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This study aims to test the validity of general relativity (GR) on kiloparsecscales by employing a newly compiled galaxy-scale strong gravitational lensing(SGL) sample. We utilize the distance sum rule within theFriedmann-Lema\^{\i}tre-Robertson-Walker metric to obtain cosmology-independentconstraints on both the parameterized post-Newtonian parameter $\gamma_{\rmPPN}$ and the spatial curvature $\Omega_{k}$, which overcomes the circularityproblem induced by the presumption of a cosmological model grounded in GR. Tocalibrate the distances in the SGL systems, we introduce a novel nonparametricapproach, Artificial Neural Network (ANN), to reconstruct a smoothdistance--redshift relation from the Pantheon+ sample of type Ia supernovae.Our results show that $\gamma_{\rm PPN}=1.16_{-0.12}^{+0.15}$ and$\Omega_k=0.89_{-1.00}^{+1.97}$, indicating a spatially flat universe with theconservation of GR (i.e., $\Omega_k=0$ and $\gamma_{\rm PPN}=1$) is basicallysupported within $1\sigma$ confidence level. Assuming a zero spatial curvature,we find $\gamma_{\rm PPN}=1.09_{-0.10}^{+0.11}$, representing an agreement withthe prediction of 1 from GR to a 9.6\% precision. If we instead assume GR holds(i.e., $\gamma_{\rm PPN}=1$), the curvature parameter constraint can be furtherimproved to be $\Omega_k=0.11_{-0.47}^{+0.78}$. These resulting constraintsdemonstrate the effectiveness of our method in testing GR on galactic scales bycombining observations of strong lensing and the distance--redshift relationreconstructed by ANN.

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