  The Lyman-$\alpha$ forest offers a unique avenue for studying the
distribution of matter in the high redshift universe and extracting precise
constraints on the nature of dark matter, neutrino masses, and other
$\Lambda$CDM extensions. However, interpreting this observable requires
accurate modelling of the thermal and ionisation state of the intergalactic
medium, and therefore resorting to computationally expensive hydrodynamical
simulations. In this work, we build a neural network that serves as a surrogate
model for rapid predictions of the one-dimensional \lya flux power spectrum
($P_{\rm 1D}$), thereby making Bayesian inference feasible for this observable.
Our emulation technique is based on modelling $P_{\rm 1D}$ as a function of the
slope and amplitude of the linear matter power spectrum rather than as a
function of cosmological parameters. We show that our emulator achieves
sub-percent precision across the full range of scales ($k_{\parallel }=0.1$ to
4Mpc$^{-1}$) and redshifts ($z=2$ to 4.5) considered, and also for three
$\Lambda$CDM extensions not included in the training set: massive neutrinos,
running of the spectral index, and curvature. Furthermore, we show that it
performs at the 1% level for ionisation and thermal histories not present in
the training set and performs at the percent level when emulating down to
$k_{\parallel}$=8Mpc$^{-1}$. These results affirm the efficacy of our emulation
strategy in providing accurate predictions even for cosmologies and
reionisation histories that were not explicitly incorporated during the
training phase, and we expect it to play a critical role in the cosmological
analysis of the DESI survey.
