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

Generalization Error Guaranteed Auto-Encoder-Based Nonlinear Model Reduction for Operator Learning