用解析方法求解地质应变率，融合物理规律提升精度
Physics-Informed Linear Model (PILM): Analytical Representations and Application to Crustal Strain Rate Estimation
- 用基函数线性组合构建可解析求解的物理模型
- 在地质数据上验证，数学正则化比物理正则化更优
- 适合需要精确解的线性反问题与不确定性系统
许多物理系统由偏微分方程（PDE）描述，从观测数据中求解PDE并估计其系数或边界条件（BCs），对理解相关现象至关重要。近年来，物理信息神经网络通过最小化PDE残差、边界条件残差和数据误差之和来求解PDE，受到广泛关注。本研究提出一种物理信息线性模型（PILM），采用基函数的线性组合表示解，实现最优解的解析表达。PILM被用于求解典型正向与反向问题，包括边界条件不确定的情况，并应用于利用大地测量数据估算地壳应变率。具体对比了基于弹性平衡的物理正则化与基于平滑性的数学正则化。从贝叶斯视角看，数学正则化表现更优。PILM提供了一个可解析求解的框架，适用于线性正向与反向问题、欠定系统及物理正则化。
原文摘要 · Abstract (English)
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.
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