Uncertainty-Aware Symbolic Regression for Galactic Kinematics: Data-Driven Modeling of the SPARC Radial Acceleration Relation

Recent applications of Symbolic Regression (SR) have demonstrated the capacity to extract phenomenological laws bridging baryonic and observed accelerations in galaxies. However, current SR implementations in this domain often rely on uniform loss functions (e.g., Ordinary Least Squares), neglecting the highly heterogeneous observational uncertainties inherent in galactic rotation curves. In this work, we introduce an uncertainty-aware SR pipeline utilizing a variance-weighted objective function. Applying this to 2,696 quality-controlled data points from 153 galaxies in the SPARC database, we demonstrate that incorporating observational error priors fundamentally alters the algorithm's evolutionary trajectory. Comparing unweighted and weighted models, we show that uncertainty weighting prevents overfitting in the densely sampled Newtonian regime. Without imposing specific rational interpolating functions, the weighted algorithm autonomously recovers the deep-MOND asymptotic limit and identifies a highly parsimonious fractional power-law scaling (g_obs ∝ g_bar^0.625) for the intermediate regime. While recognizing the limitations of fixed mass-to-light ratios and y-axis-only error weighting, we demonstrate that uncertainty-weighted SR provides a more robust framework for exploring kinematic relations in noisy astronomical data.