Physics-constrained deep kernel learning for inverse problems with noisy data

We propose a novel physics-constrained deep kernel learning (PCDKL) to estimate physical parameters and learn forward solutions for problems described by partial differential equations (PDEs) and noisy data. In this framework, a Gaussian Process (GP) with a deep kernel is constructed to model the forward solution. The posterior function samples from the GP serve as surrogates for the PDE solution. These GP posterior samples are constrained by two likelihoods: one to fit the noisy observations and the other to enforce conformity with the governing equation. To efficiently and effectively infer the deep kernel and physical parameters, we develop a stochastic estimation algorithm based on the evidence lower bound (ELBO), which serves as a posterior regularization objective function. The effectiveness of the proposed PCDKL is demonstrated through a systematic comparison with a Bayesian physics-informed neural network (B-PINN), a state-of-the-art method for solving inverse problems in PDEs with noisy observations. Our experiments show that PCDKL not only achieves forward solutions with informative uncertainty estimates comparable to B-PINN, but also yields accurate estimates of physical parameters. These results suggest that PCDKL has significant potential for uncertainty quantification in forward solutions and accurate physical parameter estimation, making it valuable for practical applications.