Physics-driven deep learning (PD-DL) models have proven to be a powerful approach for improved reconstruction of rapid MRI scans. In order to train these models in scenarios where fully-sampled reference data is unavailable, self-supervised learning has gained prominence. However, its application at high acceleration rates frequently introduces artifacts, compromising image fidelity. To mitigate this shortcoming, we propose a novel way to train PD-DL networks via carefully-designed perturbations. In particular, we enhance the k-space masking idea of conventional self-supervised learning with a novel consistency term that assesses the model's ability to accurately predict the added perturbations in a sparse domain, leading to more reliable and artifact-free reconstructions. The results obtained from the fastMRI knee and brain datasets show that the proposed training strategy effectively reduces aliasing artifacts and mitigates noise amplification at high acceleration rates, outperforming state-of-the-art self-supervised methods both visually and quantitatively.