## 7 Conclusions  
In this work, we detailed the design and evaluation of an incremental grey-box welding current prediction model for the detection of anomalous welds in a steel mill production line. This model displays two major advantages over state-of-the-art statistical models. First, it is possible to model all welding programs in a single unified model, whereas statistical approaches consist of independent models. When external factors change the underlying system, our unified model adapts more efficiently than the independent statistical models. Time and resources will be saved this way, as fewer welds will be incorrectly flagged as anomalous. Secondly, the grey-box model is based on physical laws and can be easily interpreted. This interpretability helps process engineers to trust the model and may lead to additional insights related to the welding process.  
Our evaluations utilize 15 months of data collected from the ArcelorMittal Ghent site. The results show that the incremental current prediction model performs better than a non-incremental model (around 0.1 lower percentage error) and that our model outperforms the statistical models (between 0.1 and 0.4 lower percentage error). When using an expert-defined anomaly threshold of three percent, our incremental model results in around two-thirds fewer rejected welds. A non-incremental version of our model performs similarly to the non-incremental statistical model, though our model has the advantage of being interpretable.  
Their dependency on physical knowledge forms both the strength and limiting factor of grey-box models. In this work, we achieved good results using commonly-known  
Figure 4: Cumulative number of rejected welds (on a total of 15,231 welds) over time for models trained exclusively on data before the first transition, except for the oracle model, which was trained on all weld data. Welds are rejected when the prediction error is larger than 3%. The average-based models have noticeable jumps after both transition periods; the timings of the jumps can be attributed to batches that require different welding programs. The incremental models only exhibit minor jumps from this effect.  
physical laws, but physics knowledge for other use-cases might be incomplete or not readily available to industry experts. We believe that efficient knowledge exchange between machine learning and physics experts will prove crucial for future success stories.  
Various paths for further research remain. In this work, we used an expert-based threshold for classifying anomalies. By using models aware of uncertainty, it could be possible to let the model decide this boundary. Predicting other features, such as temperature, or combining different features into a single anomaly detection model is also an interesting path. Of course, applications need not be limited to the welding domain and can be found for many industry domains. Finally, approaches to create models using incomplete physical knowledge would also have a significant impact.  
Conceptualization, A.V.Y.-D.D. and J.D.; Formal analysis, D.D.P.; Funding acquisition, S.V.H.; Investigation, D.D.P.; Methodology, D.D.P.; Project administration, A.V.Y.-D.D.; Resources, A.V.Y.-D.D. and J.D.; Software, D.D.P.; Supervision, A.V.Y.-D.D. and S.V.H.; Visualization, D.D.P.; Writing--original draft, D.D.P.; Writing--review and editing, D.D.P. and S.V.H. All authors have read and agreed to the published version of the manuscript.  
This research was funded by Flanders Innovation & Entrepreneurship (VLAIO), Ghent University and imec, and has been carried out in the framework of the VLAIO O&O TOBIAS project under grant number HBC.2017.1002.  
Not applicable.  
The authors declare no conflict of interest.