This paper addresses the need for deep learning models to integrate well-defined constraints into their outputs, driven by their application in surrogate models, learning with limited data and partial information, and scenarios requiring flexible model behavior to incorporate non-data sample information. We introduce Bayesian Entropy Neural Networks (BENN), a framework grounded in Maximum Entropy (MaxEnt) principles, designed to impose constraints on Bayesian Neural Network (BNN) predictions. BENN is capable of constraining not only the predicted values but also their derivatives and variances, ensuring a more robust and reliable model output. To achieve simultaneous uncertainty quantification and constraint satisfaction, we employ the method of multipliers approach. This allows for the concurrent estimation of neural network parameters and the Lagrangian multipliers associated with the constraints. Our experiments, spanning diverse applications such as beam deflection modeling and microstructure generation, demonstrate the effectiveness of BENN. The results highlight significant improvements over traditional BNNs and showcase competitive performance relative to contemporary constrained deep learning methods.

翻译：本文针对深度学习模型在代理模型构建、小样本与部分信息学习、以及需要融合非样本信息的灵活建模场景中，必须将明确定义的约束条件整合至模型输出的需求展开研究。我们提出了基于最大熵原理的贝叶斯熵神经网络框架，旨在对贝叶斯神经网络的预测结果施加约束。该框架不仅能够约束预测值本身，还可对其导数与方差进行约束，从而确保模型输出更具鲁棒性与可靠性。为实现不确定性量化与约束满足的同步处理，我们采用乘子法同步估计神经网络参数及约束对应的拉格朗日乘子。通过在梁挠度建模与微结构生成等多样化应用场景中的实验验证，本方法相较于传统贝叶斯神经网络展现出显著优势，并与当前主流约束深度学习方法相比具有竞争力。