arXiv Machine Learning By Luc Brogat-Motte, Joachim Bona-Pellissier, Giacomo Meanti, Lorenzo Rosasco

Fast Learning Rates for Physics-Informed Kernel Methods

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arXiv:2609. 18901v1 Announce Type: cross Abstract: In physics-informed machine learning, a target function $u^*$ is learned from noisy value observations $y_i=u^*(x_i)+ \varepsilon_i$, together with differential information, given either by noisy observations $d_j=(Du^*)(z_j)+\xi_j$ or by a known physical constraint $Du^*=v$.

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arXiv Machine Learning
Jul 30

PIKS: Universal Physics-Informed Kernel Methods

arXiv:2607. 27062v1 Announce Type: cross Abstract: Physics-informed machine learning incorporates physical principles --often expressed via differential operators-- into data-driven models.

By Joachim Bona-Pellissier, Giacomo Meanti, Matteo Santacesaria, Lorenzo Rosasco
arXiv Machine Learning
Oct 23

Random Projections for Spectral Algorithms: Optimal Rates in Sobolev Norms

The paper studies random projections (RP) for spectral algorithms, providing a comprehensive analysis of their generalization performance under broad conditions. By exploiting the RKHS embedding property and operator similarity, it establishes optimal learning rates in Sobolev norms that match minimax lower bounds up to logarithmic factors. For common RP implementations—randomized sketches and Nyström sub-sampling—the required projection dimension scales with the effective dimension, reducing computational cost while preserving statistical efficiency.

By Caixing Wang, Yue Wang
arXiv Machine Learning
Jul 8

Kernel-based Operator Learning: Error Analysis, Budget Allocation, and a Physics-Informed Extension

arXiv:2607. 06287v1 Announce Type: cross Abstract: We study kernel-based operator learning in a two-stage sampling framework, where an offline kernel regression operator learns a discretized representation of the target operator from input-output pairs and an online kernel reconstruction operator recovers the output function from predicted observations.

By R\"udiger Kempf