# Physics-informed Attention-enhanced Fourier Neural Operator for Solar Magnetic Field Extrapolations

arXiv: https://arxiv.org/abs/2510.05351 · HTML: https://mlsanity.com/paper/2510.05351/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 32 references that are in the map.
- First posted on arXiv 6 Oct 2025.

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

- [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)
- [Multi-Scale Context Aggregation by Dilated Convolutions](https://mlsanity.com/paper/1511.07122/) (2015)
- [ECA-Net: Efficient Channel Attention for Deep Convolutional Neural Networks](https://mlsanity.com/paper/1910.03151/) (2019)
- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [GNOT: A General Neural Operator Transformer for Operator Learning](https://mlsanity.com/paper/2302.14376/) (2023)
- [U-FNO -- An enhanced Fourier neural operator-based deep-learning model for multiphase flow](https://mlsanity.com/paper/2109.03697/) (2021)
- [Neural Operators for Accelerating Scientific Simulations and Design](https://mlsanity.com/paper/2309.15325/) (2023)
- [Fourier-MIONet: Fourier-enhanced multiple-input neural operators for multiphase modeling of geological carbon sequestration](https://mlsanity.com/paper/2303.04778/) (2023)
- [Machine learning in solar physics](https://mlsanity.com/paper/2306.15308/) (2023)
- [Global-local Fourier Neural Operator for Accelerating Coronal Magnetic Field Model](https://mlsanity.com/paper/2405.12754/) (2024)
- [Deep Computer Vision for Solar Physics Big Data: Opportunities and Challenges](https://mlsanity.com/paper/2409.04850/) (2024)

## Built on by (most cited first, 1 of 1)

- [Diffusion-corrected Autoregressive Fourier Neural Operator for Droplet Evolution Prediction](https://mlsanity.com/paper/2607.16238/) (2026)

## Abstract (arXiv)

We propose Physics-informed Attention-enhanced Fourier Neural Operator (PIANO) to solve the Nonlinear Force-Free Field (NLFFF) problem in solar physics. Unlike conventional approaches that rely on iterative numerical methods, our proposed PIANO directly learns the 3D magnetic field structure from 2D boundary conditions. Specifically, PIANO integrates Efficient Channel Attention (ECA) mechanisms with Dilated Convolutions (DC), which enhances the model's ability to capture multimodal input by prioritizing critical channels relevant to the magnetic field's variations. Furthermore, we apply physics-informed loss by enforcing the force-free and divergence-free conditions in the training process so that our prediction is consistent with underlying physics with high accuracy. Experimental results on the ISEE NLFFF dataset show that our PIANO not only outperforms state-of-the-art neural operators in terms of accuracy but also shows strong consistency with the physical characteristics of NLFFF data across magnetic fields reconstructed from various solar active regions. The GitHub of this project is available https://github.com/Autumnstar-cjh/PIANO