PCA-Net is a recently proposed neural operator architecture which combines principal component analysis (PCA) with neural networks to approximate operators between infinite-dimensional function spaces. The present work develops approximation theory for this approach, improving and significantly extending previous work in this direction: First, a novel universal approximation result is derived, under minimal assumptions on the underlying operator and the data-generating distribution. Then, two potential obstacles to efficient operator learning with PCA-Net are identified, and made precise through 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 to 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. A suitable smoothness criterion is shown to ensure an algebraic decay of the PCA eigenvalues. Furthermore, it is shown that PCA-Net can overcome the general curse of dimensionality for specific operators of interest, arising from the Darcy flow and the Navier-Stokes equations.

翻译：PCA-Net是一种近期提出的神经算子架构，它结合主成分分析（PCA）与神经网络来逼近无限维函数空间之间的算子。本文针对该方法建立了逼近理论，改进了并显著扩展了既有研究方向：首先，在关于底层算子和数据生成分布的最小假设下，推导出新的通用逼近结果。随后，识别了两种潜在阻碍PCA-Net高效算子学习的障碍，并通过复杂度下界精确表征：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；另一个障碍与无限维输入输出空间之间的算子空间固有复杂性相关，由此得出维数灾难的严格可量化表述。除这些下界外，本文还推导了复杂度上界。结果表明，适当的平滑性条件可保证PCA特征值的代数衰减。进一步地，针对达西流动和纳维-斯托克斯方程所导出的特定算子，证明PCA-Net能够克服一般性的维数灾难。