Machine learning with monotonic constraint for geotechnical engineering applications: an example of slope stability prediction

Machine learning (ML) algorithms have been widely applied to analyze geotechnical engineering problems due to recent advances in data science. However, flexible ML models trained with limited data can exhibit unexpected behaviors, leading to low interpretability and physical inconsistency, thus, reducing the reliability and robustness of ML models for risk forecasting and engineering applications. As input features for geotechnical engineering applications often represent physical parameters following intrinsic and often monotonic relationships, incorporating monotonicity into ML models can help ensure the physical realism of model outputs. In this study, monotonicity was introduced as a soft constraint into artificial neural network (ANN) models, and their results were compared with several benchmark ML models. During the training process, data augmentation and point-wise gradient were used to evaluate the monotonicity of model predictions, and monotonicity violations were minimized through a modified loss function. A compilation of slope stability case histories from the literature was used for model development, benchmarking their performance, and evaluating the effects of monotonicity constraints. Cross-validation procedures were used for all model performance evaluations to reduce bias in sample selections. Results showed that unconstrained ML models produced predictions that violate monotonicity in many parts of the input space. However, by adding monotonicity constraints into ANN models, monotonicity violations were effectively reduced while maintaining relatively high performance, thus providing a more robust and interpretable prediction. Using slope stability prediction as a proxy, the methods developed in this study to incorporate monotonicity constraints into ML models can be applied to many geotechnical engineering applications. The proposed approach enhances the reliability and interpretability of ML models, resulting in more accurate and consistent outcomes for real-world applications.