Physics-informed deep learning have recently emerged as an effective tool for leveraging both observational data and available physical laws. Physics-informed neural networks (PINNs) and deep operator networks (DeepONets) are two such models. The former encodes the physical laws via the automatic differentiation, while the latter learns the hidden physics from data. Generally, the noisy and limited observational data as well as the overparameterization in neural networks (NNs) result in uncertainty in predictions from deep learning models. In [1], a Bayesian framework based on the {{Generative Adversarial Networks}} (GAN) has been proposed as a unified model to quantify uncertainties in predictions of PINNs as well as DeepONets. Specifically, the proposed approach in [1] has two stages: (1) prior learning, and (2) posterior estimation. At the first stage, the GANs are employed to learn a functional prior either from a prescribed function distribution, e.g., Gaussian process, or from historical data and available physics. At the second stage, the Hamiltonian Monte Carlo (HMC) method is utilized to estimate the posterior in the latent space of GANs. However, the vanilla HMC does not support the mini-batch training, which limits its applications in problems with big data. In the present work, we propose to use the normalizing flow (NF) models in the context of variational inference, which naturally enables the minibatch training, as the alternative to HMC for posterior estimation in the latent space of GANs. A series of numerical experiments, including a nonlinear differential equation problem and a 100-dimensional Darcy problem, are conducted to demonstrate that NF with full-/mini-batch training are able to achieve similar accuracy as the ``gold rule'' HMC.

翻译：物理知识驱动的深度学习最近已成为利用观测数据和已知物理定律的有效工具。物理信息神经网络（PINNs）和深度算子网络（DeepONets）是其中的两类典型模型：前者通过自动微分编码物理定律，后者从数据中学习隐藏的物理规律。通常，含噪且有限的观测数据以及神经网络的过参数化会导致深度学习模型预测存在不确定性。文献[1]提出了一种基于生成对抗网络（GAN）的贝叶斯框架，作为量化PINNs和DeepONets预测不确定性的统一模型。具体而言，该框架包含两个阶段：（1）先验学习；（2）后验估计。在第一阶段，利用GAN从预设函数分布（如高斯过程）、历史数据或可用物理知识中学习函数先验。在第二阶段，采用哈密顿蒙特卡洛（HMC）方法在GAN潜空间中估计后验分布。然而，原始HMC方法不支持小批量训练，这限制了其在大数据问题中的应用。本研究提出在变分推断框架中使用归一化流（NF）模型作为替代方案——该方法天然支持小批量训练——用于GAN潜空间的后验估计。通过包含非线性微分方程问题和100维达西问题的系列数值实验证明，采用全批量/小批量训练的NF模型能够达到与"黄金标准"HMC方法相当的精度。