Bayesian deep learning framework for uncertainty quantification in high dimensions
This provides a scalable uncertainty quantification method for researchers and practitioners working with high-dimensional stochastic systems, though it appears incremental as it combines existing Bayesian and deep learning techniques.
The authors tackled uncertainty quantification in high-dimensional stochastic partial differential equations by developing a Bayesian deep learning framework using Bayesian neural networks and Hamiltonian Monte Carlo. The method demonstrated effectiveness in numerical examples with computational cost nearly independent of dimension, addressing the curse of dimensionality.
We develop a novel deep learning method for uncertainty quantification in stochastic partial differential equations based on Bayesian neural network (BNN) and Hamiltonian Monte Carlo (HMC). A BNN efficiently learns the posterior distribution of the parameters in deep neural networks by performing Bayesian inference on the network parameters. The posterior distribution is efficiently sampled using HMC to quantify uncertainties in the system. Several numerical examples are shown for both forward and inverse problems in high dimension to demonstrate the effectiveness of the proposed method for uncertainty quantification. These also show promising results that the computational cost is almost independent of the dimension of the problem demonstrating the potential of the method for tackling the so-called curse of dimensionality.