Considerable progress towards the achievement of
thermonuclear burn using inertial confinement fusion has been
achieved at the National Ignition Facility (NIF) in the USA in
the last few years. Other drivers, such as the Z-machine at
Sandia, are also making progress towards this goal. A burning
thermonuclear plasma would provide a unique and extreme
plasma environment; in this paper we discuss a) different
theoretical challenges involved in modelling burning plasmas
not currently considered, b) the use of novel machine learning
based methods that might help large facilities reach ignition,
and c) the connections that a burning plasma might have to
fundamental physics, including QED studies, and the replication
and exploration of conditions that last occurred in the first few
minutes after the Big Bang.