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.

翻译：物理信息统计学习（PISL）通过整合经验数据与物理知识来提升估计器的统计性能。尽管PISL方法在实践中广泛应用，但关于信息正则化如何影响统计性质的全面理论理解仍显不足。具体而言，两个基本问题尚未得到充分解决：（1）软惩罚与硬约束之间的权衡关系是什么，以及（2）相较于纯数据驱动的经验误差最小化，融入物理知识能带来何种统计增益。本文针对物理知识以线性方程表达的凸函数类中的PISL，基于小波球方法建立了适当的复杂度依赖误差率，以回答上述问题。我们证明，在适当假设下，（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，仅相差常数项；且（2）信息惩罚能有效降低模型复杂度，类似于降维，从而提升学习性能。本研究为评估凸函数类中物理信息估计器的统计性质建立了理论框架，有助于弥合统计理论与实际PISL之间的差距，并可能应用于文献中尚未探索的案例。