Differentiable belief rule-based reasoning with knowledge-guided residual learning for fault diagnosis

Abstract Fault diagnosis of industrial equipment is essential for ensuring system safety and reliable operation. However, practical operating conditions often involve uncertain information and interpretability requirements, which pose challenges to diagnostic models. The belief rule base (BRB) is suitable for industrial fault diagnosis because expert knowledge can be represented, and uncertainty can be handled. However, under complex conditions, existing BRB methods have two major limitations when collaborative optimization with deep learning is required to enhance the data-driven capability. First, the inference process is non-differentiable, so unified training with deep learning models is difficult. Second, parameter optimization relies on offline gradient-free methods, and adaptive fine-tuning around expert knowledge is lacking, which limits adaptability to complex conditions. To address these issues, a differentiable BRB inference and knowledge-guided residual learning method (D-BRB) is proposed. First, BRB inference is formulated as a numerically stable and backpropagation-compatible module, so a trainable rule inference process is achieved. Second, the expert parameters are treated as priors, and residual and interpretable updates are performed through soft reparameterization. Finally, input space calibration and sparse rule activation are introduced to improve rule matching and inference stability. Experimental results on industrial fault diagnosis tasks demonstrate that the proposed D-BRB achieves improved diagnostic accuracy while maintaining interpretability and generalization, showing strong potential for practical industrial applications.