# Identification of Crude Distillation Unit: A Comparison between Neural Network and Koopman Operator
## 2 Modeling and Methodology
### Koopman Operator Theory
Figure 8: Summary of MSE values on the Case 5 dataset. Figure 6: Summary of MSE values on the Case 3 dataset. Figure 7: Summary of MSE values on the Case 4 dataset. ## 5 Conclusions

This article presented a comparison of three different models in identifying the CDU process, namely NARX\(-\)NN, KL, and KB, conducted under different experimental condi

Figure 11: Summary of MSE values on the Case 8 dataset. Figure 9: Summary of MSE values on the Case 6 dataset. tions, such as gain, delay mismatches, nonlinearities, and disturbances. During the training process, the NARX\(-\)NN model had the lowest average Mean Squared Error (MSE), indicating a high level of initial competency. Nevertheless, KB demonstrated higher performance and robustness across different datasets used for testing, surpassing the performance of both NARX\(-\)NN and KL. Although KL showed comparable performance to KB and sometimes outperformed NARX\(-\)NN, its performance significantly deteriorated when faced with nonlinearities. The use of a bilinear term in KB was the main differentiating factor, providing a more flexible representation of system dynamics compared to the linear assumption in the KL model. This flexibility allowed KB to capture nonlinear phenomena that other models could not, resulting in superior performance, as evidenced by lower MSE values. The KB stands out as a potential option because of its improved capacity to generalize. Conceptualization, A.N.A., M.K.K., M.A., R.P. and H.S. ; Data curation, M.K.K. and R.P. ; Formal analysis, A.N.A., M.K.K. and M.A. ; Funding acquisition, M.A. ; Methodology, A.N.A., M.K.K., M.A., R.P. and H.S. ; Project administration, M.A., R.P. and H.S. ; Software, A.N.A., M.K. and R.P. ; Supervision, M.A., R.P. and H.S. ; Validation, M.K.K., M.A., R.P. and H.S. ; Visualization, A.N.A., M.K.K. and R.P. ; Writing--original draft, A.N.A. and M.K.K. ; Writing--review and editing, A.N.A., M.K.K., M.A., R.P. and H.S. All authors have read and agreed to the published version of the manuscript. The authors would like to acknowledge the support of King Fahd University of Petroleum & Minerals and the Interdisciplinary Research Center for Sustainable Energy Systems (IRC-SES). The raw data supporting the conclusions of this article will be made available by the authors on request. Author [PERSON] and [PERSON] are employed by the company Saudi Aramco. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest. ## Appendix A

The figures provided herein illustrate the performance of each model, NARX\(-\)NN, KL and KB, across various cases. For each case, the predicted outputs \((y_{1},y_{2},y_{3})\) are compared against the actual values to evaluate the accuracy and robustness of the models. ## References

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