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A new data-driven method for operator learning of stochastic differentialequations(SDE) is proposed in this paper. The central goal is to solve forwardand inverse stochastic problems more effectively using limited data. Deepoperator network(DeepONet) has been proposed recently for operator learning.Compared to other neural networks to learn functions, it aims at the problem oflearning nonlinear operators. However, it can be challenging by using theoriginal model to learn nonlinear operators for high-dimensional stochasticproblems. We propose a new multi-resolution autoencoder DeepONet model referredto as MultiAuto-DeepONet to deal with this difficulty with the aid ofconvolutional autoencoder. The encoder part of the network is designed toreduce the dimensionality as well as discover the hidden features ofhigh-dimensional stochastic inputs. The decoder is designed to have a specialstructure, i.e. in the form of DeepONet. The first DeepONet in decoder isdesigned to reconstruct the input function involving randomness while thesecond one is used to approximate the solution of desired equations. Those twoDeepONets has a common branch net and two independent trunk nets. Thisarchitecture enables us to deal with multi-resolution inputs naturally. Byadding $L_1$ regularization to our network, we found the outputs from thebranch net and two trunk nets all have sparse structures. This reduces thenumber of trainable parameters in the neural network thus making the model moreefficient. Finally, we conduct several numerical experiments to illustrate theeffectiveness of our proposed MultiAuto-DeepONet model with uncertaintyquantification.

MultiAuto-DeepONet: A Multi-resolution Autoencoder DeepONet for  Nonlinear Dimension Reduction, Uncertainty Quantification and Operator  Learning of Forward and Inverse Stochastic Problems