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Extragalactic Test of General Relativity from Strong Gravitational Lensing by using Artificial Neural Networks

This study aims to test the validity of general relativity (GR) on kiloparsec scales by employing a newly compiled galaxy-scale strong gravitational lensing (SGL) sample. We utilize the distance sum rule within the Friedmann-Lema\^ıtre-Robertson-Walker metric to obtain cosmology-independent constraints on both the parameterized post-Newtonian parameter $γ_{\rm PPN}$ and the spatial curvature $Ω_{k}$, which overcomes the circularity problem induced by the presumption of a cosmological model grounded in GR. To calibrate the distances in the SGL systems, we introduce a novel nonparametric approach, Artificial Neural Network (ANN), to reconstruct a smooth distance--redshift relation from the Pantheon+ sample of type Ia supernovae. Our results show that $γ_{\rm PPN}=1.16_{-0.12}^{+0.15}$ and $Ω_k=0.89_{-1.00}^{+1.97}$, indicating a spatially flat universe with the conservation of GR (i.e., $Ω_k=0$ and $γ_{\rm PPN}=1$) is basically supported within $1σ$ confidence level. Assuming a zero spatial curvature, we find $γ_{\rm PPN}=1.09_{-0.10}^{+0.11}$, representing an agreement with the prediction of 1 from GR to a 9.6\% precision. If we instead assume GR holds (i.e., $γ_{\rm PPN}=1$), the curvature parameter constraint can be further improved to be $Ω_k=0.11_{-0.47}^{+0.78}$. These resulting constraints demonstrate the effectiveness of our method in testing GR on galactic scales by combining observations of strong lensing and the distance--redshift relation reconstructed by ANN.