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Beyond-two-point statistics contain additional information on cosmological aswell as astrophysical and observational (systematics) parameters. In thismethodology paper we provide an end-to-end simulation-based analysis of a setof Gaussian and non-Gaussian weak lensing statistics using detailed mockcatalogues of the Dark Energy Survey. We implement: 1) second and thirdmoments; 2) wavelet phase harmonics (WPH); 3) the scattering transform (ST).Our analysis is fully based on simulations, it spans a space of seven $\nuw$CDM cosmological parameters, and it forward models the most relevant sourcesof systematics of the data (masks, noise variations, clustering of the sources,intrinsic alignments, and shear and redshift calibration). We implement aneural network compression of the summary statistics, and we estimate theparameter posteriors using a likelihood-free-inference approach. We validatethe pipeline extensively, and we find that WPH exhibits the strongestperformance when combined with second moments, followed by ST. and then bythird moments. The combination of all the different statistics further enhancesconstraints with respect to second moments, up to 25 per cent, 15 per cent, and90 per cent for $S_8$, $\Omega_{\rm m}$, and the Figure-Of-Merit ${\rmFoM_{S_8,\Omega_{\rm m}}}$, respectively. We further find that non-Gaussianstatistics improve constraints on $w$ and on the amplitude of intrinsicalignment with respect to second moments constraints. The methodologicaladvances presented here are suitable for application to Stage IV surveys fromEuclid, Rubin-LSST, and Roman with additional validation on mock catalogues foreach survey. In a companion paper we present an application to DES Year 3 data.

Dark Energy Survey Year 3 results: simulation-based cosmological  inference with wavelet harmonics, scattering transforms, and moments of weak  lensing mass maps I: validation on simulations