  Simulating the time evolution of Partial Differential Equations (PDEs) of
large-scale systems is crucial in many scientific and engineering domains such
as fluid dynamics, weather forecasting and their inverse optimization problems.
However, both classical solvers and recent deep learning-based surrogate models
are typically extremely computationally intensive, because of their local
evolution: they need to update the state of each discretized cell at each time
step during inference. Here we develop Latent Evolution of PDEs (LE-PDE), a
simple, fast and scalable method to accelerate the simulation and inverse
optimization of PDEs. LE-PDE learns a compact, global representation of the
system and efficiently evolves it fully in the latent space with learned latent
evolution models. LE-PDE achieves speed-up by having a much smaller latent
dimension to update during long rollout as compared to updating in the input
space. We introduce new learning objectives to effectively learn such latent
dynamics to ensure long-term stability. We further introduce techniques for
speeding-up inverse optimization of boundary conditions for PDEs via
backpropagation through time in latent space, and an annealing technique to
address the non-differentiability and sparse interaction of boundary
conditions. We test our method in a 1D benchmark of nonlinear PDEs, 2D
Navier-Stokes flows into turbulent phase and an inverse optimization of
boundary conditions in 2D Navier-Stokes flow. Compared to state-of-the-art deep
learning-based surrogate models and other strong baselines, we demonstrate up
to 128x reduction in the dimensions to update, and up to 15x improvement in
speed, while achieving competitive accuracy.
