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

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.