  From ecology to atmospheric sciences, many academic disciplines deal with
data characterized by intricate spatio-temporal complexities, the modeling of
which often requires specialized approaches. Generative models of these data
are of particular interest, as they enable a range of impactful downstream
applications like simulation or creating synthetic training data. Recent work
has highlighted the potential of generative adversarial nets (GANs) for
generating spatio-temporal data. A new GAN algorithm COT-GAN, inspired by the
theory of causal optimal transport (COT), was proposed in an attempt to better
tackle this challenge. However, the task of learning more complex
spatio-temporal patterns requires additional knowledge of their specific data
structures. In this study, we propose a novel loss objective combined with
COT-GAN based on an autoregressive embedding to reinforce the learning of
spatio-temporal dynamics. We devise SPATE (spatio-temporal association), a new
metric measuring spatio-temporal autocorrelation by using the deviance of
observations from their expected values. We compute SPATE for real and
synthetic data samples and use it to compute an embedding loss that considers
space-time interactions, nudging the GAN to learn outputs that are faithful to
the observed dynamics. We test this new objective on a diverse set of complex
spatio-temporal patterns: turbulent flows, log-Gaussian Cox processes and
global weather data. We show that our novel embedding loss improves performance
without any changes to the architecture of the COT-GAN backbone, highlighting
our model's increased capacity for capturing autoregressive structures. We also
contextualize our work with respect to recent advances in physics-informed deep
learning and interdisciplinary work connecting neural networks with geographic
and geophysical sciences.
