# Projection Methods for Operator Learning and Universal Approximation

arXiv: https://arxiv.org/abs/2406.12264 · HTML: https://mlsanity.com/paper/2406.12264/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 35 references that are in the map.
- First posted on arXiv 18 Jun 2024.

## Builds on (most cited first, 12 of 12)

- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Multipole Graph Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2006.09535/) (2020)
- [The Expressive Power of Neural Networks: A View from the Width](https://mlsanity.com/paper/1709.02540/) (2017)
- [Model Reduction and Neural Networks for Parametric PDEs](https://mlsanity.com/paper/2005.03180/) (2020)
- [Convolutional Neural Operators for robust and accurate learning of PDEs](https://mlsanity.com/paper/2302.01178/) (2023)
- [Operator Learning: Algorithms and Analysis](https://mlsanity.com/paper/2402.15715/) (2024)
- [A Mathematical Guide to Operator Learning](https://mlsanity.com/paper/2312.14688/) (2023)
- [Operator learning with PCA-Net: upper and lower complexity bounds](https://mlsanity.com/paper/2303.16317/) (2023)
- [Neural Integral Equations](https://mlsanity.com/paper/2209.15190/) (2022)
- [Neural Integro-Differential Equations](https://mlsanity.com/paper/2206.14282/) (2022)
- [A finite element-based physics-informed operator learning framework for spatiotemporal partial differential equations on arbitrary domains](https://mlsanity.com/paper/2405.12465/) (2024)
- [Spectral methods for Neural Integral Equations](https://mlsanity.com/paper/2312.05654/) (2023)

## Built on by (most cited first, 2 of 2)

- [Universal Approximation of Operators with Transformers and Neural Integral Operators](https://mlsanity.com/paper/2409.00841/) (2024)
- [Leray-Schauder Mappings for Operator Learning](https://mlsanity.com/paper/2410.01746/) (2024)

## Abstract (arXiv)

We obtain a new universal approximation theorem for continuous (possibly nonlinear) operators on arbitrary Banach spaces using the Leray-Schauder mapping. Moreover, we introduce and study a method for operator learning in Banach spaces Lp of functions with multiple variables, based on orthogonal projections on polynomial bases. We derive a universal approximation result for operators where we learn a linear projection and a finite dimensional mapping under some additional assumptions. For the case of p=2, we give some sufficient conditions for the approximation results to hold. This article serves as the theoretical framework for a deep learning methodology in operator learning.