Generative Adversarial Neural Operators
Md Ashiqur Rahman, Manuel A. Florez, Anima Anandkumar, Zachary E. Ross, Kamyar Azizzadenesheli
TL;DR
This work introduces Generative Adversarial Neural Operator (GANO), a framework that extends GANs to infinite-dimensional function spaces by using a generator neural operator and a discriminator neural functional. The method learns push-forward measures from input function spaces (e.g., Gaussian random fields) to output function spaces, optimizing a Wasserstein objective with a gradient penalty tailored to infinite dimensions. It leverages discretization-invariant neural operators (e.g., Fourier and U-shaped architectures) to produce function-valued outputs that can be queried at arbitrary resolutions, and it demonstrates superior capability over classical GANs on both synthetic GRF data and real-world InSAR volcano deformation data. These contributions enable robust generative modeling for functional data in physics, geoscience, and engineering, where observations are intrinsically infinite-dimensional. The work also provides practical guidance on architecture choices and training strategies, along with code and datasets for reproducibility.
Abstract
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.