Autoencoders in Function Space
This work addresses the challenge of applying autoencoders to functional data in scientific and image domains, though it is incremental as it extends existing autoencoder frameworks to function space.
The paper introduces function-space versions of autoencoders (FAE) and variational autoencoders (FVAE) to handle data viewed as functions, such as from scientific or image processing applications, enabling new uses in tasks like inpainting and superresolution. It shows that the FAE objective is well-defined in many cases where FVAE fails, allowing smoother operation across resolutions.
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.