Physics-informed machine learning combines the expressiveness of data-based approaches with the interpretability of physical models. In this context, we consider a general regression problem where the empirical risk is regularized by a partial differential equation that quantifies the physical inconsistency. We prove that for linear differential priors, the problem can be formulated as a kernel regression task. Taking advantage of kernel theory, we derive convergence rates for the minimizer of the regularized risk and show that it converges at least at the Sobolev minimax rate. However, faster rates can be achieved, depending on the physical error. This principle is illustrated with a one-dimensional example, supporting the claim that regularizing the empirical risk with physical information can be beneficial to the statistical performance of estimators.

翻译：物理信息机器学习将数据驱动方法的表达力与物理模型的可解释性相结合。在此背景下，我们考虑一个通用回归问题，其中经验风险通过量化物理不一致性的偏微分方程进行正则化。我们证明，对于线性微分先验，该问题可以表述为核回归任务。利用核理论，我们推导了正则化风险最小化器的收敛速率，并表明其收敛速度至少达到Sobolev极小极大速率。然而，根据物理误差的不同，可以实现更快的收敛速率。这一原理通过一维示例得到说明，支持了使用物理信息对经验风险进行正则化可能有助于提高估计器统计性能的观点。