标题： 物理信息线性模型（PILM）：解析表示及其在地壳应变率估计中的应用 显示英文标题

标题： Physics-Informed Linear Model (PILM): Analytical Representations and Application to Crustal Strain Rate Estimation

摘要： 许多物理系统由偏微分方程（PDEs）描述，求解这些方程并从观测数据中估计其系数或边界条件（BCs）在理解相关现象中起着至关重要的作用。 最近，一种称为物理信息神经网络的机器学习方法引起了科学界的广泛关注，该方法通过最小化PDEs、BCs和数据的残差之和来使用神经网络求解PDEs。 在本研究中，我们研究了一种物理信息线性模型（PILM），该模型使用基函数的线性组合来表示解，从而实现最优解的解析表示。 PILM被制定并验证用于包括不确定BCs情况在内的示例正问题和反问题。 此外，PILM被应用于利用大地测量数据估计地壳应变率。 具体而言，对施加弹性平衡于速度场上的物理正则化与施加平滑约束的数学正则化进行了比较。 从贝叶斯角度来看，数学正则化表现出更优的性能。 PILM提供了一个解析可解的框架，适用于线性正问题和反问题、欠定系统以及物理正则化。 显示英文摘要

摘要： Many physical systems are described by partial differential equations (PDEs), and solving these equations and estimating their coefficients or boundary conditions (BCs) from observational data play a crucial role in understanding the associated phenomena. Recently, a machine learning approach known as physics-informed neural network, which solves PDEs using neural networks by minimizing the sum of residuals from the PDEs, BCs, and data, has gained significant attention in the scientific community. In this study, we investigate a physics-informed linear model (PILM) that uses linear combinations of basis functions to represent solutions, thereby enabling an analytical representation of optimal solutions. The PILM was formulated and verified for illustrative forward and inverse problems including cases with uncertain BCs. Furthermore, the PILM was applied to estimate crustal strain rates using geodetic data. Specifically, physical regularization that enforces elastic equilibrium on the velocity fields was compared with mathematical regularization that imposes smoothness constraints. From a Bayesian perspective, mathematical regularization exhibited superior performance. The PILM provides an analytically solvable framework applicable to linear forward and inverse problems, underdetermined systems, and physical regularization.