  The deep-learning-based least squares method has shown successful results in
solving high-dimensional non-linear partial differential equations (PDEs).
However, this method usually converges slowly. To speed up the convergence of
this approach, an active-learning-based sampling algorithm is proposed in this
paper. This algorithm actively chooses the most informative training samples
from a probability density function based on residual errors to facilitate
error reduction. In particular, points with larger residual errors will have
more chances of being selected for training. This algorithm imitates the human
learning process: learners are likely to spend more time repeatedly studying
mistakes than other tasks they have correctly finished. A series of numerical
results are illustrated to demonstrate the effectiveness of our
active-learning-based sampling in high dimensions to speed up the convergence
of the deep-learning-based least squares method.
