标题： 基于小球方法的物理信息统计学习的复杂度相关误差率 显示英文标题

标题： Complexity Dependent Error Rates for Physics-informed Statistical Learning via the Small-ball Method

摘要： 物理信息统计学习（PISL）将经验数据与物理知识相结合，以提高估计量的统计性能。 尽管PISL方法在实践中被广泛使用，但对信息正则化如何影响统计特性的全面理论理解仍然缺失。 具体来说，有两个基本问题尚未得到充分解决：（1）考虑软惩罚与硬约束之间的权衡是什么，（2）与纯粹的数据驱动经验误差最小化相比，融入物理知识的统计增益是什么。 在本文中，我们通过开发基于小球方法的适当复杂度依赖的误差率，解决了在物理知识以线性方程表达的情况下，凸函数类中的PISL这些问题。 我们证明，在适当的假设下，（1）物理信息估计器的误差率与硬约束经验误差最小化器的误差率相当，仅在常数项上有所不同，并且（2）信息惩罚可以有效降低模型复杂度，类似于维度约简，从而提高学习性能。 这项工作建立了一个理论框架，用于评估凸函数类中物理信息估计器的统计特性，有助于弥合统计理论与实际PISL之间的差距，具有潜在的应用价值，适用于文献中尚未探索的情况。 显示英文摘要

摘要： Physics-informed statistical learning (PISL) integrates empirical data with physical knowledge to enhance the statistical performance of estimators. While PISL methods are widely used in practice, a comprehensive theoretical understanding of how informed regularization affects statistical properties is still missing. Specifically, two fundamental questions have yet to be fully addressed: (1) what is the trade-off between considering soft penalties versus hard constraints, and (2) what is the statistical gain of incorporating physical knowledge compared to purely data-driven empirical error minimisation. In this paper, we address these questions for PISL in convex classes of functions under physical knowledge expressed as linear equations by developing appropriate complexity dependent error rates based on the small-ball method. We show that, under suitable assumptions, (1) the error rates of physics-informed estimators are comparable to those of hard constrained empirical error minimisers, differing only by constant terms, and that (2) informed penalization can effectively reduce model complexity, akin to dimensionality reduction, thereby improving learning performance. This work establishes a theoretical framework for evaluating the statistical properties of physics-informed estimators in convex classes of functions, contributing to closing the gap between statistical theory and practical PISL, with potential applications to cases not yet explored in the literature.