  Reduced-order models based on level-set methods are widely used tools to
qualitatively capture and track the nonlinear dynamics of an interface. The aim
of this paper is to develop a physics-informed, data-driven, statistically
rigorous learning algorithm for state and parameter estimation with level-set
methods. A Bayesian approach based on data assimilation is introduced. Data
assimilation is enabled by the ensemble Kalman filter and smoother, which are
used in their probabilistic formulations. The level-set data assimilation
framework is verified in onedimensional and two-dimensional test cases, where
state estimation, parameter estimation and uncertainty quantification are
performed. The statistical performance of the proposed ensemble Kalman filter
and smoother is quantified by twin experiments. In the twin experiments, the
combined state and parameter estimation fully recovers the reference solution,
which validates the proposed algorithm. The level-set data assimilation
framework is then applied to the prediction of the nonlinear dynamics of a
forced premixed flame, which exhibits the formation of sharp cusps and
intricate topological changes, such as pinch-off events. The proposed
physics-informed statistical learning algorithm opens up new possibilities for
making reduced-order models of interfaces quantitatively predictive, any time
that reference data is available.
