Bayesian Physics Informed Neural Networks (BPINN) have received considerable attention for inferring differential equations' system states and physical parameters according to noisy observations. However, in practice, Hamiltonian Monte Carlo (HMC) used to estimate the internal parameters of BPINN often encounters troubles, including poor performance and awful convergence for a given step size used to adjust the momentum of those parameters. To improve the efficacy of HMC convergence for the BPINN method and extend its application scope to multi-scale partial differential equations (PDE), we developed a robust multi-scale Bayesian PINN (dubbed MBPINN) method by integrating multi-scale deep neural networks (MscaleDNN) and Bayesian inference. In this newly proposed MBPINN method, we reframe HMC with Stochastic Gradient Descent (SGD) to ensure the most ``likely'' estimation is always provided, and we configure its solver as a Fourier feature mapping-induced MscaleDNN. The MBPINN method offers several key advantages: (1) it is more robust than HMC, (2) it incurs less computational cost than HMC, and (3) it is more flexible for complex problems. We demonstrate the applicability and performance of the proposed method through general Poisson and multi-scale elliptic problems in one- to three-dimensional spaces. Our findings indicate that the proposed method can avoid HMC failures and provide valid results. Additionally, our method can handle complex PDE and produce comparable results for general PDE. These findings suggest that our proposed approach has excellent potential for physics-informed machine learning for parameter estimation and solution recovery in the case of ill-posed problems.

本文研究了贝叶斯物理信息神经网络（BPINN）在处理带噪声数据时收敛性差的问题。提出了一种新的多尺度贝叶斯PINN方法（MBPINN），通过将随机梯度下降（SGD）与Hamiltonian Monte Carlo（HMC）结合，显著提高了收敛性和计算效率。实验表明，该方法在处理复杂偏微分方程时具有优越的性能和灵活性，显示出在物理信息机器学习中的广泛应用潜力。

提高贝叶斯物理信息神经网络在解决带噪声多尺度偏微分方程训练收敛性的研究