# Component Fourier Neural Operator for Singularly Perturbed Differential Equations

arXiv: https://arxiv.org/abs/2409.04779 · HTML: https://mlsanity.com/paper/2409.04779/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 31 references that are in the map.
- First posted on arXiv 7 Sep 2024.

## Builds on (most cited first, 16 of 16)

- [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)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (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)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [Universal Differential Equations for Scientific Machine Learning](https://mlsanity.com/paper/2001.04385/) (2020)
- [MIONet: Learning multiple-input operators via tensor product](https://mlsanity.com/paper/2202.06137/) (2022)
- [Learning to Optimize Multigrid PDE Solvers](https://mlsanity.com/paper/1902.10248/) (2019)
- [Learning Deep Implicit Fourier Neural Operators (IFNOs) with Applications to Heterogeneous Material Modeling](https://mlsanity.com/paper/2203.08205/) (2022)
- [Generic bounds on the approximation error for physics-informed (and) operator learning](https://mlsanity.com/paper/2205.11393/) (2022)
- [The Old and the New: Can Physics-Informed Deep-Learning Replace Traditional Linear Solvers?](https://mlsanity.com/paper/2103.09655/) (2021)
- [Seismic wave propagation and inversion with Neural Operators](https://mlsanity.com/paper/2108.05421/) (2021)
- [Applications of physics informed neural operators](https://mlsanity.com/paper/2203.12634/) (2022)
- [A nonlocal physics-informed deep learning framework using the peridynamic differential operator](https://mlsanity.com/paper/2006.00446/) (2020)

## Built on by (most cited first, 7 of 7)

- [Kernel-Adaptive PI-ELMs for Forward and Inverse Problems in PDEs with Sharp Gradients](https://mlsanity.com/paper/2507.10241/) (2025)
- [SAOT: An Enhanced Locality-Aware Spectral Transformer for Solving PDEs](https://mlsanity.com/paper/2511.18777/) (2025)
- [Fourier Neural Operators for Non-Markovian Processes:Approximation Theorems and Experiments](https://mlsanity.com/paper/2507.17887/) (2025)
- [Gated X-TFC: Soft Domain Decomposition for Forward and Inverse Problems in Sharp-Gradient PDEs](https://mlsanity.com/paper/2510.01039/) (2025)
- [Discontinuous Galerkin finite element operator network for solving non-smooth PDEs](https://mlsanity.com/paper/2601.03668/) (2026)
- [DiffPDE: Masked Diffusion Language Models as PDE Solver](https://mlsanity.com/paper/2608.30532/) (2026)
- [Selective boundary condition reduction via learned error gating](https://mlsanity.com/paper/2609.08461/) (2026)

## Abstract (arXiv)

Solving Singularly Perturbed Differential Equations (SPDEs) poses computational challenges arising from the rapid transitions in their solutions within thin regions. The effectiveness of deep learning in addressing differential equations motivates us to employ these methods for solving SPDEs. In this manuscript, we introduce Component Fourier Neural Operator (ComFNO), an innovative operator learning method that builds upon Fourier Neural Operator (FNO), while simultaneously incorporating valuable prior knowledge obtained from asymptotic analysis. Our approach is not limited to FNO and can be applied to other neural network frameworks, such as Deep Operator Network (DeepONet), leading to potential similar SPDEs solvers. Experimental results across diverse classes of SPDEs demonstrate that ComFNO significantly improves accuracy compared to vanilla FNO. Furthermore, ComFNO exhibits natural adaptability to diverse data distributions and performs well in few-shot scenarios, showcasing its excellent generalization ability in practical situations.