Title: Streamlined Approximation of Encoder-Decoder Neural Operators Using Variation Spaces

Abstract: This research investigates operator learning methodologies that leverage encoder-decoder neural architectures. Drawing inspiration from the function-space theory underpinning neural networks, we propose the concept of a variation space, serving as an infinite-dimensional structural framework for nonlinear operators. This space is constructed via vector-valued measures applied directly to the input and output domains. We derive approximation bounds for two-layer encoder-decoder networks operating within the Bochner $L^q$ norm for operators belonging to this space. The derived error bound consists of three distinct components: the error associated with input encoding, the error from output encoding, and a finite-width approximation term that scales as $N^{-1/2}$. Notably, the constant associated with the approximation term remains independent of the dimensions used for input and output encoding. In scenarios where encoding errors diminish polynomially with respect to their respective dimensions, these findings establish algebraic rates for both approximation and learning. Consequently, these results offer theoretical assurances for the efficiency of neural operator learning, extending beyond the traditional confines of general Lipschitz or Fréchet differentiable operator classes.

Source: arXiv Generated at: 2026-06-02 00:00:00 UTC