Analytical phase demodulation algorithms in optical interferometry typically fail to reach the theoretical sensitivity limit set by the Cramér-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.

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