  Many physical processes in science and engineering are naturally represented
by operators between infinite-dimensional function spaces. The problem of
operator learning, in this context, seeks to extract these physical processes
from empirical data, which is challenging due to the infinite or high
dimensionality of data. An integral component in addressing this challenge is
model reduction, which reduces both the data dimensionality and problem size.
In this paper, we utilize low-dimensional nonlinear structures in model
reduction by investigating Auto-Encoder-based Neural Network (AENet). AENet
first learns the latent variables of the input data and then learns the
transformation from these latent variables to corresponding output data. Our
numerical experiments validate the ability of AENet to accurately learn the
solution operator of nonlinear partial differential equations. Furthermore, we
establish a mathematical and statistical estimation theory that analyzes the
generalization error of AENet. Our theoretical framework shows that the sample
complexity of training AENet is intricately tied to the intrinsic dimension of
the modeled process, while also demonstrating the remarkable resilience of
AENet to noise.
