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Found 8 papers, 1 papers with code
Discretization Error of Fourier Neural Operators
Operator learning is a variant of machine learning that is designed to approximate maps between function spaces from data.
Operator Learning: Algorithms and Analysis
This review article summarizes recent progress and the current state of our theoretical understanding of neural operators, focusing on an approximation theoretic point of view.
The Parametric Complexity of Operator Learning
The first contribution of this paper is to prove that for general classes of operators which are characterized only by their $C^r$- or Lipschitz-regularity, operator learning suffers from a ``curse of parametric complexity'', which is an infinite-dimensional analogue of the well-known curse of dimensionality encountered in high-dimensional approximation problems.
Error Bounds for Learning with Vector-Valued Random Features
This paper provides a comprehensive error analysis of learning with vector-valued random features (RF).
The Nonlocal Neural Operator: Universal Approximation
A popular variant of neural operators is the Fourier neural operator (FNO).
Operator learning with PCA-Net: upper and lower complexity bounds
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.
Nonlinear Reconstruction for Operator Learning of PDEs with Discontinuities
A large class of hyperbolic and advection-dominated PDEs can have solutions with discontinuities.
On the approximation of functions by tanh neural networks
We derive bounds on the error, in high-order Sobolev norms, incurred in the approximation of Sobolev-regular as well as analytic functions by neural networks with the hyperbolic tangent activation function.
