A hybrid data driven-physics constrained Gaussian process regression framework with deep kernel for uncertainty quantification

TL;DR

This paper introduces a hybrid Gaussian process regression framework that integrates physics constraints via deep kernel learning, enabling effective uncertainty quantification with limited data in high-dimensional problems.

Contribution

It proposes a novel physics-constrained GPR model using deep kernels and Boltzmann-Gibbs distribution, improving accuracy and data efficiency.

Findings

Effective uncertainty propagation in high-dimensional problems

Accurate results with limited labeled data

Enhanced model performance through physics integration

Abstract

Gaussian process regression (GPR) has been a well-known machine learning method for various applications such as uncertainty quantifications (UQ). However, GPR is inherently a data-driven method, which requires sufficiently large dataset. If appropriate physics constraints (e.g. expressed in partial differential equations) can be incorporated, the amount of data can be greatly reduced and the accuracy further improved. In this work, we propose a hybrid data driven-physics constrained Gaussian process regression framework. We encode the physics knowledge with Boltzmann-Gibbs distribution and derive our model through maximum likelihood (ML) approach. We apply deep kernel learning method. The proposed model learns from both data and physics constraints through the training of a deep neural network, which serves as part of the covariance function in GPR. The proposed model achieves good results in high-dimensional problem, and correctly propagate the uncertainty, with very limited labelled data provided.