  Neural networks have shown great potential in accelerating the solution of
partial differential equations (PDEs). Recently, there has been a growing
interest in introducing physics constraints into training neural PDE solvers to
reduce the use of costly data and improve the generalization ability. However,
these physics constraints, based on certain finite dimensional approximations
over the function space, must resolve the smallest scaled physics to ensure the
accuracy and stability of the simulation, resulting in high computational costs
from large input, output, and neural networks. This paper proposes a general
acceleration methodology called NeuralStagger by spatially and temporally
decomposing the original learning tasks into several coarser-resolution
subtasks. We define a coarse-resolution neural solver for each subtask, which
requires fewer computational resources, and jointly train them with the vanilla
physics-constrained loss by simply arranging their outputs to reconstruct the
original solution. Due to the perfect parallelism between them, the solution is
achieved as fast as a coarse-resolution neural solver. In addition, the trained
solvers bring the flexibility of simulating with multiple levels of resolution.
We demonstrate the successful application of NeuralStagger on 2D and 3D fluid
dynamics simulations, which leads to an additional $10\sim100\times$ speed-up.
Moreover, the experiment also shows that the learned model could be well used
for optimal control.
