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Many deep learning-based models have been introduced in finger vein recognition in recent years. These solutions, however, suffer from data dependency and are difficult to achieve model generalization. To address this problem, we are inspired by the idea of domain adaptation and propose a universal learning-based framework, which achieves generalization while training with limited data. To reduce differences between data distributions, a compressed U-Net is introduced as a domain mapper to map the raw region of interest image onto a target domain. The concentrated target domain is a unified feature space for the subsequent matching, in which a local descriptor model SOSNet is employed to embed patches into descriptors measuring the similarity of matching pairs. In the proposed framework, the domain mapper is an approximation to a specific extraction function thus the training is only a one-time effort with limited data. Moreover, the local descriptor model can be trained to be representative enough based on a public dataset of non-finger-vein images. The whole pipeline enables the framework to be well generalized, making it possible to enhance universality and helps to reduce costs of data collection, tuning and retraining. The comparable experimental results to state-of-the-art (SOTA) performance in five public datasets prove the effectiveness of the proposed framework. Furthermore, the framework shows application potential in other vein-based biometric recognition as well.