## 1 Introduction  
In this paper, we focus on a challenge encountered in the billion-dollar steel production industry: the optimization of the mash seam welding process used in steel mills. If a weld breaks while passing through the annealing furnace, the steel sheets become jammed, and the production line has to be stopped. To restore the line, workers have to enter the furnace after it has been cooled and vented, exposing them to a high-risk environment. As the downtime typically lasts multiple days, this also incurs major costs for the otherwise continuous production line.  
Detecting weak welds is a form of anomaly detection, i.e., anomalous measurements made during the welding process may reflect a deviation in the material or machine and indicate a poor quality weld. [PERSON] states that expectation-based anomaly detection methods are straightforward to apply in domains where data exhibits clear patterns [1]. As such, these methods are especially suited to the seam welding context since the welding process is highly repetitive and exhibits very few poor welds. Expectation-based models predict features (e.g., welding current) based on known input parameters, which are then compared with measured values. When these values differ significantly, the weld is considered anomalous and is replaced before entering the annealing furnace. Expectation-based statistical models are currently considered the state-of-the-art anomaly detection methods for mash seam welding. These models estimate a feature as the average of recent measurements made using the same welding program. Though statistical models have been adopted by the industry, they are, in fact, black-box models that do not give insight as to why a specific feature value is expected. This is unfortunate as insight in anomaly detection models (or machine learning models in general) helps users to trust the model and can lead to a better understanding of the underlying process. In contrast, white-box models are built using expert rules and physical knowledge of the process and are fully interpretable as a result. Unfortunately, theoretical knowledge is often insufficient to accurately model a complex real-world system.  
So-called grey-box models combine the strengths of both approaches. They start from theoretical knowledge but use a data-driven approach to better tune the model to real-world observations. In this work, we investigate whether a grey-box model can outperform state-of-the-art black-box statistical models for predicting welding current in mash seam welding and place these results in the context of anomaly detection.  
Results show that our model has the comparable predictive performance to statistical models when there are no changes to the underlying behavior. However, when faced with changing behavior due to machine maintenance or material-related changes, our incremental model is significantly faster to adapt than state-of-the-art statistical models, resulting in fewer rejected welds. We used 15 months of welding data collected at the AccelerMittl Ghent site for our evaluation and found that our grey-box model resulted in two-thirds fewer rejected welds compared to a statistical model using similar features.  
We believe that grey-box models can be a useful tool beyond welding machines. Their ability to provide interpretable results and adapt quickly to real-world changes could be a powerful asset in many industries.  
Our work is structured as follows. First, we provide some background information in Section 2 and list state-of-the-art weld quality assessment techniques in Section 3. Section 4 describes the data we used for our experiments and evaluations. We describe our base model in Section 5, and extend it to an incremental model in Section 6. Finally, we conclude our findings in Section 7.