Bayesian Physics Informed Neural Networks (B-PINNs) have gained significant attention for inferring physical parameters and learning the forward solutions for problems based on partial differential equations. However, the overparameterized nature of neural networks poses a computational challenge for high-dimensional posterior inference. Existing inference approaches, such as particle-based or variance inference methods, are either computationally expensive for high-dimensional posterior inference or provide unsatisfactory uncertainty estimates. In this paper, we present a new efficient inference algorithm for B-PINNs that uses Ensemble Kalman Inversion (EKI) for high-dimensional inference tasks. We find that our proposed method can achieve inference results with informative uncertainty estimates comparable to Hamiltonian Monte Carlo (HMC)-based B-PINNs with a much reduced computational cost. These findings suggest that our proposed approach has great potential for uncertainty quantification in physics-informed machine learning for practical applications.

翻译：贝叶斯物理信息神经网络（B-PINNs）在基于偏微分方程的问题中，因推断物理参数与学习正向解的能力而受到广泛关注。然而，神经网络的过参数化特性对高维后验推断构成了计算挑战。现有的推断方法（如基于粒子或变分推断的方法）要么在高维后验推断中计算代价高昂，要么无法提供令人满意的不确定性估计。本文提出了一种针对B-PINNs的高效推断算法，采用集成卡尔曼反演（EKI）处理高维推断任务。我们发现，所提方法能以大幅降低的计算成本，获得与基于哈密顿蒙特卡洛（HMC）的B-PINNs相当且包含信息量不确定性估计的推断结果。这些结果表明，本文方法在物理信息机器学习的实际应用中具有不确定性量化的巨大潜力。