The continuously growing amount of monitored data in the Industry 4.0 context requires strong and reliable anomaly detection techniques. The advancement of Digital Twin technologies allows for realistic simulations of complex machinery, therefore, it is ideally suited to generate synthetic datasets for the use in anomaly detection approaches when compared to actual measurement data. In this paper, we present novel weakly-supervised approaches to anomaly detection for industrial settings. The approaches make use of a Digital Twin to generate a training dataset which simulates the normal operation of the machinery, along with a small set of labeled anomalous measurement from the real machinery. In particular, we introduce a clustering-based approach, called Cluster Centers (CC), and a neural architecture based on the Siamese Autoencoders (SAE), which are tailored for weakly-supervised settings with very few labeled data samples. The performance of the proposed methods is compared against various state-of-the-art anomaly detection algorithms on an application to a real-world dataset from a facility monitoring system, by using a multitude of performance measures. Also, the influence of hyper-parameters related to feature extraction and network architecture is investigated. We find that the proposed SAE based solutions outperform state-of-the-art anomaly detection approaches very robustly for many different hyper-parameter settings on all performance measures.