PIKS：通用物理信息核方法

中文摘要 AI 辅助

物理信息机器学习将通常由微分算子表达的物理原理融入数据驱动模型。尽管物理信息神经网络（PINNs）在经验应用中占据主导地位，但神经网络架构的复杂性与优化景观阻碍了对应学习理论的发展。核方法则提供了具有闭式解与分析可处理性的诱人替代方案，不过现有保证主要覆盖目标属于原生再生核希尔伯特空间（RKHS）的良好设定场景，这施加了物理目标常无法满足的不切实际正则性假设。本文引入并分析了物理信息核方法（PIKS），我们证明了PIKS对线性微分约束的通用一致性，即对于高斯、马顿（Matérn）等通用核，该估计量会渐近学习目标同时满足物理约束；还在合适源条件下推导了有限样本界，分析基于将核方法的经典算子理论分析扩展至物理信息机器学习。数值实验表明，PIKS可与PINNs及传统有限元方法相媲美。

英文摘要

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.

发表机构

• Università degli Studi di Genova（热那亚大学）
• Istituto Italiano di Tecnologia（意大利技术研究院）

机构由 AI 辅助整理，请以论文原文为准。