Efficient Approximation for Encoder--Decoder Neural Operators via Variation Spaces
Provides theoretical guarantees for efficient neural operator learning, going beyond general Lipschitz or Fréchet differentiable operator classes.
This paper introduces a variation space for nonlinear operators and proves approximation bounds for encoder-decoder two-layer networks, achieving an error of order N^{-1/2} with constants independent of encoding dimensions, yielding algebraic learning rates.
We study operator learning using encoder--decoder neural networks. Inspired by the function-space theory of neural networks, we introduce a variation space as an infinite-dimensional structural class for nonlinear operators. This space is defined through vector-valued measures directly on the input and output spaces. For operators in this space, we establish approximation bounds for encoder--decoder two-layer networks in the Bochner $L^q$ norm. The resulting error bound decomposes into the input encoding error, the output encoding error, and a finite-width approximation term of order $N^{-1/2}$, with a constant independent of the input and output encoding dimensions. When the input and output encoding errors decay polynomially in the encoding dimensions, these estimates yield algebraic approximation and learning rates. The results provide an theoretical guarantees for efficient neural operator learning beyond general Lipschitz or Fréchet differentiable operator classes.