  Physics-informed machine learning combines the expressiveness of data-based
approaches with the interpretability of physical models. In this context, we
consider a general regression problem where the empirical risk is regularized
by a partial differential equation that quantifies the physical inconsistency.
We prove that for linear differential priors, the problem can be formulated as
a kernel regression task. Taking advantage of kernel theory, we derive
convergence rates for the minimizer of the regularized risk and show that it
converges at least at the Sobolev minimax rate. However, faster rates can be
achieved, depending on the physical error. This principle is illustrated with a
one-dimensional example, supporting the claim that regularizing the empirical
risk with physical information can be beneficial to the statistical performance
of estimators.
