  We systematically investigate the nonlocal Hirota equation with nonzero
boundary conditions via Riemann-Hilbert method and multi-layer physics-informed
neural networks algorithm. Starting from the Lax pair of nonzero nonlocal
Hirota equation, we first give out the Jost function, scattering matrix, their
symmetry and asymptotic behavior. Then, the Riemann-Hilbert problem with
nonzero boundary conditions are constructed and the precise formulaes of
$N$-double poles solutions and $N$-simple poles solutions are written by
determinants. Different from the local Hirota equation, the symmetry of
scattering data for nonlocal Hirota equation is completely different, which
results in disparate discrete spectral distribution. In particular, it could be
more complicated and difficult to obtain the symmetry of scattering data under
the circumstance of double poles. Besides, we also analyse the asymptotic state
of one-double poles solution as $t\rightarrow \infty$. Whereafter, the PINN
algorithm is applied to research the data-driven soliton solutions of the
nonzero nonlocal Hirota equation by using the training data obtained from the
Riemann-Hilbert method. Most strikingly, the integrable nonlocal equation is
firstly solved via PINN algorithm. As we all know, the nonlocal equations
contain the $\mathcal{PT}$ symmetry $\mathcal{P}:x\rightarrow -x,$ or
$\mathcal{T}:t\rightarrow -t,$ which are different with local ones. Adding the
nonlocal term into the NN, we can successfully solve the integrable nonlocal
Hirota equation by PINN algorithm. The numerical results indicate the algorithm
can well recover the data-driven soliton solutions of the integrable nonlocal
equation. Noteworthily, the inverse problems of the integrable nonlocal
equation are discussed for the first time through applying the PINN algorithm
to discover the parameters of the equation in terms of its soliton solution.
