Physics-informed feature engineering with fuzzy symbolic regression for predicting settling velocity in water treatment

Abstract

Predicting the settling velocity of fractal aggregates remains a challenge in water treatment, as classical models like Stokes' Law oversimplify the influence of non-sphericity, porosity, and complex morphology. Empirical and fractal-based models lack generalizability, while most machine learning models operate as black boxes, providing limited physical insight. This study proposes a Physics-Informed Machine Learning Fuzzy Symbolic Regression (PIML-SR) framework enhanced with fuzzy preprocessing to derive interpretable and physically consistent equations for settling velocity prediction. A dataset of Al-kaolinite flocs was obtained using high-speed imaging in a sedimentation column. Morphological parameters and physics-based descriptors, such as drag force and Reynolds number, were incorporated through fuzzy preprocessing, which converts normalized features into smooth membership functions to handle regime transitions and measurement uncertainty, combined with fuzzy symbolic regression. The PIML-SR model demonstrated excellent accuracy (R² > 0.99, MAE ≈ 0.015 μm/s) and robustness to up to 10 % Gaussian noise. In contrast, a baseline symbolic model (R² ≈ 0.56, MAE ≈ 556.6 μm/s) and a purely data-driven artificial neural network (R² ≈ 0.63, MAE ≈ 518.3 μm/s), both trained solely on morphological features, along with a Physics-Informed Neural Network (R² ≈ −1.93, MAE ≈ 1794.9 μm/s), all exhibited limited or poor accuracy, underscoring the critical importance of integrating physical knowledge, as achieved by the proposed fuzzy symbolic regression approach, for attaining high-fidelity, generalizable, and interpretable predictions. This represents the first application of a fuzzy-enhanced PIML-SR framework for sedimentation, providing an interpretable, physically grounded, and noise-resilient approach for optimizing sedimentation processes in water treatment.