  The paper surveys variational approaches for image reconstruction in dynamic
inverse problems. Emphasis is on methods that rely on parametrised temporal
models. These are here encoded as diffeomorphic deformations with time
dependent parameters, or as motion constrained reconstruction where the motion
model is given by a partial differential equation. The survey also includes
recent development in integrating deep learning for solving these
computationally demanding variational methods. Examples are given for 2D
dynamic tomography, but methods apply to general inverse problems.
