  In this paper, we investigate the logarithmic nonlinear Schr\"odinger (LNLS)
equation with the parity-time (PT)-symmetric harmonic potential, which is an
important physical model in many fields such as nuclear physics, quantum
optics, magma transport phenomena, and effective quantum gravity. Three types
of initial value conditions and periodic boundary conditions are chosen to
solve the LNLS equation with PT -symmetric harmonic potential via the
physics-informed neural networks (PINNs) deep learning method, and these
obtained results are compared with ones deduced from the Fourier spectral
method. Moreover, we also investigate the effectiveness of the PINNs deep
learning for the LNLS equation with PT symmetric potential by choosing the
distinct space widths or distinct optimized steps. Finally, we use the PINNs
deep learning method to effectively tackle the data-driven discovery of the
LNLS equation with PT -symmetric harmonic potential such that the coefficients
of dispersion and nonlinear terms or the amplitudes of PT -symmetric harmonic
potential can be approximately found.
