  We propose a Bayesian physics-informed neural network (B-PINN) to solve both
forward and inverse nonlinear problems described by partial differential
equations (PDEs) and noisy data. In this Bayesian framework, the Bayesian
neural network (BNN) combined with a PINN for PDEs serves as the prior while
the Hamiltonian Monte Carlo (HMC) or the variational inference (VI) could serve
as an estimator of the posterior. B-PINNs make use of both physical laws and
scattered noisy measurements to provide predictions and quantify the aleatoric
uncertainty arising from the noisy data in the Bayesian framework. Compared
with PINNs, in addition to uncertainty quantification, B-PINNs obtain more
accurate predictions in scenarios with large noise due to their capability of
avoiding overfitting. We conduct a systematic comparison between the two
different approaches for the B-PINN posterior estimation (i.e., HMC or VI),
along with dropout used for quantifying uncertainty in deep neural networks.
Our experiments show that HMC is more suitable than VI for the B-PINNs
posterior estimation, while dropout employed in PINNs can hardly provide
accurate predictions with reasonable uncertainty. Finally, we replace the BNN
in the prior with a truncated Karhunen-Lo\`eve (KL) expansion combined with HMC
or a deep normalizing flow (DNF) model as posterior estimators. The KL is as
accurate as BNN and much faster but this framework cannot be easily extended to
high-dimensional problems unlike the BNN based framework.
