Diffusion analysis with high and low concentration regions by the finite difference method, the adaptive network-based fuzzy inference system, and the bilayered neural network method

Abstract

The diffusion of molecules in aqueous solutions in the domain of membrane technology is critical in the efficiency of chemical engineering and purification processes. In this study, the diffusion in high and low concentration regions is simulated with finite difference method (FDM), and then the results of numerical computations are coupled with adaptive network-based fuzzy inference system (ANFIS) and bilayered neural network method (BNNM). Machine learning (ML) approach can individually predict diffusion phenomena across the domain based on understanding of the machine instead of the discretization of an ordinary differential equation (ODE). The findings of the ML model confirm the FDM's simulation results. In addition to numerical computation, the error of the system is computed for different iterations. The results show that by increasing the number of iterations and training datasets, all errors reduce significantly for both training and testing. BNN method is also used to train the prediction process of diffusion for a more accurate comparison. This technique is similar to ANFIS method in terms of prediction capability. According to the findings, ANFIS approach predicts diffusion slightly better than BNN method. In this regard, ANFIS technique produces R > 0.99 while BNN method produces R around 0.98. Both ML methods are accurate enough to predict diffusion throughout the domain for different time steps. The computational time for both algorithms is less than that of FDM method to predict low and high concentrations in the domain.