# Multi-output Gaussian process prediction of physical fields under linear equality constraints

arXiv: https://arxiv.org/abs/2608.25709 · HTML: https://mlsanity.com/paper/2608.25709/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 54 references that are in the map.
- First posted on arXiv 26 Aug 2026.

## Builds on (most cited first, 10 of 10)

- [A Literature Survey of Benchmark Functions For Global Optimization Problems](https://mlsanity.com/paper/1308.4008/) (2013)
- [A Survey of Constrained Gaussian Process Regression: Approaches and Implementation Challenges](https://mlsanity.com/paper/2006.09319/) (2020)
- [Linearly constrained Gaussian processes](https://mlsanity.com/paper/1703.00787/) (2017)
- [Multivariate Functional Principal Component Analysis for Data Observed on Different (Dimensional) Domains](https://mlsanity.com/paper/1509.02029/) (2015)
- [Algorithmic Linearly Constrained Gaussian Processes](https://mlsanity.com/paper/1801.09197/) (2018)
- [Gaussian process emulators for computer experiments with inequality constraints](https://mlsanity.com/paper/1606.01265/) (2016)
- [Optimal quantisation of probability measures using maximum mean discrepancy](https://mlsanity.com/paper/2010.07064/) (2020)
- [The ICSCREAM methodology: Identification of penalizing configurations in computer experiments using screening and metamodel -- Applications in thermal-hydraulics](https://mlsanity.com/paper/2004.04663/) (2020)
- [Incorporating Sum Constraints into Multitask Gaussian Processes](https://mlsanity.com/paper/2202.01793/) (2022)
- [E-LMC: Extended Linear Model of Coregionalization for Spatial Field Prediction](https://mlsanity.com/paper/2203.00525/) (2022)

## Abstract (arXiv)

We address the simultaneous prediction of multiple high-dimensional physical fields governed by linear equality constraints, a setting that arises in many real-world applications in physics machine learning. Gaussian process (GP) regression is a widely used surrogate modeling approach due to its effectiveness in small-sample regimes and its ability to provide uncertainty quantification. However, applying GP models in this setting raises two major challenges: the high dimensionality of the discretized output fields and the enforcement of the physical constraint in predictions. For the latter, a common strategy consists in deducing one output from the others via the constraint relation. Through a benchmark, we show that this deductive approach is sensitive to the arbitrary choice of which output to deduce, affecting both predictive accuracy and uncertainty quantification. Consequently, there is a need for an approach that treats all fields symmetrically while strictly respecting the underlying physics. Motivated by these limitations, we propose a robust framework for jointly modeling constrained multi-field data. Our approach first leverages a specific PCA procedure for multi-field data, coined row-wise PCA, which has the interesting property of preserving the constraint in the latent space. Since standard PCA strategies for multi-field data do not preserve such constraints, we investigate theoretically the optimality of the row-wise choice. In a second step, we consider a linearly-constrained multi-output GP approach based on a specific kernel parametrization which is trained on the latent space of row-wise PCA. The proposed framework is validated on a population dynamics problem and on an industrial CFD application, which involves the prediction of Reynolds stress tensor components under the incompressibility constraint.