# Fourier-DeepONet: Fourier-enhanced deep operator networks for full waveform inversion with improved accuracy, generalizability, and robustness

arXiv: https://arxiv.org/abs/2305.17289 · HTML: https://mlsanity.com/paper/2305.17289/

## Key facts

- Cited by 60 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 45 references that are in the map.
- First posted on arXiv 26 May 2023.

## Builds on (most cited first, 17 of 17)

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [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)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Certified Robustness to Adversarial Examples with Differential Privacy](https://mlsanity.com/paper/1802.03471/) (2018)
- [U-FNO -- An enhanced Fourier neural operator-based deep-learning model for multiphase flow](https://mlsanity.com/paper/2109.03697/) (2021)
- [Adversarial Training Methods for Semi-Supervised Text Classification](https://mlsanity.com/paper/1605.07725/) (2016)
- [MIONet: Learning multiple-input operators via tensor product](https://mlsanity.com/paper/2202.06137/) (2022)
- [Robust Learning with Jacobian Regularization](https://mlsanity.com/paper/1908.02729/) (2019)
- [Multifidelity deep neural operators for efficient learning of partial differential equations with application to fast inverse design of nanoscale heat transport](https://mlsanity.com/paper/2204.06684/) (2022)
- [Reliable extrapolation of deep neural operators informed by physics or sparse observations](https://mlsanity.com/paper/2212.06347/) (2022)
- [InversionNet: A Real-Time and Accurate Full Waveform Inversion with CNNs and continuous CRFs](https://mlsanity.com/paper/1811.07875/) (2018)
- [Fourier-MIONet: Fourier-enhanced multiple-input neural operators for multiphase modeling of geological carbon sequestration](https://mlsanity.com/paper/2303.04778/) (2023)
- [OpenFWI: Large-Scale Multi-Structural Benchmark Datasets for Seismic Full Waveform Inversion](https://mlsanity.com/paper/2111.02926/) (2021)
- [Data-driven Seismic Waveform Inversion: A Study on the Robustness and Generalization](https://mlsanity.com/paper/1809.10262/) (2018)
- [Unsupervised Learning of Full-Waveform Inversion: Connecting CNN and Partial Differential Equation in a Loop](https://mlsanity.com/paper/2110.07584/) (2021)
- [PPDONet: Deep Operator Networks for Fast Prediction of Steady-State Solutions in Disk-Planet Systems](https://mlsanity.com/paper/2305.11111/) (2023)

## Built on by (most cited first, 30 of 60)

- [Towards a Foundation Model for Partial Differential Equations: Multi-Operator Learning and Extrapolation](https://mlsanity.com/paper/2404.12355/) (2024)
- [Conformalized-DeepONet: A Distribution-Free Framework for Uncertainty Quantification in Deep Operator Networks](https://mlsanity.com/paper/2402.15406/) (2024)
- [D2NO: Efficient Handling of Heterogeneous Input Function Spaces with Distributed Deep Neural Operators](https://mlsanity.com/paper/2310.18888/) (2023)
- [MODNO: Multi Operator Learning With Distributed Neural Operators](https://mlsanity.com/paper/2404.02892/) (2024)
- [LeMON: Learning to Learn Multi-Operator Networks](https://mlsanity.com/paper/2408.16168/) (2024)
- [AgenticSciML: Collaborative Multi-Agent Systems for Emergent Discovery in Scientific Machine Learning](https://mlsanity.com/paper/2511.07262/) (2025)
- [Active operator learning with predictive uncertainty quantification for partial differential equations](https://mlsanity.com/paper/2503.03178/) (2025)
- [Efficient and generalizable nested Fourier-DeepONet for three-dimensional geological carbon sequestration](https://mlsanity.com/paper/2409.16572/) (2024)
- [An Empirical Study of Large-Scale Data-Driven Full Waveform Inversion](https://mlsanity.com/paper/2307.15388/) (2023)
- [Speeding up and reducing memory usage for scientific machine learning via mixed precision](https://mlsanity.com/paper/2401.16645/) (2024)
- [Federated scientific machine learning for approximating functions and solving differential equations with data heterogeneity](https://mlsanity.com/paper/2410.13141/) (2024)
- [TANTE: Time-Adaptive Operator Learning via Neural Taylor Expansion](https://mlsanity.com/paper/2502.08574/) (2025)
- [Temporal Neural Operator for Modeling Time-Dependent Physical Phenomena](https://mlsanity.com/paper/2504.20249/) (2025)
- [Scattering with Neural Operators](https://mlsanity.com/paper/2308.14789/) (2023)
- [Learning Physics for Unveiling Hidden Earthquake Ground Motions via Conditional Generative Modeling](https://mlsanity.com/paper/2407.15089/) (2024)
- [PI-MFM: Physics-informed multimodal foundation model for solving partial differential equations](https://mlsanity.com/paper/2512.23056/) (2025)
- [A hybrid Decoder-DeepONet operator regression framework for unaligned observation data](https://mlsanity.com/paper/2308.09274/) (2023)
- [Paired Autoencoders for Likelihood-free Estimation in Inverse Problems](https://mlsanity.com/paper/2405.13220/) (2024)
- [A physics-informed transformer neural operator for learning generalized solutions of initial boundary value problems](https://mlsanity.com/paper/2412.09009/) (2024)
- [Separable Operator Networks](https://mlsanity.com/paper/2407.11253/) (2024)
- [Micrometer: Micromechanics Transformer for Predicting Mechanical Responses of Heterogeneous Materials](https://mlsanity.com/paper/2410.05281/) (2024)
- [Conformalized Prediction of Post-Fault Voltage Trajectories Using Pre-trained and Finetuned Attention-Driven Neural Operators](https://mlsanity.com/paper/2410.24162/) (2024)
- [RED-DiffEq: Regularization by denoising diffusion models for solving inverse PDE problems with application to full waveform inversion](https://mlsanity.com/paper/2509.21659/) (2025)
- [Resolution invariant deep operator network for PDEs with complex geometries](https://mlsanity.com/paper/2402.00825/) (2024)
- [Solving Partial Differential Equations in Different Domains by Operator Learning method Based on Boundary Integral Equations](https://mlsanity.com/paper/2406.02298/) (2024)
- [RAMS: Residual-based adversarial-gradient moving sample method for scientific machine learning in solving partial differential equations](https://mlsanity.com/paper/2509.01234/) (2025)
- [Neural Born Series Operator for Biomedical Ultrasound Computed Tomography](https://mlsanity.com/paper/2312.15575/) (2023)
- [Physics and geometry informed neural operator network with application to acoustic scattering](https://mlsanity.com/paper/2406.03407/) (2024)
- [Coefficient-to-Basis Network: A Fine-Tunable Operator Learning Framework for Inverse Problems with Adaptive Discretizations and Theoretical Guarantees](https://mlsanity.com/paper/2503.08642/) (2025)
- [A Score Filter Enhanced Data Assimilation Framework for Data-Driven Dynamical Systems](https://mlsanity.com/paper/2603.14863/) (2026)

## Abstract (arXiv)

Full waveform inversion (FWI) infers the subsurface structure information from seismic waveform data by solving a non-convex optimization problem. Data-driven FWI has been increasingly studied with various neural network architectures to improve accuracy and computational efficiency. Nevertheless, the applicability of pre-trained neural networks is severely restricted by potential discrepancies between the source function used in the field survey and the one utilized during training. Here, we develop a Fourier-enhanced deep operator network (Fourier-DeepONet) for FWI with the generalization of seismic sources, including the frequencies and locations of sources. Specifically, we employ the Fourier neural operator as the decoder of DeepONet, and we utilize source parameters as one input of Fourier-DeepONet, facilitating the resolution of FWI with variable sources. To test Fourier-DeepONet, we develop three new and realistic FWI benchmark datasets (FWI-F, FWI-L, and FWI-FL) with varying source frequencies, locations, or both. Our experiments demonstrate that compared with existing data-driven FWI methods, Fourier-DeepONet obtains more accurate predictions of subsurface structures in a wide range of source parameters. Moreover, the proposed Fourier-DeepONet exhibits superior robustness when handling data with Gaussian noise or missing traces and sources with Gaussian noise, paving the way for more reliable and accurate subsurface imaging across diverse real conditions.