This paper proposes a novel uncertainty-adjusted label transition (UALT) method for weakly supervised solar panel mapping (WS-SPM) in aerial Images. In weakly supervised learning (WSL), the noisy nature of pseudo labels (PLs) often leads to poor model performance. To address this problem, we formulate the task as a label-noise learning problem and build a statistically consistent mapping model by estimating the instance-dependent transition matrix (IDTM). We propose to estimate the IDTM with a parameterized label transition network describing the relationship between the latent clean labels and noisy PLs. A trace regularizer is employed to impose constraints on the form of IDTM for its stability. To further reduce the estimation difficulty of IDTM, we incorporate uncertainty estimation to first improve the accuracy of noisy dataset distillation and then mitigate the negative impacts of falsely distilled examples with an uncertainty-adjusted re-weighting strategy. Extensive experiments and ablation studies on two challenging aerial data sets support the validity of the proposed UALT.