  The identification of states and parameters from noisy measurements of a
dynamical system is of great practical significance and has received a lot of
attention. Classically, this problem is expressed as optimization over a class
of models. This work presents such a method, where we augment the system in
such a way that there is no distinction between parameter and state
reconstruction. We pose the resulting problem as a batch problem: given the
model, reconstruct the state from a finite sequence of output measurements. In
the case the model is linear, we derive an analytical expression for the state
reconstruction given the model and the output measurements. Importantly, we
estimate the state trajectory in its entirety and do not aim to estimate just
an initial condition: that is, we use more degrees of freedom than strictly
necessary in the optimization step. This particular approach can be
reinterpreted as training of a neural network that estimates the state
trajectory from available measurements. The technology associated with neural
network optimization/training allows an easy extension to nonlinear models. The
proposed framework is relatively easy to implement, does not depend on an
informed initial guess, and provides an estimate for the state trajectory
(which incorporates an estimate for the unknown parameters) over a given finite
time horizon.
