  Partial differential equations (PDEs) are among the most universal and
parsimonious descriptions of natural physical laws, capturing a rich variety of
phenomenology and multi-scale physics in a compact and symbolic representation.
This review will examine several promising avenues of PDE research that are
being advanced by machine learning, including: 1) the discovery of new
governing PDEs and coarse-grained approximations for complex natural and
engineered systems, 2) learning effective coordinate systems and reduced-order
models to make PDEs more amenable to analysis, and 3) representing solution
operators and improving traditional numerical algorithms. In each of these
fields, we summarize key advances, ongoing challenges, and opportunities for
further development.
