Search papers, labs, and topics across Lattice.
This paper introduces a novel framework for predicting multiple high-dimensional physical fields under linear equality constraints using Gaussian process (GP) regression. By employing a row-wise PCA technique that preserves physical constraints in the latent space, the authors address the limitations of traditional GP models, which often struggle with high dimensionality and arbitrary output deductions. The proposed method demonstrates improved predictive accuracy and uncertainty quantification in benchmark applications, including population dynamics and computational fluid dynamics (CFD) scenarios involving the Reynolds stress tensor.
Predictive accuracy and uncertainty quantification improve significantly when using a row-wise PCA approach that respects physical constraints in multi-output Gaussian processes.
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.
