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In real-world industrial processes, operating modes frequently change due to varying working environments, feedstocks, and set points. There are usually no sufficient samples in the start-up stage of the new mode. Transfer learning methods can establish a satisfying prediction model for critical quality variables with a limited amount of new mode data, which enables real-time quality monitoring across modes. However, existing transfer learning methods assume a large domain-overlapping region in quality value ranges between the original and the new operating mode, which is inconsistent with reality. This paper proposes a weighted feature alignment deep transfer regression framework to address the cold-start quality prediction under operating modes switching in industrial processes. This framework introduces sample weights for the feature alignment, which promotes the alignment in the domain-overlapping region and undermines the alignment in the extrapolation region. Besides, this framework incorporates a regularization term to preserve the feature scale after feature alignment. The effectiveness and superiority of the proposed method is verified by a numerical case and two industrial processes.