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Physics-informed deep learning have recently emerged as an effective tool forleveraging 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 automaticdifferentiation, while the latter learns the hidden physics from data.Generally, the noisy and limited observational data as well as theoverparameterization in neural networks (NNs) result in uncertainty inpredictions from deep learning models. In [1], a Bayesian framework based onthe {{Generative Adversarial Networks}} (GAN) has been proposed as a unifiedmodel 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 tolearn a functional prior either from a prescribed function distribution, e.g.,Gaussian process, or from historical data and available physics. At the secondstage, the Hamiltonian Monte Carlo (HMC) method is utilized to estimate theposterior in the latent space of GANs. However, the vanilla HMC does notsupport the mini-batch training, which limits its applications in problems withbig data. In the present work, we propose to use the normalizing flow (NF)models in the context of variational inference, which naturally enables theminibatch training, as the alternative to HMC for posterior estimation in thelatent space of GANs. A series of numerical experiments, including a nonlineardifferential equation problem and a 100-dimensional Darcy problem, areconducted to demonstrate that NF with full-/mini-batch training are able toachieve similar accuracy as the ``gold rule'' HMC.

Variational inference in neural functional prior using normalizing flows: Application to differential equation and operator learning problems