ArXiv

PIKS: Universal Physics-Informed Kernel Methods

2026-07-29 · 15:53 UTC ·Joachim Bona-Pellissier, Giacomo Meanti, Matteo Santacesaria... ·1 min read
Authors
Joachim Bona-Pellissier, Giacomo Meanti, Matteo Santacesaria...
Categories
stat.ML, cs.LG
arXiv
https://arxiv.org/abs/2607.27062v1
PDF
https://arxiv.org/pdf/2607.27062v1

Brief

PIKS (Physics-Informed Kernel methodS) formulates physics-informed learning with kernel methods to avoid PINNs' optimization complexity. Focusing on linear differential operators, the authors prove asymptotic (universal) consistency for universal kernels (Gaussian, Matérn), derive finite-sample bounds under source conditions, and extend classical operator-theoretic RKHS analysis; experiments show competitiveness with PINNs and FEM.

Why it matters

PIKS (Physics-Informed Kernel methodS), introduced by Bona‑Pellissier, Meanti, Santacesaria, and Rosasco (arXiv 2026-07-29), proves universal consistency for linear differential constraints: with universal kernels (e.g., Gaussian or Matérn) the estimator asymptotically learns the target while satisfying the physical constraints.

Key details

  • The paper derives finite-sample error bounds under source conditions, extends operator-theoretic kernel analysis to physics-informed learning, and reports numerical experiments where PIKS is competitive with physics-informed neural networks (PINNs) and traditional finite element methods.
Source evidence

Abstract

Physics-informed machine learning incorporates physical principles --often expressed via differential operators-- into data-driven models. While physics-informed neural networks (PINNs) dominate empirical applications, the complexity of neural network architectures and optimization landscapes hinders the development of a corresponding learning theory. In turn, kernel methods offer an appealing alternative with closed-form solutions and analytical tractability, yet existing guarantees primarily cover the well-specified setting where the target belongs to the native Reproducing Kernel Hilbert Space (RKHS). This imposes unrealistic regularity assumptions that physical targets often fail to satisfy. In this paper, we introduce and analyze Physics-Informed Kernel methodS (PIKS). We establish the universal consistency of PIKS for linear differential constraints, proving that for universal kernels (such as Gaussian or Matérn), the estimator asymptotically learns the target while satisfying physical constraints. We further derive finite-sample bounds under suitable source conditions. Our analysis is based on extending classical operator-theoretic analysis of kernel methods to physics-informed machine learning. Numerical experiments demonstrate that PIKS can be competitive with PINNs and traditional finite element methods.