Regression on function spaces is typically limited to models with Gaussian
process priors. We introduce the notion of universal functional regression, in
which we aim to learn a prior distribution over non-Gaussian function spaces
that remains mathematically tractable for functional regression. To do this, we
develop Neural Operator Flows (OpFlow), an infinite-dimensional extension of
normalizing flows. OpFlow is an invertible operator that maps the (potentially
unknown) data function space into a Gaussian process, allowing for exact
likelihood estimation of functional point evaluations. OpFlow enables robust
and accurate uncertainty quantification via drawing posterior samples of the
Gaussian process and subsequently mapping them into the data function space. We
empirically study the performance of OpFlow on regression and generation tasks
with data generated from Gaussian processes with known posterior forms and
non-Gaussian processes, as well as real-world earthquake seismograms with an
unknown closed-form distribution.

我们介绍了一种通用函数回归方法，该方法可以学习在数学上可处理的非高斯函数空间上的先验分布，我们开发了神经算子流 (OpFlow) 来实现这种方法，OpFlow 是一个无限维扩展的归一化流，它将（潜在的未知）数据函数空间映射到高斯过程中，通过绘制高斯过程的后验样本并映射它们到数据函数空间，OpFlow 能够实现鲁棒且准确的不确定性量化，我们在回归和生成任务上对 OpFlow 的性能进行了实证研究，其中数据来自已知后验形式的高斯过程和非高斯过程，以及来自真实世界的地震地震图，其闭合分布未知。