A problem of incorporating the expert rules into machine learning models for extending the concept-based learning is formulated in the paper. It is proposed how to combine logical rules and neural networks predicting the concept probabilities. The first idea behind the combination is to form constraints for a joint probability distribution over all combinations of concept values to satisfy the expert rules. The second idea is to represent a feasible set of probability distributions in the form of a convex polytope and to use its vertices or faces. We provide several approaches for solving the stated problem and for training neural networks which guarantee that the output probabilities of concepts would not violate the expert rules. The solution of the problem can be viewed as a way for combining the inductive and deductive learning. Expert rules are used in a broader sense when any logical function that connects concepts and class labels or just concepts with each other can be regarded as a rule. This feature significantly expands the class of the proposed results. Numerical examples illustrate the approaches. The code of proposed algorithms is publicly available.

翻译：本文阐述了将专家规则融入机器学习模型以扩展基于概念学习的问题。我们提出了结合逻辑规则与预测概念概率的神经网络的方法。这一结合的首要思路是：为所有概念取值组合的联合概率分布构建约束条件，使其满足专家规则。第二个思路是将可行概率分布集表示为凸多面体形式，并利用其顶点或面。我们提供了解决该问题的若干方法，以及训练神经网络的方案——这些方案能确保概念输出概率不违反专家规则。该问题的解决方案可被视为归纳学习与演绎学习的结合方式。专家规则在此具有更广泛的含义：任何连接概念与类别标签、或概念之间相互关联的逻辑函数均可视为规则，这一特性显著扩展了所提出结果的适用范围。数值算例展示了这些方法的有效性，所提算法的代码已公开发布。