Bayesian Deep Learning Framework for Uncertainty Quantification in Stochastic Partial Differential Equations

.Bayesian physics-informed neural networks (B-PINNs) have emerged as an efficient tool for uncertainty quantification in partial differential equations (PDEs). However, their applicability has been limited to accounting for noisy data. They fail to effectively address the uncertainty arising from the randomness of physical parameters in stochastic PDEs (SPDEs). To this end, we propose a novel Bayesian deep learning framework designed for uncertainty quantification in SPDE problems. We model the solution of an SPDE using a Bayesian neural network (BNN), where the posterior distribution of the parameters is constructed by Bayes' rule. We incorporate the governing physical laws into the likelihood distribution through automatic differentiation and estimate the likelihood using a Gaussian mixture model. To effectively learn the posterior distribution of BNN parameters, we employ a Hamiltonian Monte Carlo method, an efficient method for high-dimensional sampling. We propose a parallel algorithm to mitigate the high computational cost associated with the Hamiltonian Monte Carlo, thereby enhancing its efficiency. We provide numerical examples for both forward and inverse SPDE problems to demonstrate the effectiveness of our proposed method for uncertainty quantification in SPDEs. Notably, it is demonstrated that the computational cost is almost unaffected by the number of BNN parameters and the random dimension of the problem, underscoring its computational efficiency.KeywordsBayesian neural networkHamiltonian Monte Carlostochastic partial differential equationuncertainty quantificationMSC codes60H3535R6062M45