Controlling Statistical, Discretization, and Truncation Errors in Learning Fourier Linear Operators
This work addresses foundational errors in operator learning, which is incremental as it builds on existing Fourier Neural Operator architectures.
The paper tackles the problem of learning Fourier linear operators by identifying and analyzing three main errors—statistical, truncation, and discretization—in the learning process, establishing both upper and lower bounds for these errors using a Discrete Fourier Transform-based least squares estimator.
We study learning-theoretic foundations of operator learning, using the linear layer of the Fourier Neural Operator architecture as a model problem. First, we identify three main errors that occur during the learning process: statistical error due to finite sample size, truncation error from finite rank approximation of the operator, and discretization error from handling functional data on a finite grid of domain points. Finally, we analyze a Discrete Fourier Transform (DFT) based least squares estimator, establishing both upper and lower bounds on the aforementioned errors.