Advanced physics-enhanced machine learning and empirical approaches for estimating the oxygen transfer alpha factor in activated sludge.

Accurate estimation of the α0-factor, representing oxygen transfer efficiency under non-steady-state conditions, is critical for energy-efficient aeration in wastewater treatment. This study presents four novel empirical equations, derived from three years of full-scale data across four WWTPs. The proposed formulations include both full-parameter (13 variables) and reduced-parameter (5 variables) models, incorporating linear and nonlinear structures to provide interpretable, first-principles-informed tools for engineers, especially in resource-limited settings. To further improve predictive capability, the empirical equations were integrated as physics-informed structural priors within machine learning frameworks. Among the tested models, the Physics-Enhanced Extreme Gradient Boosting Tree model paired with a nonlinear empirical equation (i.e., PhyEXGBT-10 model) achieved the best performance, reducing test root mean square error (RMSE) by 77-87% relative to the corresponding standalone empirical equation. Across 19 tank zones, PhyEXGBT-10 consistently maintained RMSE ≤ 0.029 and relative error (RE) ≤ 3.4%, with lowest errors (RMSE ≤ 0.015, RE ≤ 1.7%) in stable aerobic zones. Slightly lower accuracy was observed near redox transition zones (RE = 5.0%), where abrupt aeration changes challenge current model assumptions. Notably, the framework remained robust under limited-data conditions, achieving RE = 3.1% using only 13 samples, highlighting the stabilizing effect of physics-guided regularization. Overall, this hybrid modeling framework effectively bridges mechanistic understanding and data-driven learning, providing a practical and scalable solution for real-time aeration control, energy optimization, and system design, while ensuring accessibility through standalone empirical equations for preliminary engineering use.