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-PINNs 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 with mean field Gaussian approximation 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-Loève (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.

我们提出了一种贝叶斯物理信息神经网络（B-PINN），以解决由偏微分方程（PDE）和噪声数据描述的正向和反向非线性问题。在这种贝叶斯框架中，贝叶斯神经网络（BNN）与PNN的PINN相结合是先验的，而哈密顿蒙特卡洛（HMC）或变分推论（VI）可以作为后验的估计。B-PINN利用物理定律和分散的噪声测量来提供预测和量化不确定性来自贝叶斯框架中嘈杂的数据。与PINN相比，B-PINN除了具有不确定性量化之外，还具有避免过度拟合的能力，因此在噪声较大的情况下也能获得更准确的预测。我们对两种不同的B-PINNs后验方法（即HMC或VI）以及用于量化深度神经网络不确定性的辍学方法进行系统比较。我们的实验表明，HMC比均值高斯近似的VI更适合B-PINNs的后验估计，而PINN中采用的辍学方法几乎无法提供具有合理不确定性的准确预测。最后，我们用截断的Karhunen-Loève（KL）扩展与HMC或深度归一化流量（DNF）模型组合作为后验估计器，以取代先前的BNN。