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Estimation of pain intensity from facial expressions captured in videos has an immense potential for health care applications. Given the challenges related to subjective variations of facial expressions, and operational capture conditions, the accuracy of state-of-the-art DL models for recognizing facial expressions may decline. Domain adaptation has been widely explored to alleviate the problem of domain shifts that typically occur between video data captured across various source and target domains. Moreover, given the laborious task of collecting and annotating videos, and subjective bias due to ambiguity among adjacent intensity levels, weakly-supervised learning is gaining attention in such applications. State-of-the-art WSL models are typically formulated as regression problems, and do not leverage the ordinal relationship among pain intensity levels, nor temporal coherence of multiple consecutive frames. This paper introduces a new DL model for weakly-supervised DA with ordinal regression that can be adapted using target domain videos with coarse labels provided on a periodic basis. The WSDA-OR model enforces ordinal relationships among intensity levels assigned to target sequences, and associates multiple relevant frames to sequence-level labels. In particular, it learns discriminant and domain-invariant feature representations by integrating multiple instance learning with deep adversarial DA, where soft Gaussian labels are used to efficiently represent the weak ordinal sequence-level labels from target domain. The proposed approach was validated using RECOLA video dataset as fully-labeled source domain data, and UNBC-McMaster shoulder pain video dataset as weakly-labeled target domain data. We have also validated WSDA-OR on BIOVID and Fatigue datasets for sequence level estimation.