Gaussian Process Regression with Soft Equality Constraints
Didem Kochan, Xiu Yang · Preprints.org · 2025
This study introduces a novel Gaussian process (GP) regression framework that probabilistically enforces physical constraints, with a particular focus on equality conditions. The GP model is trained using the quantum-inspired Hamiltonian Monte Carlo (QHMC) algorithm, which efficiently samples from a wide range of distributions by allowing a particle's mass matrix to vary according to a probability distribution. By integrating QHMC into the GP regression with probabilistic handling of the constraints, the approach balances the computational cost and accuracy in the resulting GP model as the probabilistic nature of the method contributes to shorter execution times compared with existing GP-based approaches. Additionally, an adaptive learning algorithm is introduced to optimize the selection of constraint locations, further enhancing the method's flexibility. The effectiveness and efficiency of this approach are demonstrated through several applications: estimating hyperparameters for high-dimensional GP models under noisy conditions and reconstructing a sparsely observed steady-state heat transport problem. The numerical results indicate that the proposed approach accelerates the process while maintaining the accuracy.