Operator learning with PCA-Net: upper and lower complexity bounds

• URL: http://arxiv.org/abs/2303.16317v5
• Date: Fri, 13 Oct 2023 22:28:39 GMT
• Title: Operator learning with PCA-Net: upper and lower complexity bounds
• Authors: Samuel Lanthaler
• Abstract summary: PCA-Net is a recently proposed neural operator architecture which combines principal component analysis with neural networks. A novel universal approximation result is derived, under minimal assumptions on the underlying operator. 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.
• Score: 2.375038919274297
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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