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Automatic estimation of pain intensity from facial expressions in videos has an immense potential in health care applications. However, domain adaptation (DA) is needed to alleviate the problem of domain shifts that typically occurs between video data captured in source and target do-mains. Given the laborious task of collecting and annotating videos, and the subjective bias due to ambiguity among adjacent intensity levels, weakly-supervised learning (WSL)is gaining attention in such applications. Yet, most state-of-the-art WSL models are typically formulated as regression problems, and do not leverage the ordinal relation between intensity levels, nor the temporal coherence of multiple consecutive frames. This paper introduces a new deep learn-ing model for weakly-supervised DA with ordinal regression(WSDA-OR), where videos in target domain have coarse la-bels provided on a periodic basis. The WSDA-OR model enforces ordinal relationships among the intensity levels as-signed to the target sequences, and associates multiple relevant frames to sequence-level labels (instead of a single frame). In particular, it learns discriminant and domain-invariant feature representations by integrating multiple in-stance learning with deep adversarial DA, where soft Gaussian labels are used to efficiently represent the weak ordinal sequence-level labels from the target domain. The proposed approach was validated on the RECOLA video dataset as fully-labeled source domain, and UNBC-McMaster video data as weakly-labeled target domain. We have also validated WSDA-OR on BIOVID and Fatigue (private) datasets for sequence level estimation. Experimental results indicate that our approach can provide a significant improvement over the state-of-the-art models, allowing to achieve a greater localization accuracy.