Using proper orthogonal decomposition for data-driven model error correction for heat conduction problems with uncertain boundary conditions

In applications such as digital twins, models must combine high accuracy with low computational cost. Reduced order modelling, often involving dimensionality reduction, together with data-driven model error correction, can make it possible to achieve both objectives. We focus on the use of proper orthogonal decomposition (POD), together with sample solutions from the available physics-based models, to perform dimensionality reduction for reduced-order models, and on considerations that must be made when these are to be corrected using a data-driven model. If only sample solutions from the best available full-order model are used in the POD procedure, the accuracy of the corrected models may be limited before the data-driven error-correcting model is even trained, because the true solution fields cannot be represented in the POD subspace. To overcome this problem, we propose expanding the POD subspace based on sample solutions reflecting anticipated uncertainty in the full order model. The proposed method is demonstrated using two examples involving heat conduction with uncertain boundary conditions. The results show that the expanded POD subspaces allow the corrected models to outperform the original full-order model, while the same is not true for the original POD subspaces.