## 1 Introduction  
Hydrogen, a globally clean energy source, is applied in fuel cells. However, currently, the production of hydrogen is mainly conducted using fossil fuels, resulting in mixtures that contain impurities [1]. Various techniques are available for purifying and separating hydrogen [2, 3, 4, 5]. PSA represents a breakthrough technology in the field of gas separation and adsorption, capable of performing adsorption and desorption processes through variations in pressure. The traditional methods and simplified process make it highly suitable for gas separation and purification in industry environments and laboratories [6, 7, 8]. [PERSON] al. conducted a comparative analysis of hydrogen separation using PSA and palladium membrane technologies in a polygeneration system for fuel-rich HCCI engines. The results indicated that PSA serves as an excellent method for purifying hydrogen [9]. Achieving optimal purification efficiency in the PSA process is complex, including the steps of selecting adsorbents [10; 11], designing adsorption bed structures [12; 13], and determining the number of cyclic operation steps [14]. As PSA technology continues to mature and achieve higher accuracy, coupled with the growing demand for efficient PSA separations, developing effective optimization methods has become crucial. Various mathematical models and optimization theories have been employed in an attempt to optimize the PSA process. [PERSON] and colleagues introduced a novel optimization method for PSA units featuring an optimal Pareto front, effectively balancing the need to maximize productivity while minimizing energy consumption [15]. [PERSON] and colleagues developed a unique steam-purge PSA cycle for pre-combustion CO\({}_{2}\) capture, using genetic algorithm (GA) optimization to optimize energy consumption while maximizing productivity [16]. [PERSON] and co-authors formulated a graphical approach for optimizing both purification recovery and feed purity, demonstrating that the method can precisely and conveniently optimize both parameters simultaneously [17]. Several studies have explored the optimization and prediction of hydrogen production through the application of response surface methodology (RSM). [PERSON] and colleagues formulated a regression model utilizing RSM and achieved precise predictions for the optimization target [18]. In our preliminary research work, we introduced the Box-Behnken design (BBD) method, which is grounded in RSM, to optimize the PSA model. This approach allowed us to obtain quadratic regression models for the purity, recovery, and productivity of hydrogen, thereby significantly reducing computation time [19]. However, due to complex nonlinear scenarios, using only an RSM quadratic polynomial in the BBD sometimes cannot achieve the required prediction accuracy [20]. In contrast to RSM, artificial neural networks (ANNs) function as mathematical models that mimic the functioning of human neurons, exhibiting excellent prediction capabilities for nonlinear and unrelated data. As machine learning is widely applied in engineering and science to solve complex problems, the use of ANNs can significantly cut down the computational time and empower the execution of more intricate optimization tasks. To investigate the efficacy of ANN models, [PERSON] and colleagues conducted a comparison between Bayesian regularization and ANN algorithms. The findings reveal that ANN models are particularly adept at validating and comparing with process models, demonstrating their precision and potential in optimizing biomass gasification processes, with an impressive R\({}_{2}\) value of 0.99 [21]. [PERSON] and colleagues designed an ANN model aimed at optimizing the efficiency of the PSA process for generating high-purity hydrogen. Their findings suggest that the ANN could accurately simulate dynamic PSA behavior and performance [22]. [PERSON] and colleagues used an ANN as an alternative model for optimizing VPSA, achieving the integration of H\({}_{2}\) purification and CO\({}_{2}\) capture.