Bayesian network is a frequently-used method for fault detection and
diagnosis in industrial processes. The basis of Bayesian network is structure
learning which learns a directed acyclic graph (DAG) from data. However, the
search space will scale super-exponentially with the increase of process
variables, which makes the data-driven structure learning a challenging
problem. To this end, the DAGs with NOTEARs methods are being well studied not
only for their conversion of the discrete optimization into continuous
optimization problem but also their compatibility with deep learning framework.
Nevertheless, there still remain challenges for NOTEAR-based methods: 1) the
infeasible solution results from the gradient descent-based optimization
paradigm; 2) the truncation operation to promise the learned graph acyclic. In
this work, the reason for challenge 1) is analyzed theoretically, and a novel
method named DAGs with Tears method is proposed based on mix-integer
programming to alleviate challenge 2). In addition, prior knowledge is able to
incorporate into the new proposed method, making structure learning more
practical and useful in industrial processes. Finally, a numerical example and
an industrial example are adopted as case studies to demonstrate the
superiority of the developed method.

提出了一种混合整数规划的新方法，名为 DAGs with Tears，用于在工业过程中更实用和有用地学习结构，并演示了该方法的优越性。