解决科学机器学习中物理信息核的半正定性问题
Resolving positive semi-definiteness in physics-informed kernels for scientific machine learning
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中文总结 AI 辅助
本文针对科学机器学习中物理信息核的半正定性问题，确立了m阶线性微分算子对应的基核需m次连续可微的充分条件，指出相关不稳定性源于数值问题，为算子信息核方法提供了统一有效性基础。
中文摘要 AI 辅助
许多现代机器学习模型可被理解为基于核的函数空间模型，包括高斯过程和神经正切核。在科学机器学习中，微分算子正被越来越多地用于将物理结构直接编码到这类模型中。然而，这些构造在何种条件下是有效的机器学习模型（即保持半正定性），以及观测到的不稳定性是否源于不适定建模或数值效应，仍不清楚。本文中，我们为半正定性确立了一个简单且充分的条件：对于m阶线性微分算子，基核必须是m次连续可微的。至关重要的是，该保证适用于具有非常系数甚至不连续系数的算子，这类例子在物理系统中普遍存在，包括扩散、材料弹性、非均匀介质中的波传播以及量子系统。我们得出结论，剩余的不稳定性可归因于数值问题，为算子信息核方法提供了统一的有效性基础。
英文摘要
Many modern machine learning models can be understood as kernel-based function-space models, including Gaussian processes and neural tangent kernels. In scientific machine learning, differential operators are increasingly used to encode physical structure directly into such models. However, it has remained unclear under which conditions these constructions are valid machine learning models, i.e. preserve positive semi-definiteness, and whether observed instabilities arise from ill-posed modeling or numerical effects. Here, we establish a simple and sufficient condition for positive semi-definiteness: for linear differential operators of order m, the base kernel must be m-times continuously differentiable. Crucially, this guarantee holds for operators with non-constant and even discontinuous coefficients. Examples are ubiquitous in physical systems, including diffusion, material elasticity, wave propagation in inhomogeneous media, and quantum systems. We conclude that remaining instabilities are attributable to numerical issues, providing a unifying validity foundation for operator-informed kernel methods.
