## Abstract

Removal of dyes from wastewater is a challenging task for scientists and environmentalists. This work has studied a mathematical model characterizing the typical staining process within sewage systems. Two widely used nanoparticles, *ZnO*, and \(TiO_{2}\), are used to remove dyes from wastewater. The BET (Brunauer, Emmett, and Teller) method determines the pore diameter d. The mathematical model of the phenomenon is modeled as a highly nonlinear partial differential equation (HNDE), detailed in a semi-infinite domain. In the present study, a hybridization of the Levenberg-Marquardt Backpropagation and Supervised Neural Network (LMB-SNN) is utilized to find the model’s surrogate solutions. The Runge-Kutta of the order four (RK4) technique is used to create reference solutions. We have analyzed our surrogate solution models by considering eight different scenarios. The stability and equilibrium of the mathematical model are checked by varying physical quantities like the ratio of final pressure to initial pressure. Our candidate solutions are divided into training, testing, and experimental categories to establish the reliability of our machine learning procedure. Comparative studies of statistical values based on mean squared error function (MSEF), effectiveness, regression plots, and failure histograms confirm the efficiency of the (LMB-SNN) scheme.

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## Data Availability Statement

This manuscript has associated data in a data repository. [Authors’ comment: All surrogate models are included in Figures 10, 11, 12, and 13. These models can be used for the reproduction of our results. No further data is associated with these models. For further queries, interested readers may contact the corresponding author for elaborations.]

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Kamal, M., Sulaiman, M. & Alshammari, F.S. Quantitative features analysis of a model for separation of dissolved substances from a fluid flow by using a hybrid heuristic.
*Eur. Phys. J. Plus* **137**, 1062 (2022). https://doi.org/10.1140/epjp/s13360-022-03226-0

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DOI: https://doi.org/10.1140/epjp/s13360-022-03226-0