Fast Learning Rates for Physics-Informed Kernel Methods

arXiv:2609.18901v1 Announce Type: cross Abstract: In physics-informed machine learning, a target function u^* is learned from noisy value observations y_i=u^(x_i)+ arepsilon_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 Dhat u.

Source: arXiv cs.LG | 2026-09-17