PIKS: Universal Physics-Informed Kernel Methods

2026-07-29

AI Summary

This work addresses the lack of theoretical guarantees in existing physics-informed machine learning approaches, such as physics-informed neural networks (PINNs), and the restrictive regularity assumptions of classical kernel methods, which require the target function to reside in a reproducing kernel Hilbert space (RKHS)—an assumption often violated in real-world physical systems. To overcome these limitations, we propose Physics-Informed Kernel Methods (PIKMs), which embed physical laws expressed as linear differential operators directly into a kernel framework, yielding data-driven models that inherently satisfy physical constraints and admit closed-form solutions. We establish, for the first time, the universal consistency of PIKMs under commonly used kernels (e.g., Gaussian and Matérn), thereby relaxing the RKHS assumption, and derive finite-sample error bounds. Theoretically, PIKMs asymptotically converge to the true physical solution, and numerical experiments demonstrate their performance is competitive with both PINNs and finite element methods.

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.

Problem

• physics-informed machine learning
• kernel methods
• Reproducing Kernel Hilbert Space
• universal consistency
• differential constraints

Innovation

• Physics-Informed Kernel Methods
• Universal Consistency
• Reproducing Kernel Hilbert Space
• Differential Constraints
• Operator-Theoretic Analysis