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

TL;DR

This paper introduces a physics-informed linear model (PILM) that provides analytical solutions for PDE-based problems and demonstrates its application to crustal strain rate estimation using geodetic data.

Contribution

The study develops and verifies a PILM framework that offers analytical solutions for linear PDE problems and compares physical and mathematical regularization methods in geophysical applications.

Findings

PILM enables analytical solutions for forward and inverse PDE problems.

Physical regularization enforces elastic equilibrium, improving solution interpretability.

Mathematical regularization with smoothness constraints performs better from a Bayesian perspective.

Abstract

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.