  The state of turbulent, minimal-channel flow is estimated from
spatio-temporal sparse observations of the velocity, using both a
physics-informed neural network (PINN) and adjoint-variational data
assimilation (4DVar). The performance of PINN is assessed against the benchmark
results from 4DVar. The PINN is efficient to implement, takes advantage of
automatic differentiation to evaluate the governing equations, and does not
require the development of an adjoint model. In addition, the flow evolution is
expressed in terms of the network parameters which have a far smaller dimension
than the predicted trajectory in state space or even just the initial condition
of the flow. Provided adequate observations, network architecture and training,
the PINN can yield satisfactory estimates of the the flow field, both for the
missing velocity data and the entirely unobserved pressure field. However,
accuracy depends on the network architecture, and the dependence is not known a
priori. In comparison to 4DVar estimation which becomes progressively more
accurate over the observation horizon, the PINN predictions are generally less
accurate and maintain the same level of errors throughout the assimilation time
window. Another notable distinction is the capacity to accurately forecast the
flow evolution: while the 4DVar prediction depart from the true flow state
gradually and according to the Lyapunov exponent, the PINN is entirely
inaccurate immediately beyond the training time horizon unless re-trained. Most
importantly, while 4DVar satisfies the discrete form of the governing equations
point-wise to machine precision, in PINN the equations are only satisfied in an
$L^2$ sense.
