# Local approximation of operators

arXiv: https://arxiv.org/abs/2202.06392 · HTML: https://mlsanity.com/paper/2202.06392/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 74 references that are in the map.
- First posted on arXiv 13 Feb 2022.

## Builds on (most cited first, 11 of 11)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [The Random Feature Model for Input-Output Maps between Banach Spaces](https://mlsanity.com/paper/2005.10224/) (2020)
- [Deep vs. shallow networks : An approximation theory perspective](https://mlsanity.com/paper/1608.03287/) (2016)
- [Deep Nonparametric Estimation of Operators between Infinite Dimensional Spaces](https://mlsanity.com/paper/2201.00217/) (2022)
- [Kernel based analysis of massive data](https://mlsanity.com/paper/2003.13226/) (2020)
- [Function approximation with zonal function networks with activation functions analogous to the rectified linear unit functions](https://mlsanity.com/paper/1709.08174/) (2017)
- [A witness function based construction of discriminative models using Hermite polynomials](https://mlsanity.com/paper/1901.02975/) (2019)
- [A function approximation approach to the prediction of blood glucose levels](https://mlsanity.com/paper/2105.05893/) (2021)
- [A manifold learning approach for gesture recognition from micro-Doppler radar measurements](https://mlsanity.com/paper/2110.01670/) (2021)

## Built on by (most cited first, 8 of 8)

- [Operator Learning: Algorithms and Analysis](https://mlsanity.com/paper/2402.15715/) (2024)
- [Transformers as Neural Operators for Solutions of Differential Equations with Finite Regularity](https://mlsanity.com/paper/2405.19166/) (2024)
- [IAE-Net: Integral Autoencoders for Discretization-Invariant Learning](https://mlsanity.com/paper/2203.05142/) (2022)
- [Nonlinear functional regression by functional deep neural network with kernel embedding](https://mlsanity.com/paper/2401.02890/) (2024)
- [Cauchy Random Features for Operator Learning in Sobolev Space](https://mlsanity.com/paper/2503.00300/) (2025)
- [Tractability of approximation by general shallow networks](https://mlsanity.com/paper/2308.03230/) (2023)
- [Regularized Random Fourier Features and Finite Element Reconstruction for Operator Learning in Sobolev Space](https://mlsanity.com/paper/2512.17884/) (2025)
- [Using Linearized Optimal Transport to Predict the Evolution of Stochastic Particle Systems](https://mlsanity.com/paper/2408.01857/) (2024)

## Abstract (arXiv)

Many applications, such as system identification, classification of time series, direct and inverse problems in partial differential equations, and uncertainty quantification lead to the question of approximation of a non-linear operator between metric spaces mathfrakX and mathfrakY. We study the problem of determining the degree of approximation of such operators on a compact subset K_mathfrakXsubset mathfrakX using a finite amount of information. If F: K_mathfrakX→ K_mathfrakY, a well established strategy to approximate F(F) for some F∈ K_mathfrakX is to encode F (respectively, F(F)) in terms of a finite number d (repectively m) of real numbers. Together with appropriate reconstruction algorithms (decoders), the problem reduces to the approximation of m functions on a compact subset of a high dimensional Euclidean space ℝd, equivalently, the unit sphere mathbbSd embedded in ℝd+1. The problem is challenging because d, m, as well as the complexity of the approximation on mathbbSd are all large, and it is necessary to estimate the accuracy keeping track of the inter-dependence of all the approximations involved. In this paper, we establish constructive methods to do this efficiently; i.e., with the constants involved in the estimates on the approximation on mathbbSd being O(d1/6). We study different smoothness classes for the operators, and also propose a method for approximation of F(F) using only information in a small neighborhood of F, resulting in an effective reduction in the number of parameters involved.