物理约束下的模型泛化能力由代数簇维数决定，而非参数量。
Understanding Generalization in Physics Informed Models through Affine Variety Dimensions
- 提出统一残差形式，融合配点与变分法，支持不完整物理约束
- 证明泛化性能取决于物理约束对应的代数簇维数
- 适用于线性与非线性系统，适合关注模型可解释性的研究者
物理信息机器学习正因其能通过融入物理知识提升统计性能和样本效率而受到广泛关注。然而，现有理论分析常假设在非混合设置下具备完全先验知识，忽视了观测数据的整合，并多局限于线性系统，难以应对现实应用中普遍存在的非线性特性。为此，我们提出一种统一的残差形式，融合配点法与变分法，可在混合学习设置中引入不完整且复杂的物理约束。在此框架下，我们证明：物理信息回归在混合设置中的泛化性能由对应物理约束的仿射簇维数决定，而非参数数量。该结论统一适用于线性与非线性方程。我们还提出一种近似该维数的方法，并通过实验验证了理论结果。
原文摘要 · Abstract (English)
Physics-informed machine learning is gaining significant traction for enhancing statistical performance and sample efficiency through the integration of physical knowledge. However, current theoretical analyses often presume complete prior knowledge in non-hybrid settings, overlooking the crucial integration of observational data, and are frequently limited to linear systems, unlike the prevalent nonlinear nature of many real-world applications. To address these limitations, we introduce a unified residual form that unifies collocation and variational methods, enabling the incorporation of incomplete and complex physical constraints in hybrid learning settings. Within this formulation, we establish that the generalization performance of physics-informed regression in such hybrid settings is governed by the dimension of the affine variety associated with the physical constraint, rather than by the number of parameters. This enables a unified analysis that is applicable to both linear and nonlinear equations. We also present a method to approximate this dimension and provide experimental validation of our theoretical findings.
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