  The Deep Operator Networks~(DeepONet) is a fundamentally different class of
neural networks that we train to approximate nonlinear operators, including the
solution operator of parametric partial differential equations (PDE). DeepONets
have shown remarkable approximation and generalization capabilities even when
trained with relatively small datasets. However, the performance of DeepONets
deteriorates when the training data is polluted with noise, a scenario that
occurs very often in practice. To enable DeepONets training with noisy data, we
propose using the Bayesian framework of replica-exchange Langevin diffusion.
Such a framework uses two particles, one for exploring and another for
exploiting the loss function landscape of DeepONets. We show that the proposed
framework's exploration and exploitation capabilities enable (1) improved
training convergence for DeepONets in noisy scenarios and (2) attaching an
uncertainty estimate for the predicted solutions of parametric PDEs. In
addition, we show that replica-exchange Langeving Diffusion (remarkably) also
improves the DeepONet's mean prediction accuracy in noisy scenarios compared
with vanilla DeepONets trained with state-of-the-art gradient-based
optimization algorithms (e.g. Adam). To reduce the potentially high
computational cost of replica, in this work, we propose an accelerated training
framework for replica-exchange Langevin diffusion that exploits the neural
network architecture of DeepONets to reduce its computational cost up to 25%
without compromising the proposed framework's performance. Finally, we
illustrate the effectiveness of the proposed Bayesian framework using a series
of experiments on four parametric PDE problems.
