  On the forefront of scientific computing, Deep Learning (DL), i.e., machine
learning with Deep Neural Networks (DNNs), has emerged a powerful new tool for
solving Partial Differential Equations (PDEs). It has been observed that DNNs
are particularly well suited to weakening the effect of the curse of
dimensionality, a term coined by Richard E. Bellman in the late `50s to
describe challenges such as the exponential dependence of the sample
complexity, i.e., the number of samples required to solve an approximation
problem, on the dimension of the ambient space. However, although DNNs have
been used to solve PDEs since the `90s, the literature underpinning their
mathematical efficiency in terms of numerical analysis (i.e., stability,
accuracy, and sample complexity), is only recently beginning to emerge. In this
paper, we leverage recent advancements in function approximation using
sparsity-based techniques and random sampling to develop and analyze an
efficient high-dimensional PDE solver based on DL. We show, both theoretically
and numerically, that it can compete with a novel stable and accurate
compressive spectral collocation method. In particular, we demonstrate a new
practical existence theorem, which establishes the existence of a class of
trainable DNNs with suitable bounds on the network architecture and a
sufficient condition on the sample complexity, with logarithmic or, at worst,
linear scaling in dimension, such that the resulting networks stably and
accurately approximate a diffusion-reaction PDE with high probability.
