Deep unfolding networks have recently emerged as a promising approach for synthetic aperture radar (SAR) imaging. However, baseline unfolding networks, typically derived from iterative reconstruction algorithms such as the alternating direction method of multipliers (ADMM), lack generalization capability across scenes, primarily because their regularizers are empirically designed rather than learned from data. In this study, we introduce a learnable regularizer into the unfolding network and propose a SAR imaging network with adaptive regularization (ARSAR-Net), which aims to generalize across heterogeneous scenes including offshore ships, islands, urban areas, and mountainous terrain. Furthermore, two variants of ARSAR-Net are developed, targeting improved imaging efficiency and reconstruction quality, respectively. Extensive validation through simulated and real-data experiments demonstrates three key advantages of ARSAR-Net: (1) a 50% increase in imaging speed over existing unfolding networks, (2) a PSNR gain of up to 2.0 dB in imaging quality, and (3) enhanced adaptability to complex scenes. These advancements establish a new paradigm for computationally efficient and generalizable SAR imaging systems.