  We apply the Physics Informed Neural Network (PINN) to the problem of
wildfire fire-front modelling. We use the PINN to solve the level-set equation,
which is a partial differential equation that models a fire-front through the
zero-level-set of a level-set function. The result is a PINN that simulates a
fire-front as it propagates through the spatio-temporal domain. We show that
popular optimisation cost functions used in the literature can result in PINNs
that fail to maintain temporal continuity in modelled fire-fronts when there
are extreme changes in exogenous forcing variables such as wind direction. We
thus propose novel additions to the optimisation cost function that improves
temporal continuity under these extreme changes. Furthermore, we develop an
approach to perform data assimilation within the PINN such that the PINN
predictions are drawn towards observations of the fire-front. Finally, we
incorporate our novel approaches into a Bayesian PINN (B-PINN) to provide
uncertainty quantification in the fire-front predictions. This is significant
as the standard solver, the level-set method, does not naturally offer the
capability for data assimilation and uncertainty quantification. Our results
show that, with our novel approaches, the B-PINN can produce accurate
predictions with high quality uncertainty quantification on real-world data.
