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）与神经网络来逼近无限维函数空间之间的算子。本文发展了该方法的逼近理论，改进并显著扩展了该方向的先前工作：首先，在算子和数据生成分布的最小假设条件下，推导了新颖的普适逼近结果。随后，识别了使用PCA-Net进行高效算子学习的两个潜在障碍，并通过下复杂度界精确刻画：第一个障碍与输出分布的复杂度相关，表现为PCA特征值的缓慢衰减；第二个障碍涉及无限维输入与输出空间之间算子空间的固有复杂度，由此得出了严格且可量化的维度灾难表述。除这些下界外，还推导了上复杂度界。证明了合适的平滑性准则可确保PCA特征值的代数衰减。此外，研究表明，对于源自达西流和纳维-斯托克斯方程的特定算子，PCA-Net能够克服一般性的维度灾难。