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.
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