  The computational complexity of classical numerical methods for solving
Partial Differential Equations (PDE) scales significantly as the resolution
increases. As an important example, climate predictions require fine
spatio-temporal resolutions to resolve all turbulent scales in the fluid
simulations. This makes the task of accurately resolving these scales
computationally out of reach even with modern supercomputers. As a result,
current numerical modelers solve PDEs on grids that are too coarse (3km to
200km on each side), which hinders the accuracy and usefulness of the
predictions. In this paper, we leverage the recent advances in Implicit Neural
Representations (INR) to design a novel architecture that predicts the
spatially continuous solution of a PDE given a spatial position query. By
augmenting coordinate-based architectures with Graph Neural Networks (GNN), we
enable zero-shot generalization to new non-uniform meshes and long-term
predictions up to 250 frames ahead that are physically consistent. Our Mesh
Agnostic Neural PDE Solver (MAgNet) is able to make accurate predictions across
a variety of PDE simulation datasets and compares favorably with existing
baselines. Moreover, MAgNet generalizes well to different meshes and
resolutions up to four times those trained on.
