  In this study, a novel physics-data-driven Bayesian method named Heat
Conduction Equation assisted Bayesian Neural Network (HCE-BNN) is proposed. The
HCE-BNN is constructed based on the Bayesian neural network, it is a
physics-informed machine learning strategy. Compared with the existed pure data
driven method, to acquire physical consistency and better performance of the
data-driven model, the heat conduction equation is embedded into the loss
function of the HCE-BNN as a regularization term. Hence, the proposed method
can build a more reliable model by physical constraints with less data. The
HCE-BNN can handle the forward and inverse problems consistently, that is, to
infer unknown responses from known partial responses, or to identify boundary
conditions or material parameters from known responses. Compared with the exact
results, the test results demonstrate that the proposed method can be applied
to both heat conduction forward and inverse problems successfully. In addition,
the proposed method can be implemented with the noisy data and gives the
corresponding uncertainty quantification for the solutions.
