The main contribution of this paper is to develop a hierarchical Bayesian formulation of PINNs for linear inverse problems, which is called BPINN-IP. The proposed methodology extends PINN to account for prior knowledge on the nature of the expected NN output, as well as its weights. Also, as we can have access to the posterior probability distributions, naturally uncertainties can be quantified. Also, variational inference and Monte Carlo dropout are employed to provide predictive means and variances for reconstructed images. Un example of applications to deconvolution and super-resolution is considered, details of the different steps of implementations are given, and some preliminary results are presented.

翻译：本文的主要贡献在于为线性逆问题提出了一种分层贝叶斯框架下的物理信息神经网络（PINN）方法，称为BPINN-IP。所提出的方法扩展了PINN，使其能够融入关于预期神经网络输出的先验知识及其权重的先验信息。此外，由于我们可以获得后验概率分布，因此能够自然地量化不确定性。同时，本文采用变分推断和蒙特卡洛丢弃法来为重建图像提供预测均值与方差。文中以去卷积和超分辨率作为应用示例，给出了不同实现步骤的详细说明，并展示了一些初步结果。