# Pseudo-Differential Neural Operator: Generalized Fourier Neural Operator for Learning Solution Operators of Partial Differential Equations

arXiv: https://arxiv.org/abs/2201.11967 · HTML: https://mlsanity.com/paper/2201.11967/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 39 references that are in the map.
- First posted on arXiv 28 Jan 2022.

## Builds on (most cited first, 18 of 18)

- [Gaussian Error Linear Units (GELUs)](https://mlsanity.com/paper/1606.08415/) (2016)
- [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)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [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)
- [Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data](https://mlsanity.com/paper/1901.06314/) (2019)
- [Adaptive Fourier Neural Operators: Efficient Token Mixers for Transformers](https://mlsanity.com/paper/2111.13587/) (2021)
- [Multiwavelet-based Operator Learning for Differential Equations](https://mlsanity.com/paper/2109.13459/) (2021)
- [Bayesian Deep Convolutional Encoder-Decoder Networks for Surrogate Modeling and Uncertainty Quantification](https://mlsanity.com/paper/1801.06879/) (2018)
- [Prediction of Aerodynamic Flow Fields Using Convolutional Neural Networks](https://mlsanity.com/paper/1905.13166/) (2019)
- [Learning to Control PDEs with Differentiable Physics](https://mlsanity.com/paper/2001.07457/) (2020)
- [Learning Operators with Coupled Attention](https://mlsanity.com/paper/2201.01032/) (2022)
- [Solving PDE-constrained Control Problems Using Operator Learning](https://mlsanity.com/paper/2111.04941/) (2021)
- [A Deep Neural Network Surrogate for High-Dimensional Random Partial Differential Equations](https://mlsanity.com/paper/1806.02957/) (2018)
- [Trend to Equilibrium for the Kinetic Fokker-Planck Equation via the Neural Network Approach](https://mlsanity.com/paper/1911.09843/) (2019)
- [The model reduction of the Vlasov-Poisson-Fokker-Planck system to the Poisson-Nernst-Planck system via the Deep Neural Network Approach](https://mlsanity.com/paper/2009.13280/) (2020)

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

- [KANO: Kolmogorov-Arnold Neural Operator](https://mlsanity.com/paper/2509.16825/) (2025)
- [Neural operators struggle to learn complex PDEs in pedestrian mobility: Hughes model case study](https://mlsanity.com/paper/2504.18267/) (2025)
- [GNRK: Graph Neural Runge-Kutta method for solving partial differential equations](https://mlsanity.com/paper/2310.00618/) (2023)
- [Reformulating Neural Operators in d+1 Dimensions for Embedding Evolution](https://mlsanity.com/paper/2505.11766/) (2025)
- [LFNO: Bridging Laplace and Fourier via Transient-Steady Decomposition](https://mlsanity.com/paper/2606.07601/) (2026)
- [Euclidean Fourier Neural Operators](https://mlsanity.com/paper/2608.28425/) (2026)
- [More Sensors Only One Field: Rethinking Continual Spatio-Temporal Forecasting](https://mlsanity.com/paper/2609.31325/) (2026)
- [Low-Rank Kolmogorov-Arnold Generator Flows for Interpretable Operator Learning](https://mlsanity.com/paper/2610.13028/) (2026)

## Abstract (arXiv)

Learning the mapping between two function spaces has garnered considerable research attention. However, learning the solution operator of partial differential equations (PDEs) remains a challenge in scientific computing. Fourier neural operator (FNO) was recently proposed to learn solution operators, and it achieved an excellent performance. In this study, we propose a novel textitpseudo-differential integral operator (PDIO) to analyze and generalize the Fourier integral operator in FNO. PDIO is inspired by a pseudo-differential operator, which is a generalized differential operator characterized by a certain symbol. We parameterize this symbol using a neural network and demonstrate that the neural network-based symbol is contained in a smooth symbol class. Subsequently, we verify that the PDIO is a bounded linear operator, and thus is continuous in the Sobolev space. We combine the PDIO with the neural operator to develop a textitpseudo-differential neural operator (PDNO) and learn the nonlinear solution operator of PDEs. We experimentally validate the effectiveness of the proposed model by utilizing Darcy flow and the Navier-Stokes equation. The obtained results indicate that the proposed PDNO outperforms the existing neural operator approaches in most experiments.