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.

本研究提出了一种名为 GANO 的生成对抗神经操作器模型，可用于学习无限维函数空间上的概率。通过使用 Wasserstein 标准对其进行实例化，证明了其在真实世界函数数据上的优越性能。