Figure 2.

A schematic of the VDE. Features, x_t, at some timepoint, t, are fed into the network in order to make a prediction of the state of the system, $x_{t+\tau }^{\text{'}}$, at a future timepoint, t + τ, where τ is some Markovian lag time. As with a traditional VAE, the network can be subdivided into three parts: the encoder network; variational layer, Λ; and the decoder network, as labeled. Our encoder network is a DNN with non-linear activation functions in the hidden layers, which eventually bottlenecks into the one-dimensional latent space, z_t. The latent space is then slightly perturbed with Gaussian noise by the Λ-layer to generate z′, as described by Kingma and Welling¹⁶. Finally, the decoder network, also a DNN, mirrors the encoder network in architecture by using z′ to generate $x_{t+\tau }^{\text{'}}$, a prediction of how the system will evolve after one lag time of τ.