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，它通过结合主成分分析（PCA）与神经网络来逼近无限维函数空间之间的算子。本研究为该方法的逼近理论提供了新的发展，改进了并显著拓展了这一方向的前期工作：首先，在仅对底层算子与数据生成分布施加最小假设的条件下，推导出了一种新颖的通用逼近结果。随后，识别出两种可能阻碍基于PCA-Net的有效算子学习的障碍，并通过下界复杂性分析精确刻画了这些障碍：其一与输出分布的复杂性有关，该复杂性通过PCA特征值的缓慢衰减进行度量；另一个则涉及无限维输入-输出空间之间算子空间的固有复杂性，从而严谨且量化地阐述了维度灾难现象。除下界结果外，本文还推导了上界复杂性。研究表明，适当的光滑性条件可保证PCA特征值呈现代数衰减。进一步，本文证明PCA-Net能够克服由Darcy流与Navier-Stokes方程所导出的特定算子中普遍存在的维度灾难。