IB-UQ: Information bottleneck based uncertainty quantification for neural function regression and neural operator learning

In this paper, a novel framework is established for uncertainty quantification via information bottleneck (IB-UQ) for scientific machine learning tasks, including deep neural network (DNN) regression and neural operator learning (DeepONet). Specifically, we first employ the General Incompressible-Flow Networks (GIN) model to learn a "wide" distribution fromnoisy observation data. Then, following the information bottleneck objective, we learn a stochastic map from input to some latent representation that can be used to predict the output. A tractable variational bound on the IB objective is constructed with a normalizing flow reparameterization. Hence, we can optimize the objective using the stochastic gradient descent method. IB-UQ can provide both mean and variance in the label prediction by explicitly modeling the representation variables. Compared to most DNN regression methods and the deterministic DeepONet, the proposed model can be trained on noisy data and provide accurate predictions with reliable uncertainty estimates on unseen noisy data. We demonstrate the capability of the proposed IB-UQ framework via several representative examples, including discontinuous function regression, real-world dataset regression and learning nonlinear operators for diffusion-reaction partial differential equation.