Autoencoders in Function Space

TL;DR

This paper introduces function-space autoencoders and variational autoencoders, analyzing their theoretical foundations and demonstrating their application to scientific data processing tasks like inpainting and superresolution.

Contribution

It develops well-defined function-space autoencoder frameworks and explores their integration with neural operators for advanced scientific data modeling.

Findings

FVAE objective requires data-model compatibility, limiting applicability.

FAE objective is broadly applicable and well-defined in many scenarios.

Neural operator-based autoencoders enable new scientific data applications.

Abstract

Autoencoders have found widespread application in both their original deterministic form and in their variational formulation (VAEs). In scientific applications and in image processing it is often of interest to consider data that are viewed as functions; while discretisation (of differential equations arising in the sciences) or pixellation (of images) renders problems finite dimensional in practice, conceiving first of algorithms that operate on functions, and only then discretising or pixellating, leads to better algorithms that smoothly operate between resolutions. In this paper function-space versions of the autoencoder (FAE) and variational autoencoder (FVAE) are introduced, analysed, and deployed. Well-definedness of the objective governing VAEs is a subtle issue, particularly in function space, limiting applicability. For the FVAE objective to be well defined requires compatibility of the data distribution with the chosen generative model; this can be achieved, for example, when the data arise from a stochastic differential equation, but is generally restrictive. The FAE objective, on the other hand, is well defined in many situations where FVAE fails to be. Pairing the FVAE and FAE objectives with neural operator architectures that can be evaluated on any mesh enables new applications of autoencoders to inpainting, superresolution, and generative modelling of scientific data.