A Kronecker product accelerated efficient sparse Gaussian Process (E-SGP) for flow emulation

In this paper, we introduce an efficient sparse Gaussian process (E-SGP) for the surrogate modelling of fluid mechanics. This novel Bayesian machine learning algorithm allows efficient model training using databases of different structures. It is a further development of the approximated sparse GP algorithm, combining the concept of efficient GP (E-GP) and variational energy free sparse Gaussian process (VEF-SGP). The developed E-SGP approach exploits the arbitrariness of inducing points and the monotonically increasing nature of the objective function with respect to the number of inducing points in VEF-SGP. By specifying the inducing points on the orthogonal grid/input subspace and using the Kronecker product, E-SGP significantly improves computational efficiency without imposing any constraints on the covariance matrix or increasing the number of parameters that need to be optimised during training. The E-SGP algorithm developed in this paper outperforms E-GP not only in scalability but also in model quality in terms of mean standardized logarithmic loss (MSLL). The computational complexity of E-GP suffers from the cubic growth regarding the growing structured training database. However, E-SGP maintains computational efficiency whilst the resolution of the model, (i.e., the number of inducing points) remains fixed. The examples show that E-SGP produces more accurate predictions in comparison with E-GP when the model resolutions are similar in both. E-GP benefits from more training data but comes with higher computational demands, while E-SGP achieves a comparable level of accuracy but is more computationally efficient, making E-SGP a potentially preferable choice for fluid mechanic problems. Furthermore, E-SGP can produce more reasonable estimates of model uncertainty, whilst E-GP is more likely to produce over-confident predictions.