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.

由于数据科学的最新进展，机器学习（ML）算法已被广泛应用于分析岩土工程问题。然而，使用有限数据训练的灵活机器学习模型可能会表现出意想不到的行为，导致可解释性低和物理不一致，从而降低机器学习模型在风险预测和工程应用中的可靠性和鲁棒性。由于岩土工程应用的输入特征通常表示遵循内在且通常单调关系的物理参数，因此将单调性纳入 ML 模型有助于确保模型输出的物理真实性。在这项研究中，单调性作为软约束被引入人工神经网络（ANN）模型中，并将其结果与几个基准机器学习模型进行了比较。在训练过程中，使用数据增强和逐点梯度来评估模型预测的单调性，并通过修改的损失函数最小化单调性违规。文献中边坡稳定性案例历史的汇编被用于模型开发，对其性能进行基准测试，并评估单调性约束的影响。所有模型性能评估均使用交叉验证程序，以减少样本选择中的偏差。结果表明，无约束的机器学习模型产生的预测违反了输入空间许多部分的单调性。然而，通过在 ANN 模型中添加单调性约束，可以有效减少单调性违规，同时保持相对较高的性能，从而提供更稳健和可解释的预测。使用边坡稳定性预测作为代理，本研究中开发的将单调性约束纳入机器学习模型的方法可以应用于许多岩土工程应用。所提出的方法增强了机器学习模型的可靠性和可解释性，从而为现实世界的应用带来更准确和一致的结果。