  Deep kernel learning is a promising combination of deep neural networks and
nonparametric function learning. However, as a data driven approach, the
performance of deep kernel learning can still be restricted by scarce or
insufficient data, especially in extrapolation tasks. To address these
limitations, we propose Physics Informed Deep Kernel Learning (PI-DKL) that
exploits physics knowledge represented by differential equations with latent
sources. Specifically, we use the posterior function sample of the Gaussian
process as the surrogate for the solution of the differential equation, and
construct a generative component to integrate the equation in a principled
Bayesian hybrid framework. For efficient and effective inference, we
marginalize out the latent variables in the joint probability and derive a
collapsed model evidence lower bound (ELBO), based on which we develop a
stochastic model estimation algorithm. Our ELBO can be viewed as a nice,
interpretable posterior regularization objective. On synthetic datasets and
real-world applications, we show the advantage of our approach in both
prediction accuracy and uncertainty quantification.
