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.

翻译：科学与工程中的许多物理过程天然由无限维函数空间之间的算子表示。在此背景下，算子学习问题旨在从经验数据中提取这些物理过程，但由于数据具有无限或高维特性，这一任务极具挑战性。模型降维作为应对该挑战的核心环节，可同时降低数据维度与问题规模。本文通过探究基于自编码器的神经网络（AENet），利用非线性低维结构实现模型降维。AENet首先学习输入数据的潜在变量，随后学习从这些潜在变量到对应输出数据的变换。数值实验验证了AENet能够准确学习非线性偏微分方程的解算子。此外，我们建立了分析AENet泛化误差的数学与统计估计理论。理论框架表明，训练AENet的样本复杂度与被建模过程的内蕴维度紧密相关，同时揭示了AENet对噪声的显著鲁棒性。