  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. We propose in
this paper, a Variational Bayes DeepONet (VB-DeepONet) for operator learning,
which can alleviate these limitations of DeepONet architecture to a great
extent and give user additional information regarding the associated
uncertainty at the prediction stage. The key idea behind neural networks set in
Bayesian framework is that, the weights and bias of the neural network are
treated as probability distributions instead of point estimates and, Bayesian
inference is used to update their prior distribution. Now, to manage the
computational cost associated with approximating the posterior distribution,
the proposed VB-DeepONet uses \textit{variational inference}. Unlike Markov
Chain Monte Carlo schemes, variational inference has the capacity to take into
account high dimensional posterior distributions while keeping the associated
computational cost low. Different examples covering mechanics problems like
diffusion reaction, gravity pendulum, advection diffusion have been shown to
illustrate the performance of the proposed VB-DeepONet and comparisons have
also been drawn against DeepONet set in deterministic framework.
