experimental data—one of the key barriers to developing robust DTs in biotechnology. We effectively managed system complexity as the dynamics evolved over time, adapting to significant shifts in behaviour. Additionally, we successfully estimated numerous unknown and uncertain parameters in the dynamic models, further enhancing the model’s accuracy and predictive capabilities

Starting with a scaffold—an ODE-based system representing bioreactor dynamics and serving as a generic framework—we used PINNs to upgrade the scaffold into a DT by integrating real-world experimental data. Leveraging the power of PINNs to adhere to the underlying physics of the process (i.e., scaffold) while reducing the need for labelled data, we were able to capture state-space knowledge, building predictive potential and incentivizing data-driven intelligence. We demonstrate our twinning approach on both a batch reactor (continuous case) and a fed-batch reactor (discontinuous case). We showcase that even with minimal data (e.g., just 2 points), the integration of process knowledge into PINNs enables successful twinning. Additional data further refines the model, evolving the scaffold into a fully functional DT. Using PINNs, we were able to estimate the unknown kinetic parameters of the bioreactor dynamics and perform accurate short- and long-term predictions, which are particularly valuable in process optimization and control. By leveraging PINNs, we overcame the need for the complete data trajectory and full knowledge of system changes to build an accurate model. Through twinning, we developed a more robust model, even without prior training on discrete changes. The scaffold’s inherited system knowledge allows the model to predict discrete changes and transitions.

In conclusion, our approach demonstrates the potential of PINN powered DTs to transform bioprocess modelling, even in the face of limited data and complex system dynamics. By integrating domain knowledge into the scaffold and allowing the model to adapt to changes in real time, we can develop a flexible and scalable framework for bioreactor optimization. This work highlights the capability of DT not only to enhance predictive accuracy but also to streamline the estimation of complex kinetic parameters, paving the way for more efficient, data-driven biomanufacturing processes.