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

arXiv: https://arxiv.org/abs/2312.10023 · HTML: https://mlsanity.com/paper/2312.10023/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 38 references that are in the map.
- First posted on arXiv 13 Dec 2023.

## Builds on (most cited first, 19 of 19)

- [Machine Learning for Fluid Mechanics](https://mlsanity.com/paper/1905.11075/) (2019)
- [Gaussian Processes for Big Data](https://mlsanity.com/paper/1309.6835/) (2013)
- [Scalable Variational Gaussian Process Classification](https://mlsanity.com/paper/1411.2005/) (2014)
- [When Gaussian Process Meets Big Data: A Review of Scalable GPs](https://mlsanity.com/paper/1807.01065/) (2018)
- [Kernel Interpolation for Scalable Structured Gaussian Processes (KISS-GP)](https://mlsanity.com/paper/1503.01057/) (2015)
- [Practical Deep Learning with Bayesian Principles](https://mlsanity.com/paper/1906.02506/) (2019)
- [Machine Learning of Linear Differential Equations using Gaussian Processes](https://mlsanity.com/paper/1701.02440/) (2017)
- [Solving and Learning Nonlinear PDEs with Gaussian Processes](https://mlsanity.com/paper/2103.12959/) (2021)
- [Numerical Gaussian Processes for Time-dependent and Non-linear Partial Differential Equations](https://mlsanity.com/paper/1703.10230/) (2017)
- [Adversarial Examples, Uncertainty, and Transfer Testing Robustness in Gaussian Process Hybrid Deep Networks](https://mlsanity.com/paper/1707.02476/) (2017)
- [Product Kernel Interpolation for Scalable Gaussian Processes](https://mlsanity.com/paper/1802.08903/) (2018)
- [Kernel Interpolation for Scalable Online Gaussian Processes](https://mlsanity.com/paper/2103.01454/) (2021)
- [Efficient Modeling of Latent Information in Supervised Learning using Gaussian Processes](https://mlsanity.com/paper/1705.09862/) (2017)
- [Physics-Aware Gaussian Processes in Remote Sensing](https://mlsanity.com/paper/2012.07986/) (2020)
- [Applying Machine Learning to Study Fluid Mechanics](https://mlsanity.com/paper/2110.02083/) (2021)
- [Structured Bayesian Gaussian process latent variable model: applications to data-driven dimensionality reduction and high-dimensional inversion](https://mlsanity.com/paper/1807.04302/) (2018)
- [Gaussian Process Latent Force Models for Learning and Stochastic Control of Physical Systems](https://mlsanity.com/paper/1709.05409/) (2017)
- [Data-Centric Engineering: integrating simulation, machine learning and statistics. Challenges and Opportunities](https://mlsanity.com/paper/2111.06223/) (2021)
- [Efficient State-Space Inference of Periodic Latent Force Models](https://mlsanity.com/paper/1310.6319/) (2013)

## Abstract (arXiv)

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.