  Effective data-driven PDE forecasting methods often rely on fixed spatial and
/ or temporal discretizations. This raises limitations in real-world
applications like weather prediction where flexible extrapolation at arbitrary
spatiotemporal locations is required. We address this problem by introducing a
new data-driven approach, DINo, that models a PDE's flow with continuous-time
dynamics of spatially continuous functions. This is achieved by embedding
spatial observations independently of their discretization via Implicit Neural
Representations in a small latent space temporally driven by a learned ODE.
This separate and flexible treatment of time and space makes DINo the first
data-driven model to combine the following advantages. It extrapolates at
arbitrary spatial and temporal locations; it can learn from sparse irregular
grids or manifolds; at test time, it generalizes to new grids or resolutions.
DINo outperforms alternative neural PDE forecasters in a variety of challenging
generalization scenarios on representative PDE systems.
