We train graph neural networks on halo catalogues from Gadget N-body simulations to perform field-level likelihood-free inference of cosmological parameters. The catalogues contain $\lesssim$5,000 halos with masses $\gtrsim 10^{10}~h^{-1}M_\odot$ in a periodic volume of $(25~h^{-1}{\rm Mpc})^3$; every halo in the catalogue is characterized by several properties such as position, mass, velocity, concentration, and maximum circular velocity. Our models, built to be permutationally, translationally, and rotationally invariant, do not impose a minimum scale on which to extract information and are able to infer the values of $\Omega_{\rm m}$ and $\sigma_8$ with a mean relative error of $\sim6\%$, when using positions plus velocities and positions plus masses, respectively. More importantly, we find that our models are very robust: they can infer the value of $\Omega_{\rm m}$ and $\sigma_8$ when tested using halo catalogues from thousands of N-body simulations run with five different N-body codes: Abacus, CUBEP$^3$M, Enzo, PKDGrav3, and Ramses. Surprisingly, the model trained to infer $\Omega_{\rm m}$ also works when tested on thousands of state-of-the-art CAMELS hydrodynamic simulations run with four different codes and subgrid physics implementations. Using halo properties such as concentration and maximum circular velocity allow our models to extract more information, at the expense of breaking the robustness of the models. This may happen because the different N-body codes are not converged on the relevant scales corresponding to these parameters.

使用图神经网络在Halo目录上进行训练，实现宇宙学参数无似然场水平的推断。该文章表明，其模型在使用多个不同的N-body模拟和代码运作的Halo目录进行测试时非常健壮，同时探讨了使用多种参数会受到精度和健壮性的权衡问题。

暗物质晕的稳健场级推断