# Generative Adversarial Neural Operators

arXiv: https://arxiv.org/abs/2205.03017 · HTML: https://mlsanity.com/paper/2205.03017/

## Key facts

- Cited by 52 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 31 references that are in the map.
- First posted on arXiv 6 May 2022.

## Builds on (most cited first, 20 of 20)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Improved Training of Wasserstein GANs](https://mlsanity.com/paper/1704.00028/) (2017)
- [Unsupervised Representation Learning with Deep Convolutional Generative Adversarial Networks](https://mlsanity.com/paper/1511.06434/) (2015)
- [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [Neural Processes](https://mlsanity.com/paper/1807.01622/) (2018)
- [U-FNO -- An enhanced Fourier neural operator-based deep-learning model for multiphase flow](https://mlsanity.com/paper/2109.03697/) (2021)
- [U-NO: U-shaped Neural Operators](https://mlsanity.com/paper/2204.11127/) (2022)
- [Neural Stochastic Differential Equations: Deep Latent Gaussian Models in the Diffusion Limit](https://mlsanity.com/paper/1905.09883/) (2019)
- [Neural SDEs as Infinite-Dimensional GANs](https://mlsanity.com/paper/2102.03657/) (2021)
- [The Random Feature Model for Input-Output Maps between Banach Spaces](https://mlsanity.com/paper/2005.10224/) (2020)
- [Generative Models as Distributions of Functions](https://mlsanity.com/paper/2102.04776/) (2021)
- [Banach Wasserstein GAN](https://mlsanity.com/paper/1806.06621/) (2018)
- [Seismic wave propagation and inversion with Neural Operators](https://mlsanity.com/paper/2108.05421/) (2021)
- [Approximation and Convergence Properties of Generative Adversarial Learning](https://mlsanity.com/paper/1705.08991/) (2017)
- [Wasserstein GANs with Gradient Penalty Compute Congested Transport](https://mlsanity.com/paper/2109.00528/) (2021)

## Built on by (most cited first, 30 of 52)

- [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)
- [Score-based Diffusion Models in Function Space](https://mlsanity.com/paper/2302.07400/) (2023)
- [Multi-Grid Tensorized Fourier Neural Operator for High-Resolution PDEs](https://mlsanity.com/paper/2310.00120/) (2023)
- [Functional Flow Matching](https://mlsanity.com/paper/2305.17209/) (2023)
- [Diffusion Generative Models in Infinite Dimensions](https://mlsanity.com/paper/2212.00886/) (2022)
- [Training neural operators to preserve invariant measures of chaotic attractors](https://mlsanity.com/paper/2306.01187/) (2023)
- [Multilevel Diffusion: Infinite Dimensional Score-Based Diffusion Models for Image Generation](https://mlsanity.com/paper/2303.04772/) (2023)
- [Guided Diffusion Sampling on Function Spaces with Applications to PDEs](https://mlsanity.com/paper/2505.17004/) (2025)
- [Conditional Optimal Transport on Function Spaces](https://mlsanity.com/paper/2311.05672/) (2023)
- [Approximate Bayesian Neural Operators: Uncertainty Quantification for Parametric PDEs](https://mlsanity.com/paper/2208.01565/) (2022)
- [Variational Autoencoding Neural Operators](https://mlsanity.com/paper/2302.10351/) (2023)
- [FunDiff: Diffusion Models over Function Spaces for Physics-Informed Generative Modeling](https://mlsanity.com/paper/2506.07902/) (2025)
- [Deep Operator Learning Lessens the Curse of Dimensionality for PDEs](https://mlsanity.com/paper/2301.12227/) (2023)
- [Stochastic Process Learning via Operator Flow Matching](https://mlsanity.com/paper/2501.04126/) (2025)
- [Broadband Ground Motion Synthesis via Generative Adversarial Neural Operators: Development and Validation](https://mlsanity.com/paper/2309.03447/) (2023)
- [Principled Approaches for Extending Neural Architectures to Function Spaces for Operator Learning](https://mlsanity.com/paper/2506.10973/) (2025)
- [Mesh-Informed Neural Operator : A Transformer Generative Approach](https://mlsanity.com/paper/2506.16656/) (2025)
- [Probabilistic operator learning: generative modeling and uncertainty quantification for foundation models of differential equations](https://mlsanity.com/paper/2509.05186/) (2025)
- [Universal Functional Regression with Neural Operator Flows](https://mlsanity.com/paper/2404.02986/) (2024)
- [PaCMO: Partner Dependent Human Motion Generation in Dyadic Human Activity using Neural Operators](https://mlsanity.com/paper/2211.16210/) (2022)
- [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)
- [Generative Adversarial Reduced Order Modelling](https://mlsanity.com/paper/2305.15881/) (2023)
- [Scattering with Neural Operators](https://mlsanity.com/paper/2308.14789/) (2023)
- [Generative Neural Operators of Log-Complexity Can Simultaneously Solve Infinitely Many Convex Programs](https://mlsanity.com/paper/2508.14995/) (2025)
- [Challenges and opportunities for machine learning in multiscale computational modeling](https://mlsanity.com/paper/2303.12261/) (2023)
- [Flow Straight and Fast in Hilbert Space: Functional Rectified Flow](https://mlsanity.com/paper/2509.10384/) (2025)
- [Functional Mean Flow in Hilbert Space](https://mlsanity.com/paper/2511.12898/) (2025)
- [FNOPE: Simulation-based inference on function spaces with Fourier Neural Operators](https://mlsanity.com/paper/2505.22573/) (2025)

## Abstract (arXiv)

We propose the generative adversarial neural operator (GANO), a generative model paradigm for learning probabilities on infinite-dimensional function spaces. The natural sciences and engineering are known to have many types of data that are sampled from infinite-dimensional function spaces, where classical finite-dimensional deep generative adversarial networks (GANs) may not be directly applicable. GANO generalizes the GAN framework and allows for the sampling of functions by learning push-forward operator maps in infinite-dimensional spaces. GANO consists of two main components, a generator neural operator and a discriminator neural functional. The inputs to the generator are samples of functions from a user-specified probability measure, e.g., Gaussian random field (GRF), and the generator outputs are synthetic data functions. The input to the discriminator is either a real or synthetic data function. In this work, we instantiate GANO using the Wasserstein criterion and show how the Wasserstein loss can be computed in infinite-dimensional spaces. We empirically study GANO in controlled cases where both input and output functions are samples from GRFs and compare its performance to the finite-dimensional counterpart GAN. We empirically study the efficacy of GANO on real-world function data of volcanic activities and show its superior performance over GAN.