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 a "curse of parametric complexity", an infinite-dimensional analogue of the well-known curse of dimensionality encountered in high-dimensional approximation problems. In addition to these lower bounds, upper complexity bounds are finally 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 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，其将主成分分析（PCA）与神经网络相结合以逼近无穷维函数空间之间的算子。本文针对该方法建立了逼近理论，改进了该方向的先前工作并进行了重要拓展：首先，在关于底层算子和数据生成分布的最小假设下，推导了新的通用逼近结果；随后，识别出PCA-Net高效算子学习的两个潜在障碍，并通过下复杂度界加以精确刻画：第一个障碍与输出分布的复杂度相关，由PCA特征值缓慢衰减衡量；第二个障碍涉及无穷维输入与输出空间之间算子空间的固有着复杂度，这严格量化地引出了"参数复杂度灾难"的论述——即高维逼近问题中熟知的维数灾难在无穷维情形下的对应物。除这些下界外，本文最终推导了上复杂度界。研究表明，合适的平滑性准则能确保PCA特征值的代数衰减，并进一步证明，对于来自达西渗流和纳维-斯托克斯方程的相关特定算子，PCA-Net可克服上述通用灾难。