Frequency-enhanced AutoEncoding decomposed transformer (FEADformer) for total phosphorus prediction in industrial...

Accurate total phosphorus (TP) control is critical for the stable operation of industrial recirculating cooling water (RCW) systems. However, industrial TP series inherently suffer from high missing rates, low sampling frequencies, and strong non-stationary fluctuations. Existing forecasting models experience severe accuracy decline under these harsh constraints. This study aims to develop a robust prediction framework that overcomes long-sequence dependency loss and extreme data missingness to support intelligent dosing control. We propose the Frequency-Enhanced AutoEncoding Decomposed Transformer (FEADformer). It captures multi-scale periodicity with linear complexity via a wavelet-based frequency-enhanced attention mechanism and mitigates the impact of incomplete data using a dedicated Autoencoding Decomposition Module (AD). Across five independent runs on the 2015-2024 industrial dataset, FEADformer achieved competitive predictive performance across the evaluated univariate and multivariate settings. At the 7-day horizon, it delivered 51.0% and 46.7% average Mean Squared Error (MSE) reductions against the top-five competitive baselines in univariate and multivariate forecasting, respectively, outperforming recent models like TimesNet. Evaluation on five public benchmarks further supported the architectural applicability of FEADformer across diverse time-series domains; however, direct cross-facility transfer of the trained model among industrial water-treatment systems remains to be validated. A six-month prospective forecasting verification within the same RCW facility assessed temporal and seasonal stability, while a separate two-month operational control trial increased the TP compliance rate from 70% to 83% and reduced chemical consumption by 10%. These results support the potential applicability of FEADformer to industrial TP forecasting within the evaluated facility and operating conditions.