  When introducing physics-constrained deep learning solutions to the
volumetric super-resolution of scientific data, the training is challenging to
converge and always time-consuming. We propose a new hierarchical sampling
method based on octree to solve these difficulties. In our approach, scientific
data is preprocessed before training, and a hierarchical octree-based data
structure is built to guide sampling on the latent context grid. Each leaf node
in the octree corresponds to an indivisible subblock of the volumetric data.
The dimensions of the subblocks are different, making the number of sample
points in each randomly cropped training data block to be adaptive. We
reconstruct the octree at intervals according to loss distribution to perform
the multi-stage training. With the Rayleigh-B\'enard convection problem, we
deploy our method to state-of-the-art models. We constructed adequate
experiments to evaluate the training performance and model accuracy of our
method. Experiments indicate that our sampling optimization improves the
convergence performance of physics-constrained deep learning super-resolution
solutions. Furthermore, the sample points and training time are significantly
reduced with no drop in model accuracy. We also test our method in training
tasks of other deep neural networks, and the results show our sampling
optimization has extensive effectiveness and applicability. The code is
publicly available at https://github.com/xinjiewang/octree-based_sampling.
