  A new data-driven method for operator learning of stochastic differential
equations(SDE) is proposed in this paper. The central goal is to solve forward
and inverse stochastic problems more effectively using limited data. Deep
operator network(DeepONet) has been proposed recently for operator learning.
Compared to other neural networks to learn functions, it aims at the problem of
learning nonlinear operators. However, it can be challenging by using the
original model to learn nonlinear operators for high-dimensional stochastic
problems. We propose a new multi-resolution autoencoder DeepONet model referred
to as MultiAuto-DeepONet to deal with this difficulty with the aid of
convolutional autoencoder. The encoder part of the network is designed to
reduce the dimensionality as well as discover the hidden features of
high-dimensional stochastic inputs. The decoder is designed to have a special
structure, i.e. in the form of DeepONet. The first DeepONet in decoder is
designed to reconstruct the input function involving randomness while the
second one is used to approximate the solution of desired equations. Those two
DeepONets has a common branch net and two independent trunk nets. This
architecture enables us to deal with multi-resolution inputs naturally. By
adding $L_1$ regularization to our network, we found the outputs from the
branch net and two trunk nets all have sparse structures. This reduces the
number of trainable parameters in the neural network thus making the model more
efficient. Finally, we conduct several numerical experiments to illustrate the
effectiveness of our proposed MultiAuto-DeepONet model with uncertainty
quantification.
