  Closure problems are omnipresent when simulating multiscale systems, where
some quantities and processes cannot be fully prescribed despite their effects
on the simulation's accuracy. Recently, scientific machine learning approaches
have been proposed as a way to tackle the closure problem, combining
traditional (physics-based) modeling with data-driven (machine-learned)
techniques, typically through enriching differential equations with neural
networks. This paper reviews the different reduced model forms, distinguished
by the degree to which they include known physics, and the different objectives
of a priori and a posteriori learning. The importance of adhering to physical
laws (such as symmetries and conservation laws) in choosing the reduced model
form and choosing the learning method is discussed. The effect of spatial and
temporal discretization and recent trends toward discretization-invariant
models are reviewed. In addition, we make the connections between closure
problems and several other research disciplines: inverse problems, Mori-Zwanzig
theory, and multi-fidelity methods. In conclusion, much progress has been made
with scientific machine learning approaches for solving closure problems, but
many challenges remain. In particular, the generalizability and
interpretability of learned models is a major issue that needs to be addressed
further.
