Physics-informed methods have gained a great success in analyzing data with partial differential equation (PDE) constraints, which are ubiquitous when modeling dynamical systems. Different from the common penalty-based approach, this work promotes adherence to the underlying physical mechanism that facilitates statistical procedures. The motivating application concerns modeling fluorescence recovery after photobleaching, which is used for characterization of diffusion processes. We propose a physics-encoded regression model for handling spatio-temporally distributed data, which enables principled interpretability, parsimonious computation and efficient estimation by exploiting the structure of solutions of a governing evolution equation. The rate of convergence attaining the minimax optimality is theoretically demonstrated, generalizing the result obtained for the spatial regression. We conduct simulation studies to assess the performance of our proposed estimator and illustrate its usage in the aforementioned real data example.

翻译：物理信息方法在分析受偏微分方程约束的数据方面取得了巨大成功，这类约束在动态系统建模中普遍存在。与常见的基于惩罚项的方法不同，本研究旨在促进对底层物理机制的遵循，从而辅助统计过程。本研究的动机应用涉及光漂白后荧光恢复的建模，该技术用于表征扩散过程。我们提出了一种用于处理时空分布数据的物理编码回归模型，该模型通过利用控制演化方程解的结构，实现了有原则的可解释性、简约的计算和高效的估计。理论上证明了达到极小极大最优性的收敛速率，推广了空间回归中获得的结果。我们进行了模拟研究以评估所提出估计器的性能，并在上述真实数据示例中说明了其用法。