Physics-informed temporal convolutional network with auxiliary process-state trajectories for effluent quality prediction

Accurate forecasting of effluent chemical oxygen demand (COD) and total nitrogen (TN) is essential for operational optimization, energy management, and carbon-footprint assessment in wastewater treatment plants. Conventional mechanistic models typically involve numerous parameters and require extensive calibration, whereas purely data-driven models often lack physical consistency and process interpretability. To address these limitations, this study proposes a physics-informed temporal convolutional network (PI-TCN). Unlike existing approaches that approximate unmeasured mechanistic states using empirical surrogates or a limited number of reaction rates, PI-TCN introduces an auxiliary process-state trajectory branch alongside the primary prediction branch to represent the temporal evolution of latent states associated with organic matter degradation, nitrification, denitrification, and organic nitrogen transformation. Activated Sludge Model (ASM)-inspired kinetics are further incorporated to formulate mass-balance, kinetic-residual, and weak-supervision constraints. Results obtained using Benchmark Simulation Model No 2 (BSM2) data show that PI-TCN achieved R² values of 0.9031 and 0.9854 for COD and TN prediction, respectively, while reducing the error metrics by 14.31-23.06% relative to a purely data-driven TCN. The proposed framework integrates effluent-quality prediction with ASM-inspired soft constraints, while the auxiliary trajectories represent non-identifiable, physics-constrained latent process states rather than direct reconstructions of ASM states.