A variational condition for minimal-residual latent representations

Autoencoders are a useful unsupervised-learning architecture that can be used to build surrogate models of systems governed by partial differential equations. In this article, we address two key questions underpinning this procedure: whether the reconstructed output satisfies the partial differential equation, and whether other latent vectors not corresponding to the encoding of any training data satisfy the same equation. Our results spell out some relevant conditions, and clarify the different impact of three main design decisions (architecture, training criterion, and choice of training solutions) on the final result.