PID4LaTe: a physics-informed deep learning model for lake multi-depth temperature prediction
Lake temperature plays a pivotal role in the physical and chemical processes in the water. It has a significant impact on the distribution of lake organisms. In lake temperature modelling, the physics-based models have the shortcomings of parameter calibration and generalization difficulties. The data-driven models are highly data dependent. This makes hybrid models an effective solution at present. In this paper, we explore the spatial and temporal co-evolution process for multi-depth lake temperature and propose a Physics-Informed Deep learning model for Lake multi-depth Temperature prediction, PID4LaTe. It consists of three sub-models, two of which are long short-term memory (LSTM) models for spatial and temporal prediction respectively, and the other is a physical model, the General Lake Model. The physical model offers simulation data based on its rich knowledge for data-driven model learning, while guaranteeing consistency between model results and physical mechanisms. We compared PID4LaTe with the Process-Based model (PB), the Deep Learning model (DL) and the Physics-Guided Recurrent Neural Networks model (PGRNN), and used Root Mean Square Error (RMSE), Mean Square Error (MSE), Mean Absolute Error (MAE) and Mean Absolute Percentage Error (MAPE) to assess the effectiveness of the models. Extensive experiments show the superiority of our hybrid model for lake multi-depth prediction over PB, DL and PGRNN with RMSE of 0.798, MSE of 0.644, MAE of 0.567 and MAPE of 4.367% in Mendota Lake and RMSE of 1.099, MSE of 1.261, MAE of 0.783 and MAPE of 7.936% in Sparkling Lake. The two-cases study indicates that the hybrid model combining the physics-based model with the data-driven model is a promising technique for multi-depth lake temperature predicting. This study provides a reference method for accurate prediction of temperature at multiple depths in lakes.
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