  We present simulation-based cosmological $w$CDM inference using Dark Energy
Survey Year 3 weak-lensing maps, via neural data compression of weak-lensing
map summary statistics: power spectra, peak counts, and direct map-level
compression/inference with convolutional neural networks (CNN). Using
simulation-based inference, also known as likelihood-free or implicit
inference, we use forward-modelled mock data to estimate posterior probability
distributions of unknown parameters. This approach allows all statistical
assumptions and uncertainties to be propagated through the forward-modelled
mock data; these include sky masks, non-Gaussian shape noise, shape measurement
bias, source galaxy clustering, photometric redshift uncertainty, intrinsic
galaxy alignments, non-Gaussian density fields, neutrinos, and non-linear
summary statistics. We include a series of tests to validate our inference
results. This paper also describes the Gower Street simulation suite: 791
full-sky PKDGRAV dark matter simulations, with cosmological model parameters
sampled with a mixed active-learning strategy, from which we construct over
3000 mock DES lensing data sets. For $w$CDM inference, for which we allow
$-1<w<-\frac{1}{3}$, our most constraining result uses power spectra combined
with map-level (CNN) inference. Using gravitational lensing data only, this
map-level combination gives $\Omega_{\rm m} = 0.283^{+0.020}_{-0.027}$, ${S_8 =
0.804^{+0.025}_{-0.017}}$, and $w < -0.80$ (with a 68 per cent credible
interval); compared to the power spectrum inference, this is more than a factor
of two improvement in dark energy parameter ($\Omega_{\rm DE}, w$) precision.
