From comparison to integration: Enhancing forward osmosis performance prediction with mathematical and RBF neural network models

Reliable prediction of forward osmosis (FO) performance requires advanced models that can handle the complex, nonlinear interactions within operational conditions. This study developed, compared, and integrated mathematical and radial basis function neural network (RBFNN) models to predict the performance of pilot-scale plate-and-frame FO system. RBFNN demonstrates strong generalization capabilities for capturing nonlinear relationships, making it particularly effective in noisy experimental environments typical of FO applications. Both models demonstrated high accuracy within the experimental data ranges (R² > 0.97). The mathematical model provided consistent predictions and insights into internal module dynamics, while the RBFNN exhibited high computational efficiency. However, the RBFNN showed limitations in predicting recovery accuracy for operational ranges with insufficient data. To address this, we introduced a data distance index to assess the reliability of RBFNN predictions, particularly in extrapolation scenarios. We then integrated the approaches using the mathematical model for data imputation to expand the RBFNN's training dataset. The integrated model, retrained with augmented data, achieved an R² of 0.9920 and an RMSE of 0.3414 LMH for water flux prediction. This approach not only provides more reliable predictions but also enhances the understanding of key FO performance parameters through Shapley Additive exPlanations (SHAP) analysis. This synergistic method facilitates efficient FO system design and operation by optimizing process parameters under diverse conditions. The current study highlights the effectiveness of combining physics-based modeling with machine learning in membrane technology, improving the robustness of predictive tools for water treatment applications. Furthermore, the data distance index offers significant implications for evaluating prediction reliability in other processes with limited training data.