A physically constrained deep learning framework for total nitrogen prediction: A case study of a reservoir in North China.

Accurate forecasting of total nitrogen (TN), a key driver of eutrophication, is critical for effective water environment management. However, due to the dynamic nature of hydrological and meteorological conditions, water quality data commonly exhibit non-stationarity and distribution shift, leading to significantly reduced generalization performance of existing data-driven models during practical deployment. To address this challenge, this paper proposes a novel physically constrained deep learning framework that explicitly incorporates biogeochemical knowledge of the nitrogen cycle into model design to enhance robustness. The framework synergistically optimizes three key components: (1) Feature Engineering: A dual-criterion selection mechanism based on correlation and stability is developed to identify robust predictors of TN; (2) Model Structure: Reversible Instance Normalization (RevIN) is incorporated to mitigate distribution shift between training and testing phases; (3) Loss Function: A physically constrained joint loss function is proposed. By adaptively identifying the dominant biogeochemical driver among three candidate environmental proxies, it applies a soft directional penalty during training, thereby encouraging physical consistency while preserving point-wise accuracy. Extensive experiments using real-world datasets from a reservoir in North China demonstrate the superior performance of our framework (RMSE = 0.1172 mg/L, R2 = 0.9269). The proposed framework significantly outperforms the state-of-the-art DLinear, reducing prediction error RMSE by approximately 30.7%. Ablation experiments further quantify the contribution of each component, confirming the pivotal role of RevIN in handling non-stationarity. This study provides a systematic solution to address distribution shift in water quality time-series forecasting.