Efficient Approximation for Encoder--Decoder Neural Operators via Variation Spaces

2026-05-31

AI Summary

This work addresses the problem of efficient, dimension-independent approximation by encoder–decoder neural operators through a novel theoretical framework grounded in variational spaces. The approach characterizes classes of nonlinear operators via vector-valued measures on input and output function spaces, thereby circumventing restrictive assumptions such as Lipschitz continuity or Fréchet differentiability commonly imposed in prior analyses. Leveraging Bochner space techniques, the study establishes an $L^q$-norm approximation error bound for two-layer encoder–decoder networks, where the bounding constant is independent of the encoding dimension. The error decomposes into contributions from input encoding, output encoding, and a finite-width term scaling as $N^{-1/2}$. When the encoding errors decay polynomially, the overall scheme achieves algebraic approximation rates and corresponding learning rates.

Abstract

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.

Problem

• neural operators
• encoder-decoder networks
• approximation theory
• variation spaces
• Bochner norm

Innovation

• variation spaces
• encoder-decoder neural operators
• approximation theory
• Bochner L^q norm
• algebraic learning rates