Neural Networks (NNs) (or 'deep' neural networks (DNNs)) have found great success in many applications across all fields of engineering, and in particular have found recent success in the field of Photonics. In this work we discuss the application of NNs to optical interferometry for the purpose of quantitative phase imaging (QPI). We show that NNs are capable of quantifying the optical pathlength difference in an interferogram with sensitivities that achieve the fundamental limit given by the Cramér-Rao bound (CRB). As an application, we consider a particular QPI technique known as wavelength shifting interferometry (WSI) which obtains the OPL by acquiring multiple interferograms at different, evenly spaced wavenumbers. Traditional phase demodulation algorithms for WSI fail to reach the theoretical OPL sensitivity limit set by the CRB. We have designed NNs which are capable of achieving this bound across a wide range of OPL differences. The NNs are trained on simulated data, and then applied to experimental data. In both simulation and experiment, the NNs outperform the existing analytical demodulation techniques and provide highly sensitive signal demodulation in cases where the analytical approach fails. Thus, NNs provide better performance and more flexibility in the design and use of a WSI system. We expect that the techniques developed in this work can be extended to other two-beam interference based QPI system.