  Analytical phase demodulation algorithms in optical interferometry typically
fail to reach the theoretical sensitivity limit set by the Cram\'er-Rao bound
(CRB). We show that deep neural networks (DNNs) can perform efficient phase
demodulation by achieving or exceeding the CRB by significant margins when
trained with new information that is not utilized by conventional algorithms,
such as noise statistics and parameter constraints. As an example, we developed
and applied DNNs to wavelength shifting interferometry. When trained with noise
statistics, the DNNs outperform the conventional algorithm in terms of phase
sensitivity and achieve the traditional three parameter CRB. Further, by
incorporating parameter constraints into the training sets, they can exceed the
traditional CRB. For well confined parameters, the phase sensitivity of the
DNNs can even approach a fundamental limit we refer to as the single parameter
CRB. Such sensitivity improvement can translate into significant increase in
single-to-noise ratio without hardware modification, or be used to relax
hardware requirements.
