  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.
