Variational autoencoders with latent high-dimensional steady geometric flows for dynamics

We develop Riemannian approaches to variational autoencoders (VAEs) for PDE-type ambient data with regularizing geometric latent dynamics, which we refer to as VAE-DLM, or VAEs with dynamical latent manifolds. We redevelop the VAE framework such that manifold geometries, subject to a geometric flow, embedded in Euclidean space are learned in the intermediary latent space developed by encoders and decoders. We reformulate the traditional evidence lower bound (ELBO) loss with a considerate choice of prior. We develop a linear geometric flow with a steady-state regularizing term. This geometric flow requires only automatic differentiation of one time derivative, and can be solved in moderately high dimensions in a physics-informed approach, allowing more expressive latent representations. We discuss how this flow can be formulated as a gradient flow, and maintains entropy away from metric singularity. This, along with an eigenvalue penalization condition, helps ensure the manifold is sufficiently large in measure, nondegenerate, and a canonical geometry, which contribute to a robust representation. Our methods focus on the modified multi-layer perceptron architecture with tanh activations for the manifold encoder-decoder. We demonstrate, on our datasets of interest, our methods perform at least as well as the traditional VAE, and oftentimes better. Our methods can outperform a standard VAE and a VAE endowed with our proposed architecture by up to 25% reduction in out-of-distribution (OOD) error and potentially greater. We highlight our method on ambient PDEs whose solutions maintain minimal variation in late times over its solution. Our approaches are particularly favorable with severe OOD effect. We provide empirical justification towards how latent Riemannian manifolds improve robust learning for external dynamics with VAEs.