Empirical-Constraint Physics-Informed Neural Network for Predicting the Height of Water-Conducting Fracture Zone

Accurate prediction of water-conducting fracture zone (WCFZ) height is essential for preventing mine water hazards. Existing empirical, numerical, and artificial intelligence methods are limited by applicability, parameter sensitivity, or insufficient incorporation of engineering prior knowledge. This study develops a physics-informed neural network (PINN) using 107 measured datasets from North China-type coalfields. Overburden lithology, mining height, mining depth, and working face length are used as inputs. Empirical formulas from China’s “Three-Under” coal mining regulations are converted into interval constraints and incorporated into the loss function, while Bayesian optimization determines key hyperparameters. The model converges stably and achieves coefficients of determination above 0.93 for both training and testing sets, with root mean square error values of 5.717 m and 4.330 m for the training and testing sets, respectively, and mean absolute error values below 4 m. It outperforms empirical formulas, backpropagation neural networks, and deep neural networks without the empirical-constraint term. For the 11201 working face in a Shanxi mine, the predicted WCFZ height is 85.51 m, with a relative error of 1.71%. These results support the use of soft empirical constraints to improve prediction accuracy and engineering reliability, providing technical support for WCFZ height prediction and mine water hazard prevention in North China-type coalfields.