Physics-Informed Linear Model (PILM): Analytical Representations and Application to Crustal Strain Rate Estimation
This work addresses the challenge of solving PDEs in physical systems like geophysics, offering an analytically solvable framework, but it is incremental as it builds on existing physics-informed neural network approaches by using linear models.
The authors tackled the problem of solving partial differential equations (PDEs) and estimating coefficients from data by proposing a physics-informed linear model (PILM) that uses linear combinations of basis functions for analytical solutions, and applied it to estimate crustal strain rates, finding that mathematical regularization outperformed physical regularization from a Bayesian perspective.
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.