Uncertainty Quantification of Stochastic Partial Differential Equations Based on Bayesian Neural Networks

Stochastic Partial Differential Equations (SPDEs) are extensively applied across a multitude of real-world contexts, particularly within the realms of engineering and finance. Given that the input parameters for SPDEs are inherently stochastic, ensuring reliable and robust predictions for the physical systems in question necessitates rigorous uncertainty quantification (UQ). Recently, a multitude of approaches utilizing neural networks to solve SPDEs have emerged, showcasing promising results. However, the majority of these methods operate under the presumption of precisely known parameters. In practical applications, information about parameters can only be obtained through a limited number of sensors. Therefore, we have combined the variational form of SPDEs with Bayesian Neural Networks (BNN) to quantify uncertainties in SPDEs. This approach circumvents the need for second-order derivative calculations, significantly enhancing the accuracy of UQ through the direct application of BNN. Specifically, we employ BNN to model the solution of the SPDE, integrating the governing equations into the likelihood function via automatic differentiation, and employing the Hamiltonian Monte Carlo (HMC) technique to draw samples from the posterior distribution. Numerical experiments indicate that our approach substantially enhances the accuracy of UQ for SPDEs.