VB-DeepONet: A Bayesian operator learning framework for uncertainty quantification

Abstract

Neural network based data-driven operator learning schemes have shown tremendous potential in computational mechanics. DeepONet is one such neural network architecture which has gained widespread appreciation owing to its excellent prediction capabilities. Having said that, being set in a deterministic framework exposes DeepONet architecture to the risk of overfitting, poor generalization and in its unaltered form, it is incapable of quantifying the uncertainties associated with its predictions. To address these challenges, we propose a novel Bayesian operator learning framework referred to as the Variational Bayes DeepONet (VB-DeepONet). VB-DeepONet is rooted in Bayesian statistics and hence, (a) is less prone to overfitting as compared to its deterministic counterpart, (b) has better generalization, and (c) yields predictive uncertainty which is instrumental when decision making is concerned. VB-DeepONet exploits variational inference and hence has the capacity to take into account high dimensional posterior distributions while keeping the associated computational cost reasonable. Different examples covering mechanics problems like diffusion reaction, gravity pendulum, advection diffusion have been considered to illustrate the performance of the proposed VB-DeepONet and comparisons have been drawn against DeepONet set in deterministic framework, Proper Orthogonal Decomposition based Gaussian Process and DenseED. The results obtained illustrate the efficacy of the proposed approach in solving uncertainty quantification problems.