  We introduce a mathematical formulation of feature-informed data assimilation
(FIDA). In FIDA, the information about feature events, such as shock waves,
level curves, wavefronts and peak value, in dynamical systems are used for the
estimation of state variables and unknown parameters. The observation operator
in FIDA is a set-valued functional, which is fundamentally different from the
observation operators in conventional data assimilation. Demonstrated in three
example, FIDA problems introduced in this note exist in a wide spectrum of
applications in science and engineering.
