9:00 AM - 9:15 AM
[SGD03-01] Three-Dimensional Crustal Deformation Analysis Using Physics-Informed Deep Learning
Crustal deformation caused by plate motions and earthquakes is described using the dislocation model. Traditionally, analytical solutions and numerical methods such as finite difference and finite element methods have been developed for modeling these phenomena. In recent years, physics-informed neural networks (PINNs), which utilize neural networks (NNs) to solve partial differential equations (PDEs), have been proposed (Raissi et al., 2019) and applied across various fields of natural sciences. In crustal deformation analysis, Okazaki et al. (2022) applied PINNs to the forward analysis of coseismic deformation in the anti-plane problem. Furthermore, Okazaki et al. (2025) extended this approach to forward analysis and synthetic inversion analysis in the in-plane problem.
In this study, we applied PINNs to analyze coseismic crustal deformation in three-dimensional structures and perform inverse analysis using real data. The governing equations (equilibrium equations and constitutive laws) are formulated in terms of displacement and stress as variables, while stress is uniquely determined by displacement in an elastic medium. Therefore, in the forward analysis, we examined two different representations for the NN output variables: displacement and displacement–stress representation. The displacement representation has three degrees of freedom and strictly satisfies the constitutive law; however, since it involves second-order derivatives, differentiation in PINNs is computationally expensive and increases the complexity of the loss function. In contrast, the displacement–stress representation has nine degrees of freedom and only approximately satisfies the constitutive law, but it involves only first-order derivatives, making optimization in PINNs easier. A comparative analysis in a homogeneous half-space showed that the displacement–stress representation yielded more efficient and higher-accuracy solutions. Therefore, we adopted this representation in the subsequent analyses.
In the forward analysis, removing rigid body motion requires boundary conditions at infinity, where displacement and its derivatives must asymptotically approach zero. In PINNs, this condition is approximately enforced by extending the computational domain sufficiently and imposing it at the outer boundaries. To assess the impact of this condition on solution accuracy, we conducted a comparative analysis in which, instead of enforcing boundary conditions at the outer boundaries, we used ground truth displacements at four points within the domain as supervised training data. The results showed that incorporating a small number of supervised displacements reduced estimation errors by an order of magnitude compared to enforcing external boundary conditions. This suggests that the primary source of error is the challenge of constraining rigid motion rather than internal deformation.
In the inversion analysis, we estimated the slip distribution on the prespecified fault surface from GNSS displacement data. First, we confirmed fair performance in synthetic tests, although sparser observations and larger noises degrade accuracy. Next, we analyzed the coseismic deformation of the 2008 Iwate-Miyagi inland earthquake. The analysis utilized F5 solutions from the GEONET network at 51 observation points, and the fault plane was defined based on the model by Ozawa et al. (2008) in a homogeneous half-space. The estimated slip distribution included a reverse faulting component with a left-lateral strike-slip component. The maximum slip was estimated to be 4.49 m, and the seismic moment was at 1.58×1019 Nm (Mw = 6.73). These results are consistent with previous studies, although slightly smaller in magnitude. Furthermore, we are extending the analysis using a realistic subsurface structure model based on the Japan Integrated Velocity Structure Model (Koketsu et al., 2012), which will also be discussed in the presentation.
In this study, we applied PINNs to analyze coseismic crustal deformation in three-dimensional structures and perform inverse analysis using real data. The governing equations (equilibrium equations and constitutive laws) are formulated in terms of displacement and stress as variables, while stress is uniquely determined by displacement in an elastic medium. Therefore, in the forward analysis, we examined two different representations for the NN output variables: displacement and displacement–stress representation. The displacement representation has three degrees of freedom and strictly satisfies the constitutive law; however, since it involves second-order derivatives, differentiation in PINNs is computationally expensive and increases the complexity of the loss function. In contrast, the displacement–stress representation has nine degrees of freedom and only approximately satisfies the constitutive law, but it involves only first-order derivatives, making optimization in PINNs easier. A comparative analysis in a homogeneous half-space showed that the displacement–stress representation yielded more efficient and higher-accuracy solutions. Therefore, we adopted this representation in the subsequent analyses.
In the forward analysis, removing rigid body motion requires boundary conditions at infinity, where displacement and its derivatives must asymptotically approach zero. In PINNs, this condition is approximately enforced by extending the computational domain sufficiently and imposing it at the outer boundaries. To assess the impact of this condition on solution accuracy, we conducted a comparative analysis in which, instead of enforcing boundary conditions at the outer boundaries, we used ground truth displacements at four points within the domain as supervised training data. The results showed that incorporating a small number of supervised displacements reduced estimation errors by an order of magnitude compared to enforcing external boundary conditions. This suggests that the primary source of error is the challenge of constraining rigid motion rather than internal deformation.
In the inversion analysis, we estimated the slip distribution on the prespecified fault surface from GNSS displacement data. First, we confirmed fair performance in synthetic tests, although sparser observations and larger noises degrade accuracy. Next, we analyzed the coseismic deformation of the 2008 Iwate-Miyagi inland earthquake. The analysis utilized F5 solutions from the GEONET network at 51 observation points, and the fault plane was defined based on the model by Ozawa et al. (2008) in a homogeneous half-space. The estimated slip distribution included a reverse faulting component with a left-lateral strike-slip component. The maximum slip was estimated to be 4.49 m, and the seismic moment was at 1.58×1019 Nm (Mw = 6.73). These results are consistent with previous studies, although slightly smaller in magnitude. Furthermore, we are extending the analysis using a realistic subsurface structure model based on the Japan Integrated Velocity Structure Model (Koketsu et al., 2012), which will also be discussed in the presentation.