Unsupervised learning with functional data is an emerging paradigm of machine learning research with applications to computer vision, climate modeling and physical systems. A natural way of modeling functional data is by learning operators between infinite dimensional spaces, leading to discretization invariant representations that scale independently of the sample grid resolution. Here we present Variational Autoencoding Neural Operators (VANO), a general strategy for making a large class of operator learning architectures act as variational autoencoders. For this purpose, we provide a novel rigorous mathematical formulation of the variational objective in function spaces for training. VANO first maps an input function to a distribution over a latent space using a parametric encoder and then decodes a sample from the latent distribution to reconstruct the input, as in classic variational autoencoders. We test VANO with different model set-ups and architecture choices for a variety of benchmarks. We start from a simple Gaussian random field where we can analytically track what the model learns and progressively transition to more challenging benchmarks including modeling phase separation in Cahn-Hilliard systems and real world satellite data for measuring Earth surface deformation.

翻译：以功能数据进行不受监督的学习是计算机视觉、气候建模和物理系统应用的机械学习研究的新兴范例。功能数据模型的自然方式是,在无限维空间之间学习操作员,从而导致与样本网格分辨率不相容的分解异的表达方式。在这里,我们介绍了使一大批操作员学习结构成为变式自动自动解码操作员的一般战略(VANO),这是使一大批操作员学习结构成为变异自动解码器的一般战略。为此,我们提供了一种新型的严格数学配置,说明用于培训功能空间的变异性目标。VANO首先用参数编码器绘制了一个输入函数,用于在潜在空间上进行分布,然后解码潜在分布的样本,以重建输入,如典型的变式自动解码器一样。我们用不同的模型集和结构选择来测试各种基准。我们从一个简单的高斯随机场开始,我们可以分析模型学到了什么,并逐步过渡到更具挑战性的基准,包括Chn-Hillard系统中的阶段分离和用于测量地表变形的实际世界卫星数据。