  Wide binaries play a crucial role in analyzing the birth environment of stars
and the dynamical evolution of clusters. When wide binaries are located at
greater distances, their companions may overlap in the observed images,
becoming indistinguishable and resulting in unresolved wide binaries, which are
difficult to detect using traditional methods. Utilizing deep learning, we
present a method to identify unresolved wide binaries by analyzing the
point-spread function (PSF) morphology of telescopes. Our trained model
demonstrates exceptional performance in differentiating between single stars
and unresolved binaries with separations ranging from 0.1 to 2 physical pixels,
where the PSF FWHM is ~2 pixels, achieving an accuracy of 97.2% for simulated
data from the Chinese Space Station Telescope. We subsequently tested our
method on photometric data of NGC 6121 observed by the Hubble Space Telescope.
The trained model attained an accuracy of 96.5% and identified 18 wide binary
candidates with separations between 7 and 140 au. The majority of these wide
binary candidates are situated outside the core radius of NGC 6121, suggesting
that they are likely first-generation stars, which is in general agreement with
the results of Monte Carlo simulations. Our PSF-based method shows great
promise in detecting unresolved wide binaries and is well suited for
observations from space-based telescopes with stable PSF. In the future, we aim
to apply our PSF-based method to next-generation surveys such as the China
Space Station Optical Survey, where a larger-field-of-view telescope will be
capable of identifying a greater number of such wide binaries.
