# PIKS: Universal Physics-Informed Kernel Methods

arXiv: https://arxiv.org/abs/2607.27062 · HTML: https://mlsanity.com/paper/2607.27062/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 3 of its 94 references that are in the map.
- First posted on arXiv 29 Jul 2026.

## Builds on (most cited first, 3 of 3)

- [Universal Differential Equations for Scientific Machine Learning](https://mlsanity.com/paper/2001.04385/) (2020)
- [Challenges in Training PINNs: A Loss Landscape Perspective](https://mlsanity.com/paper/2402.01868/) (2024)
- [Fast kernel methods: Sobolev, physics-informed, and additive models](https://mlsanity.com/paper/2509.02649/) (2025)

## Built on by (most cited first, 1 of 1)

- [Fast Learning Rates for Physics-Informed Kernel Methods](https://mlsanity.com/paper/2609.18901/) (2026)

## Abstract (arXiv)

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 Mat'ern), 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.