# Fourier Neural Operator Networks: A Fast and General Solver for the Photoacoustic Wave Equation

arXiv: https://arxiv.org/abs/2108.09374 · HTML: https://mlsanity.com/paper/2108.09374/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 43 references that are in the map.
- First posted on arXiv 20 Aug 2021.

## Builds on (most cited first, 10 of 10)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Zero-Shot Learning -- A Comprehensive Evaluation of the Good, the Bad and the Ugly](https://mlsanity.com/paper/1707.00600/) (2017)
- [Solving ill-posed inverse problems using iterative deep neural networks](https://mlsanity.com/paper/1704.04058/) (2017)
- [Learning to Optimize Multigrid PDE Solvers](https://mlsanity.com/paper/1902.10248/) (2019)
- [Fully Dense UNet for 2D Sparse Photoacoustic Tomography Artifact Removal](https://mlsanity.com/paper/1808.10848/) (2018)
- [Model based learning for accelerated, limited-view 3D photoacoustic tomography](https://mlsanity.com/paper/1708.09832/) (2017)
- [Deep Learning for Photoacoustic Tomography from Sparse Data](https://mlsanity.com/paper/1704.04587/) (2017)
- [Limited View and Sparse Photoacoustic Tomography for Neuroimaging with Deep Learning](https://mlsanity.com/paper/1911.04357/) (2019)

## Built on by (most cited first, 10 of 10)

- [Neural Operators for Accelerating Scientific Simulations and Design](https://mlsanity.com/paper/2309.15325/) (2023)
- [Towards Multi-spatiotemporal-scale Generalized PDE Modeling](https://mlsanity.com/paper/2209.15616/) (2022)
- [Clifford Neural Layers for PDE Modeling](https://mlsanity.com/paper/2209.04934/) (2022)
- [Physics-Informed Machine Learning: A Survey on Problems, Methods and Applications](https://mlsanity.com/paper/2211.08064/) (2022)
- [Towards Stability of Autoregressive Neural Operators](https://mlsanity.com/paper/2306.10619/) (2023)
- [Out-of-distributional risk bounds for neural operators with applications to the Helmholtz equation](https://mlsanity.com/paper/2301.11509/) (2023)
- [Improved generalization with deep neural operators for engineering systems: Path towards digital twin](https://mlsanity.com/paper/2301.06701/) (2023)
- [Latent Mamba Operator for Partial Differential Equations](https://mlsanity.com/paper/2505.19105/) (2025)
- [Neural Operator: Is data all you need to model the world? An insight into the paradigm of data-driven scientific ML](https://mlsanity.com/paper/2301.13331/) (2023)
- [CoNO: Complex Neural Operator for Continous Dynamical Physical Systems](https://mlsanity.com/paper/2406.02597/) (2024)

## Abstract (arXiv)

Simulation tools for photoacoustic wave propagation have played a key role in advancing photoacoustic imaging by providing quantitative and qualitative insights into parameters affecting image quality. Classical methods for numerically solving the photoacoustic wave equation relies on a fine discretization of space and can become computationally expensive for large computational grids. In this work, we apply Fourier Neural Operator (FNO) networks as a fast data-driven deep learning method for solving the 2D photoacoustic wave equation in a homogeneous medium. Comparisons between the FNO network and pseudo-spectral time domain approach demonstrated that the FNO network generated comparable simulations with small errors and was several orders of magnitude faster. Moreover, the FNO network was generalizable and can generate simulations not observed in the training data.