(579g) Hybrid Data-Driven Models with Physics-Informed Loss for Process Monitoring and Control

Emphasis has been placed on combining process and data-driven knowledge for modeling chemical processes using Machine Learning (ML) [1]. Hybrid models have been proposed as a solution due to their distinct advantages over conventional ML models. Such models incorporate knowledge obtained from physical phenomena and equations or process flowsheets to improve their generalizability [2]. Improved extrapolation capability, increased accuracy in the presence of noise, and improved efficiency in model development are some of the advantages of such models. Physics-informed neural networks (PINNs) are a type of hybrid model that use physics-based knowledge, such as in the form of partial differential equations (PDEs), to generate residuals from neural network predictions. The residuals are incorporated as soft constraints in the optimization objective of training the neural network, and the physics-informed loss serves to regularize the model, guiding it toward conforming to the PDEs. In our previous works, we have shown that PINNs can be successfully used to model complex adsorption processes such as Pressure Swing Adsorption (PSA) [3] and chromatographic separations [4]. In this work, we intend to develop PINNs considering temporal correlations and in a form compatible with a typical process monitoring and control framework.

Most PINNs used in literature have been for cases where automatic differentiation is employed to calculate gradients with respect to time, considering time as an explicit input. This approach may pose challenges for chemical processes, which typically exhibit temporal correlations. Moreover, to render the model compatible with a monitoring and control framework, it would be advantageous to develop a recursive model that integrates previous measurements and predictions with inputs to obtain future prediction values. Nonlinear Auto-Regressive models with eXogenous inputs (NARX) and Recurrent Neural Networks (RNN) [5] are possible choices for this task. NARX models are easier to develop and train while RNNs require more training effort and utilize a larger amount of computational resources. Depending on measurement frequency, the choice of a model must be made. For example, if frequent measurements are available, a NARX model may be suitable since inputs needed for predictions are available at regular intervals. However, if the sampling frequency is low, RNNs may need to be used.

Including process knowledge in these models is essential to leverage the advantages of PINNs. In the absence of complete knowledge about the process, PINNs can be used to estimate the unknown parameters while training the models [6]. While training the neural networks, process parameters in residual equations can be set up as trainable parameters to estimate their values from measurement data. Furthermore, utilizing the residual equations within PINNs facilitates monitoring of the process data and changing process parameters. In such cases, the model needs to be continuously updated to accommodate the evolving conditions. Once the model loses accuracy/fidelity beyond a threshold, this can be set up as an inverse problem to estimate the value of the evolved parameter and re-training can be conducted.

In this work, we propose different physics-informed data-driven models in a form compatible with process control frameworks to predict process behavior, especially for systems with spatiotemporal variations. For the case of high sampling frequency, a Physics-Informed Nonlinear Auto-Regressive model with eXogenous inputs (PI-NARX) model is developed using measurements at previous time instances. To incorporate physics-based knowledge, gradients in PDE equations need to be calculated using an appropriate numerical technique on model predictions, since time is not an explicit input and automatic differentiation can no longer be employed. Finite differencing is implemented to generate physics-based residuals. The difference between predictions and the corresponding inputs is used to obtain the temporal gradients. For the cases where the sampling frequency is low, the PI-NARX model cannot be used since the measurements at all previous time instances aren’t available as inputs to the model. A physics-informed Encoder-Decoder Recurrent Neural Network (PI-RNN) architecture is proposed which makes use of available measurements to predict values of process variables at all time instants even at which measurements are not available. These models employ a recurrent layer in the encoder and decoder part each, to capture the temporal dependency. The current prediction and the prediction from previous time instants can be utilized to compute time-based gradients to obtain the residuals for incorporating physics-based constraints. Development of such models is more difficult and time-consuming since the gradients need to be backpropagated through time in addition to the inputs.

The methods outlined above will be employed to model both the reactor unit of the benchmark Tennessee Eastman Process (TEP) and the adsorption step of a PSA process for post-combustion carbon capture. TEP is a complex process consisting of different unit operations used for testing and validating control and optimization algorithms. In this work, only the reactor unit is considered for modeling but the process can be extended to data from other unit operations. Time series data from the reactor unit of the TEP can be modeled using PI-NARX and PI-RNN models. PSA is a complex nonlinear cyclic process that is used to separate a mixture of gases using an adsorbent material. Data from PSA is also obtained as a time series, and each step of PSA in a cycle can also be considered as a batch process. The improved performance of employing a recurrent model over conventional PINN will be demonstrated. To demonstrate the utility of the models for parameter estimation, it will be assumed that complete knowledge about the process parameters is unavailable. Consequently, the parameters will be estimated while training the model using simulation data. Moreover, the use of PI-NARX and PI-RNN models for model updating and maintenance will also be showcased. This will involve considering an evolving parameter, such as the activation energy of reaction in the TEP reactor and adsorption isotherm parameters for PSA. The changes in model predictions will be monitored using residual values, by comparing them with the base value. Once a threshold value is breached, model re-training will be triggered, estimating the changed parameter in the process.

Model validation and updating is conducted on the adsorption step of a four-step PSA for post-combustion carbon capture using conventional PINN. It is assumed that the parameter bed voidage changes over time, causing a deviation in model predictions. The residual loss is used to monitor the process and the model is updated once a threshold is reached. Results are plotted in the figure for monitoring the gas phase composition of carbon dioxide measured at the outlet of the bed (x = L, where L is the bed length) and at the end of the adsorption step (t = t_ads where t_ads is the adsorption time). It can be seen that the updated PINN tracks the parameter changes and improves model predictions. In this work, PI-NARX and PI-RNN will be used instead of conventional PINNs, and their utilities for reactors and PSA will be demonstrated.

References

[1] N. Sharma and Y. A. Liu, “A hybrid science-guided machine learning approach for modeling chemical processes: A review”, AIChE Journal, vol. 68, issue 5, p. 1-19. 2022.

[2] E. Gallup, T. Gallup and K. Powell, “Physics-guided neural networks with engineering domain knowledge for hybrid process modeling”, Computers & Chemical Engineering, vol. 170, p. 108111. 2023.

[3] S. G. Subraveti, Z. Li, V. Prasad and A. Rajendran, “Physics-Based Neural Networks for Simulation and Synthesis of Cyclic Adsorption Processes”, Ind. Eng. Chem. Res., vol. 61, no. 11, p. 4095–4113. 2022.

[4] S. G. Subraveti, Z. Li, V. Prasad and A. Rajendran, “Can a Computer “Learn” Nonlinear Chromatography?: Experimental Validation of Physics-Based Deep Neural Networks for the Simulation of Chromatographic Processes”, Ind. Eng. Chem. Res., vol. 62, no. 14, p. 5929–5944. 2023.

[5] Y. Zheng, C. Hu, X. Wang and Z. Wu, “Physics-informed recurrent neural network modeling for predictive control of nonlinear processes”, Journal of Process Control, vol. 128, p. 103005. 2023.

[6] M. Raissi, P. Perdikaris and G. E. Karniadakis, “Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations”, Journal of Computational Physics, vol. 378, p. 686-707. 2019.