  We extend the Physics-Informed Echo State Network (PI-ESN) framework to
reconstruct the evolution of an unmeasured state (hidden state) in a chaotic
system. The PI-ESN is trained by using (i) data, which contains no information
on the unmeasured state, and (ii) the physical equations of a prototypical
chaotic dynamical system. Non-noisy and noisy datasets are considered. First,
it is shown that the PI-ESN can accurately reconstruct the unmeasured state.
Second, the reconstruction is shown to be robust with respect to noisy data,
which means that the PI-ESN acts as a denoiser. This paper opens up new
possibilities for leveraging the synergy between physical knowledge and machine
learning to enhance the reconstruction and prediction of unmeasured states in
chaotic dynamical systems.
