To address the overly conservative predictions of empirical formulas and the lack of physical mechanism guidance in purely data-driven models for predicting the local scour depth around bridge piers, a multilayer perceptron (PSI-MLP) model based on coupled physics-informed and symbolic regression constraints was developed. In the PSI-MLP model, the HEC-18 equation and the Melville scour formula were introduced into the standard multilayer perceptron framework as physical constraints, explicit equations extracted from data by the symbolic regression algorithm were combined as additional constraints,and the loss function integrates data loss, physical loss, and symbolic regression. With extreme gradient boosting, support vector machine, and purely data-driven multilayer perceptron models as benchmark models, the performance differences between the PSI-MLP model and each benchmark model were comparatively analyzed from multiple dimensions of prediction accuracy, model interpretability, and physical consistency. Meanwhile, based on 56 groups of field-measured engineering data, the cross-scale extrapolation and generalization abilities of the PSI-MLP model in practical engineering scenarios were analyzed. The results show that the root mean square error, coefficient of determination, and Kling-Gupta efficiency coefficient of the PSI-MLP model on the test set are 0.234, 0.88, and 0.928, respectively. Compared with the benchmark models, the root mean square error decreases by up to 34.8%, and the coefficient of determination and Kling-Gupta efficiency coefficient increase by up to 16.4% and 25.9%, respectively. The decision logic of the PSI-MLP model is consistent with hydrodynamic theories. The relative error of the PSI-MLP model is less than 25% under 71.4% of the working conditions, which is superior to the traditional empirical formulas and benchmark models. The coupled constraint mechanism of physical information and symbolic regression can improve the adaptability and prediction stability of the model under complex engineering-scale extrapolation conditions.