Incorporating sensor measurements using data assimilation and machine learning to improve the accuracy of thermal finite element models

Abstract

Complex systems are commonly encountered in engineering. Such systems have a degree of unpredictability, which can lead to undesirable results. The digital twin concept has been proposed as a method to obtain more information about the system so that its behaviour can be better predicted. To construct a digital twin of a system, a high-fidelity model that predicts the evolution of the system over time is required. Classically, such high fidelity models would be either physics-based or data-driven, though both of these approaches have disadvantages that may make them unsuitable for application in a digital twin. A hybrid model is a combination of physics-based and data-driven models that seeks to exploit the advantages of both approaches, and is a promising candidate for producing a model that is suitable for application in a digital twin.

This work investigates the training of hybrid models of real engineering systems. It considers systems that are dynamic and that are partially observed. Physics-based models of these systems are available in the form of partial differential equations that are solved numerically using the finite element method. To reduce the computational cost associated with such models, surrogate models are constructed. The construction of surrogate models involves a dimensionality reduction step in the form of proper orthogonal decomposition (POD), as well as a prediction step that involves the training of a data-driven model to predict the evolution of the POD coefficients over time. Hybrid models are then trained using the surrogate models as their physics-based component. The training of hybrid models takes place using a combination of data assimilation and machine learning. The machine learning-data assimilation (ML-DA) algorithm that is documented in the literature is used, together with two algorithms proposed in this work, called the data assimilation-observation (DA-O) and per-step DA-O algorithms.

It is the nature of the application of this methodology of training hybrid models that allows this work to make a contribution relative to the published literature. Previous uses of data assimilation in the training of hybrid models perform investigations using simplified problems that allow direct use of the physics-based model in the hybrid model. Meanwhile, the increased computational demands of the real engineering problem considered in this work necessitates the use of a surrogate model of the physics-based component of the hybrid model. Surrogate models have been previously applied in hybrid models constructed to solve engineering problems. However, these approaches do not apply naturally to applications such as digital twins where observations are continuously available. The use of data assimilation in this work allows it to address this shortcoming.

The proposed methodology of training hybrid models is evaluated using two case studies. The first case study involves a thermal simulation model of a small section of the freeboard of a process converter. Surrogate models of the simulation model are constructed using different data-driven function approximation techniques, such as Gaussian process regression, Support Vector Machines (SVMs) and neural networks. These surrogate models are then used to train hybrid models using simulated observations and the DA-O, per-step DA-O and ML-DA algorithms. When 30 of the 29077 nodes of the simulation model are observed, the per-step DA-O algorithm produces the best hybrid model in terms of a root mean square error (RMSE) metric calculated using the analysis states estimated during data assimilation. The ML-DA algorithm which performed next best is, however, easier to apply to different numbers of observed nodes and is potentially less sensitive to observation noise. The ML-DA algorithm is subsequently used to investigate the effect of using different numbers of observed nodes. While there is a benefit to using a greater number of observed nodes, the trained hybrid models still outperform the physics-based model when as few as two of the 29077 nodes of the simulation model are observed. These results indicate that the training of hybrid models for sparsely observed systems is feasible.

The second case study considered in this work involves a thermal half model of the process converter freeboard for which actual sensor observations are available. Surrogate models are again constructed using different data-driven function approximation techniques, and these are used in the training of hybrid models. Only the ML-DA algorithm is now used for the training of hybrid models. Simulated sensor observations are used at first to understand whether improvements in predictive performance that hybrid models make relative to physics-based models on the observed nodes extend to larger subsets of the nodes of the system. It is found that when the performance of the hybrid models is evaluated in terms of the RMSE calculated using analysis states estimated during data assimilation, it is possible that improvements in performance are made on the observed nodes but not on larger subsets of the nodes. When the RMSE is instead calculated using predictions of the evolution of the system over 30 time steps, the performance of the hybrid models on the observed nodes correlates with their performance on larger subsets of the nodes. When real observations are used to train hybrid models, the trained hybrid models improve on the performance of physics-based models on the observed nodes in terms of the RMSE calculated using analysis states and in terms of the RMSE calculated using 30 time step predictions. The improvement in performance on the latter could indicate that this improvement on the observed nodes extends to larger subsets of nodes of the system. There are, however, other possible explanations for this improvement.