这篇论文给核方法在物理信息学习里补上了理论短板，证明用高斯核就能渐近满足约束，还跟PINNs比了效果，搞理论的值得看。
本论文提出PIKS（物理信息核方法），在通用核（如高斯核或Matérn核）下证明了对线性微分约束的普适一致性，即估计器能渐近学习目标并满足物理约束。作者推导了在适当源条件下的有限样本界，将经典核方法的算子理论分析扩展至物理信息机器学习。数值实验表明，PIKS在多个问题上与PINNs及传统有限元方法具有竞争力。
PIKS: Universal Physics-Informed Kernel Methods
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.