# PODNO: Proper Orthogonal Decomposition Neural Operators

arXiv: https://arxiv.org/abs/2504.18513 · HTML: https://mlsanity.com/paper/2504.18513/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 53 references that are in the map.
- First posted on arXiv 25 Apr 2025.

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

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Multipole Graph Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2006.09535/) (2020)
- [Model Reduction and Neural Networks for Parametric PDEs](https://mlsanity.com/paper/2005.03180/) (2020)
- [The Random Feature Model for Input-Output Maps between Banach Spaces](https://mlsanity.com/paper/2005.10224/) (2020)
- [Nonlocality and Nonlinearity Implies Universality in Operator Learning](https://mlsanity.com/paper/2304.13221/) (2023)
- [A DeepONet multi-fidelity approach for residual learning in reduced order modeling](https://mlsanity.com/paper/2302.12682/) (2023)
- [DeepParticle: learning invariant measure by a deep neural network minimizing Wasserstein distance on data generated from an interacting particle method](https://mlsanity.com/paper/2111.01356/) (2021)
- [A variational neural network approach for glacier modelling with nonlinear rheology](https://mlsanity.com/paper/2209.02088/) (2022)

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

- [Preconditioned One-Step Generative Modeling for Bayesian Inverse Problems in Function Spaces](https://mlsanity.com/paper/2603.14798/) (2026)
- [Inertial Manifold Neural Operator for Dissipative Time-Dependent Partial Differential Equations](https://mlsanity.com/paper/2608.23546/) (2026)

## Abstract (arXiv)

In this paper, we introduce Proper Orthogonal Decomposition Neural Operators (PODNO) for solving partial differential equations (PDEs) dominated by high-frequency components. Building on the structure of Fourier Neural Operators (FNO), PODNO replaces the Fourier transform with (inverse) orthonormal transforms derived from the Proper Orthogonal Decomposition (POD) method to construct the integral kernel. Due to the optimality of POD basis, the PODNO has potential to outperform FNO in both accuracy and computational efficiency for high-frequency problems. From analysis point of view, we established the universality of a generalization of PODNO, termed as Generalized Spectral Operator (GSO). In addition, we evaluate PODNO's performance numerically on dispersive equations such as the Nonlinear Schrodinger (NLS) equation and the Kadomtsev-Petviashvili (KP) equation.