Fast Learning Rates for Physics-Informed Kernel Methods
Abstract
In physics-informed machine learning, a target function u* is learned from noisy value observations yi=u*(xi)+ εi, together with differential information, given either by noisy observations dj=(Du*)(zj)+ξj 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 û 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 û = 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 û and Dû.
Related
- Compressed Multiscale Kernel Collocation and Physics-Informed Kernel Learning for Elliptic BVPs
- Minimax Rates for imperfect Physics-Informed Regression
- Physics-Informed Mixed Integer Programming Methods for Optimal Patch Antennas
- Learning Hamiltonians from Trajectory Data with Kernel Methods
- Comparison of neural and spline representations for physics-informed learning
BibTeX
@misc{brogatmotte2026fast,
title = {{Fast Learning Rates for Physics-Informed Kernel Methods}},
author = {Luc Brogat-Motte and Joachim Bona-Pellissier and Giacomo Meanti and Lorenzo Rosasco},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.18901},
eprint = {2609.18901},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.18901},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Fast Learning Rates for Physics-Informed Kernel Methods
AU - Luc Brogat-Motte
AU - Joachim Bona-Pellissier
AU - Giacomo Meanti
AU - Lorenzo Rosasco
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.18901
UR - https://arxiv.org/abs/2609.18901
AB - In physics-informed machine learning, a target function u^* is learned from noisy value observations y_i=u^*(x_i)+ ε_i, together with differential information, given either by noisy observations d_j=(Du^*)(z_j)+ξ_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 û 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 û = 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 û and Dû.
ER -