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We propose a Bayesian physics-informed neural network (B-PINN) to solve bothforward and inverse nonlinear problems described by partial differentialequations (PDEs) and noisy data. In this Bayesian framework, the Bayesianneural network (BNN) combined with a PINN for PDEs serves as the prior whilethe Hamiltonian Monte Carlo (HMC) or the variational inference (VI) could serveas an estimator of the posterior. B-PINNs make use of both physical laws andscattered noisy measurements to provide predictions and quantify the aleatoricuncertainty arising from the noisy data in the Bayesian framework. Comparedwith PINNs, in addition to uncertainty quantification, B-PINNs obtain moreaccurate predictions in scenarios with large noise due to their capability ofavoiding overfitting. We conduct a systematic comparison between the twodifferent 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-PINNsposterior estimation, while dropout employed in PINNs can hardly provideaccurate predictions with reasonable uncertainty. Finally, we replace the BNNin the prior with a truncated Karhunen-Lo\`eve (KL) expansion combined with HMCor a deep normalizing flow (DNF) model as posterior estimators. The KL is asaccurate as BNN and much faster but this framework cannot be easily extended tohigh-dimensional problems unlike the BNN based framework.

B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data