# Identification of Crude Distillation Unit: A Comparison between Neural Network and Koopman Operator
## 2 Modeling and Methodology
### Koopman Operator Theory
To validate the efficiency of the proposed methods in capturing the dynamics of the CDU, training and validation were performed on the nominal dataset (Case 1), and each model was tested on the remaining datasets. Figure 3: Crude distillation unit diagram. ## 4 Results and Discussion

To ensure a fair comparison among the models, the hyperparameters of each model were fine-tuned using Bayesian optimization, implemented with the KerasTuner API. The objective was to minimize the validation MSE. Table 3 outlines the specific hyperparameters and their respective search ranges. The results are summarized in Table 4. In what follows, the Koopman linear and bilinear models are referred to as KL and KB, respectively. The results based on the training dataset are summarized in Figure 4, and those for the test datasets are provided within Figures 5-11. The plots in Figures 12-13 in Appendix A depict the predicted outputs \((y_{1},y_{2},y_{3})\) vs. the actual values for each case. As can be noticed from Figure 4, the NAR\(-\)NN model had the lowest average MSE (0.0081) on the training dataset, suggesting that NAR\(-\)NN fitted the training data well and indicating good initial performance in the identification of the CDU. In comparison, \begin{table}
\begin{tabular}{l l} \hline \hline
**Inputs** & \\ \hline \(u_{1}\) & Column reflux \\ \(u_{2}\) & Middle draw \\ \(u_{3}\) & Column bottom tray temperature \\ \(d\) & Feed to column \\ \hline
**Outputs** & \\ \hline \(y_{1}\) & Overhead composition \\ \(y_{2}\) & Middle-draw composition \\ \(y_{3}\) & Bottoms’ composition \\ \hline \hline \end{tabular}
\end{table} 
Table 1: List of inputs and outputs. \begin{table}
\begin{tabular}{l l l} \hline \hline
**Model** & **Hyperparameter** & **Range** \\ \hline KL / KB & Network width & [16; 256] \\  & Network depth & [1; 32] \\  & No. of lifted dynamics & [4; 64] \\ \hline NARX & Network width & [4; 256] \\  & Network depth & [1; 32] \\  & Delay & [1; 4] \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Hyperparameter search space. \begin{table}
\begin{tabular}{l l l} \hline \hline
**Model** & **Hyperparameter** & **Range** \\ \hline KL / KB & Network width & [16; 256] \\  & Network depth & [1; 32] \\  & No. of lifted dynamics & [4; 64] \\ \hline NARX & Network width & [4; 256] \\  & Network depth & [1; 32] \\  & Delay & [1; 4] \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Hyperparameters tuning results. slightly better than the KL model, where the latter showed the highest average MSE of 0.0127. However, across all test scenarios, KB consistently achieved superior performance with the lowest average MSE, indicating better generalization ability and robustness in capturing the dynamics of the CDU across different experimental conditions. The KL model performed competitively with KB and often outperformed NARX\(-\)NN. While benefiting from exogenous inputs and past outputs is powerful for NARX\(-\)NN, it also adds more complexity to the model. This may be sensitive in terms of model parameters and can lead to overfitting, causing poorer generalization in capturing the dynamics under different experimental conditions. In cases where nonlinearities were present in the inputs, both the KL and KB models experienced a noticeable degradation in performance. This degradation was primarily due to the assumption that the control inputs remained unchanged within the lifted dynamics. However, the KL model exhibited the poorest performance; conversely, the inclusion of a bilinear term in the KB model allowed for a more flexible representation of the system dynamics. This flexibility enabled the KB model to capture certain nonlinear behaviors that the linear KL model could not, leading to its superior performance. Based on these observations, one may conclude that under different experimental conditions, the NARX\(-\)NN model dramatically failed to provide predictions, while the KB model continued to make good predictions. Figure 4: Summry of MSE values on the Case 1 dataset. Figure 5: Summary of MSE values on the Case 2 dataset.