Fast Learning Rates for Physics-Informed Kernel Methods

In physics-informed machine learning, a target function $u^*$ is learned from noisy value observations $y_i=u^*(x_i)+ \varepsilon_i$, together with differential information, given either by noisy observations $d_j=(Du^*)(z_j)+\xi_j$ or by a known physical constraint $Du^*=v$. We consider the setting where $D$ is a linear differential operator and analyze a physics-informed kernel estimator $\hat u$ combining $n$ value observations and $m$ differential observations. In this context, we ask how much can differential information improve predictions, and how does this improvement depend quantitatively on $n$, $m$, and $D$. We prove finite-sample bounds, supported by numerical simulations, revealing a two-regime structure for the prediction error. When $m$ is limited, the rate depends jointly on $n$ and $m$; when $m$ exceeds a problem-dependent threshold, the rate saturates and matches the oracle rate obtained when the perfect constraint $D \hat u = Du^*$ is imposed. Examples are discussed for Sobolev spaces which are reproducing kernel Hilbert spaces and include partial Laplacian constraints on the torus and gradient observations on bounded domains. These examples illustrate the range of possible learning rate improvements --- from the standard nonparametric $n^{-1/4}$ to the parametric rate $n^{-1/2}$. Finally, we derive physically consistent rates in a stronger norm that jointly controls the errors in $\hat u$ and $D\hat u$.