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

$N$-double poles solutions for nonlocal Hirota equation with nonzero boundary conditions using Riemann-Hilbert method and PINN algorithm