Solar panel mapping from high-resolution aerial images is becoming increasingly crucial to grid planning and operation, where weakly supervised approach has been explored. To cope with the noisy nature of pseudo-labels (PLs) generated by weakly supervised object localization, we propose an effective uncertainty-aware forward correction (UA-FC) method to learn clean predictions from the noisy PLs. The proposed method consists of two steps: heteroscedastic uncertainty estimation and forward correction procedure. The purpose of the first step is to produce uncertainty as an indicator for the instance-dependent noise. The second step includes a target mapping network to produce clean predictions and a transition function to model the relationship between clean predictions and noisy PLs. As estimating every probability of one class flipped into another is difficult and time-consuming, we introduce heteroscedastic uncertainty as a measurement and propose an uncertainty-based transformation function to map clean predictions into noisy ones. By minimizing the errors between the noisy predictions and noisy labels, the target mapping network is able to offer clean predictions close to the actual objects. Extensive experiments on an aerial dataset reveal that the proposed method outperforms other state-of-the-art methods by a large margin, especially in recovering the boundary of the objects.