Neural operators are gaining attention in computational science and engineering. PCA-Net is a recently proposed neural operator architecture which combines principal component analysis (PCA) with neural networks to approximate an underlying operator. The present work develops approximation theory for this approach, improving and significantly extending previous work in this direction. In terms of qualitative bounds, this paper derives a novel universal approximation result, under minimal assumptions on the underlying operator and the data-generating distribution. In terms of quantitative bounds, two potential obstacles to efficient operator learning with PCA-Net are identified, and made rigorous through the derivation of lower complexity bounds; the first relates to the complexity of the output distribution, measured by a slow decay of the PCA eigenvalues. The other obstacle relates the inherent complexity of the space of operators between infinite-dimensional input and output spaces, resulting in a rigorous and quantifiable statement of the curse of dimensionality. In addition to these lower bounds, upper complexity bounds are derived; first, a suitable smoothness criterion is shown to ensure a algebraic decay of the PCA eigenvalues. Then, it is shown that PCA-Net can overcome the general curse of dimensionality for specific operators of interest, arising from the Darcy flow and Navier-Stokes equations.

翻译：神经算子在计算科学与工程领域正获得越来越多的关注。PCA-Net是近期提出的一种神经算子架构，它结合主成分分析（PCA）与神经网络来近似底层算子。本文针对该方法发展了逼近理论，改进了该方向上的已有工作并进行了显著扩展。在定性界方面，本文在算子与数据生成分布的最小假设下，推导了一个全新的通用逼近结果。在定量界方面，我们识别了两个可能阻碍PCA-Net高效算子学习的问题，并通过下界复杂度推导使其严格化：第一个障碍与输出分布的复杂性相关，具体表现为PCA特征值的缓慢衰减；另一个障碍涉及无限维输入与输出空间之间算子空间的内在复杂性，从而形成了严格且可量化的维度灾难表述。除了这些下界外，本文还推导了上界复杂度：首先，证明一个合适的平滑性准则可确保PCA特征值呈代数衰减。随后，本文证明PCA-Net能够克服由达西流和纳维-斯托克斯方程导出的特定算子的通用维度灾难。