  We present the application of the physics-informed neural network (PINN)
approach in Bayesian formulation. We have adopted the Bayesian neural network
framework to obtain posterior densities from Laplace approximation. For each
model or fit, the evidence is computed, which is a measure that classifies the
hypothesis. The optimal solution is the one with the highest value of evidence.
We have proposed a modification of the Bayesian algorithm to obtain
hyperparameters of the model. We have shown that within the Bayesian framework,
one can obtain the relative weights between the boundary and equation
contributions to the total loss. Presented method leads to predictions
comparable to those obtained by sampling from the posterior distribution within
the Hybrid Monte Carlo algorithm (HMC). We have solved heat, wave, and Burger's
equations, and the results obtained are in agreement with the exact solutions,
demonstrating the effectiveness of our approach. In Burger's equation problem,
we have demonstrated that the framework can combine information from
differential equations and potential measurements. All solutions are provided
with uncertainties (induced by the model's parameter dependence) computed
within the Bayesian framework.
