  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.
