Multi-output Gaussian processes for the reconstruction of homogenized cross-sections

Special Issue on ‘Overview of recent advances in HPC simulation methods for nuclear applications’, edited by Andrea Zoia, Elie Saikali, Cheikh Diop and Cyrille de Saint Jean

Open Access

Table 1.

Features of the multi-output Gaussian process models studied in this work.

Model	Advantages	Shortcomings	Refs.
ICM	No approximation.	Computational costs are cubic in the number of tasks – although variants of the model lifting this bottleneck using a low-rank cross-task correlation matrix could be devised.	[19, 25]
	Simple expressions and covariance structure.
	Closed-form LOO expressions are available (see 4.3).	Lesser expressiveness: all latent processes share the same kernel (same lengthscales, etc.).
	Accommodates any noise term, thus can be used straightforwardly for uncertainty propagation.
	Can handle missing data [22] (situation where not all tasks are observed at every input training point).	Requires advanced implementations for large-scale variance computation (see [24]) and other advanced functionalities (inducing points approximations, etc.).
	The training data can be updated easily (or even with rank-1 updates [23]), without retraining the model.

VLMC	Computational costs are linear in the number of regression tasks.	“Black-box” model, in which the relationship between the training data and predictions is lost.	[20], [26], [27, 28]
	Very flexible: the latent processes are independent, the loss function (lower bound of the MLL) factors over points and tasks and can be modified in various ways.	In particular, impossible to derive closed-form LOO expressions, or to easily update the training dataset.
	Natively implements an inducing points approximation, which breaks the cubic complexity in the number of datapoints.	Heavily parametrized (all pseudo-input points are parameters).
	Accommodates any noise term, thus can be used straightforwardly for uncertainty propagation.	Difficult to ponder the various available approximations, in particular for obtaining well-calibrated uncertainties.
	Trivially handles settings where the outputs are observed at different input locations.

PLMC	No approximation, but a constrained noise model.	Convergence of training is slower and/or less stable in some settings.	[21, 24, 29]
	Computational costs are linear in the number of regression tasks.	The constraint put on the noise term prevents usage for uncertainty propagation at the moment.
	Flexible: the latent processes are independent conditionally upon the training data, the loss function (MLL) factors over points and tasks (except for some cheap terms).
	Can trivially implement any approximation devised for single-output Gaussian processes, by wrapping the latent processes with it.	Cannot accommodate non-zero mean functions or missing observations at the moment.
	Closed-form LOO expressions are available (see 4.3).
	The training data can be updated easily (or even with rank-1 updates [23]), without retraining the model.

Lazy-LMC	Requires no training.	Non-rigorous model, unable to deliver reliable uncertainty estimations.	2.3, [24]
	Computational costs are linear in the number of regression tasks.	Accuracy not guaranteed if the magnitude of the data noise was misspecified.
	Closed-form LOO expressions are available (see 4.3).
	The training data can be updated easily (or even with rank-1 updates [23]), without retraining the model.	Prone to conditioning issues.
	Can trivially implement any approximation devised for single-output Gaussian processes, by wrapping the latent processes with it.

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.