Grunnleggende konsepter
The discretization error that arises from approximating Fourier Neural Operators on a grid rather than on a continuum can be bounded theoretically and empirically shown to decrease at an algebraic rate as the grid resolution increases.
Sammendrag
The paper analyzes the discretization error that arises when Fourier Neural Operators (FNOs) are implemented on a grid rather than on a continuum. FNOs are a type of operator learning architecture that parameterize the model directly in function space, generalizing deep neural networks.
Key highlights:
- The authors derive a theoretical bound on the L2 norm of the discretization error, showing it decreases at a rate of N^-s as the grid resolution N increases, where s is the regularity of the input function.
- Numerical experiments validate the theoretical results, demonstrating the predicted error decay rates for FNOs with random weights as well as trained FNO models.
- The experiments highlight the importance of using smooth activation functions like GeLU that preserve regularity, in contrast to non-smooth functions like ReLU.
- The authors provide guidelines for mitigating discretization error in practice, such as using periodic positional encodings.
- An adaptive subsampling algorithm is proposed to speed up FNO training by adjusting the grid resolution during the optimization process.
Overall, the paper provides a comprehensive theoretical and empirical analysis of discretization error in Fourier Neural Operators, offering insights to improve the performance and efficiency of this important operator learning architecture.
Statistikk
The discretization error decreases at a rate of N^-s, where N is the grid resolution and s is the regularity of the input function.
Sitater
"The discretization error that results from performing a single convolution on a grid rather than on a continuum depends on the regularity, or smoothness, of the input function in the Sobolev sense."
"Because the smooth GeLU (Gaussian Error Linear Unit) activation preserve regularity, while the (non-differentiable) ReLU activations do not, the analysis in this paper is confined to the former and extends to other smooth activation functions."