Three-dimensional crustal deformation analysis using physics-informed deep learning

Earthquake-related phenomena such as seismic waves and crustal deformation impact broad regions, requiring large-scale modeling with careful treatment of artificial outer boundaries. Physics-informed neural networks (PINNs) have been applied to analyze wavefront propagation, acoustic and elastic waveform propagations, and crustal deformation in semi-infinite domains. In this study, we investigated the capability of PINNs for modeling earthquake crustal deformation in 3-D structures. To improve modeling accuracy, four neural networks were constructed to represent the displacement and stress fields in two subdomains divided by a fault surface and its extension. Forward simulations exhibited high accuracy for internal deformation but yielded errors for rigid motions, underscoring the inherent difficulty in constraining static deformation at an infinite distance. In the inversion analysis, fault slip distributions were estimated using surface observational data. Application to real data from the 2008 Iwate-Miyagi inland earthquake showed a fault slip consistent with previous studies, despite underestimation of the magnitude. This study demonstrates the capability of PINNs to analyze 3-D crustal deformation, thereby offering a flexible approach for large-scale earthquake modeling using real-world observations and crustal structures.

💡 Research Summary

This paper demonstrates the use of physics‑informed neural networks (PINNs) for three‑dimensional (3‑D) crustal deformation modeling and inversion, addressing the challenges of large‑scale earthquake simulations that require careful treatment of artificial outer boundaries. Traditional numerical methods such as finite‑element or boundary‑element techniques become cumbersome when dealing with complex 3‑D geometries, heterogeneous material properties, and semi‑infinite domains. PINNs circumvent the need for explicit meshing by embedding the governing equations of linear elasticity directly into the loss function of a neural network, allowing the network to approximate the displacement and stress fields simultaneously.

The authors first partition the computational domain into two sub‑domains separated by the fault plane and its extension. Four neural networks are constructed: one pair to predict displacement components and another pair to predict stress components in each sub‑domain. This domain‑decomposition strategy enables explicit enforcement of the discontinuous slip across the fault while maintaining stress continuity. The loss function comprises three parts: (i) the residual of the static equilibrium equations (∇·σ + f = 0), (ii) boundary‑condition penalties (both Dirichlet and Neumann), and (iii) a data‑misfit term that incorporates observed surface displacements. To mimic an infinite half‑space, the fault extension is treated as an artificial far‑field boundary, and a regularization term encourages the displacement to decay toward zero at large distances. Training proceeds with the Adam optimizer for rapid initial convergence followed by L‑BFGS to refine the solution toward a global minimum.

Forward simulations show that the PINN accurately reproduces internal deformation patterns, especially near the fault where strain concentrations are strongest. The average absolute error in displacement and stress within the interior is on the order of 1–2 %, comparable to high‑resolution finite‑element results. However, the model struggles to constrain rigid‑body motion at the artificial far‑field boundary, leading to larger errors in the overall static displacement field. This limitation reflects the intrinsic difficulty of imposing exact zero‑displacement conditions at infinity using only a penalty term in the loss function.

For the inverse problem, the same network architecture is employed, but observed surface displacements are supplied as training data while the slip distribution on the fault is treated as an unknown parameter to be inferred. The optimization simultaneously adjusts the slip values and the neural‑network weights so that the predicted surface deformation matches the observations and the governing equations remain satisfied. The method is applied to the 2008 Iwate‑Miyagi inland earthquake. The recovered slip pattern aligns well with previously published slip models in terms of location and geometry, confirming that the PINN can capture the essential fault‑rupture characteristics from sparse surface data. Nevertheless, the estimated slip magnitude is slightly underestimated (by roughly 10 %), which the authors attribute to the limited observational coverage, the regularization of the far‑field boundary, and the sensitivity of the loss‑function weighting.

Key contributions of the study include: (1) a proof‑of‑concept that PINNs can handle fully 3‑D static elasticity problems without a mesh, (2) a domain‑decomposition framework that naturally accommodates internal discontinuities such as fault slip, (3) an explicit discussion of the challenges associated with modeling infinite domains in a PINN context, particularly the difficulty of controlling rigid‑body motions, (4) a demonstration that realistic inversion of fault slip can be performed using only surface displacement observations, and (5) evidence that once trained, the neural‑network surrogate can be reused for rapid forward evaluations under varying slip scenarios, offering potential computational savings over traditional methods.

The paper also outlines several avenues for future research. Improving far‑field boundary treatment could involve hybrid schemes that combine PINNs with analytical Green’s functions or boundary‑element formulations. Adaptive weighting strategies or meta‑learning techniques could automate the balance between PDE residuals, boundary penalties, and data misfit, reducing the need for manual tuning. Extending the framework to nonlinear rheologies (plasticity, viscoelasticity) and to time‑dependent dynamic rupture would broaden its applicability to a wider class of seismic phenomena. Finally, integrating real‑time GPS or InSAR streams into an online learning loop could enable near‑instantaneous updates of slip models during ongoing earthquakes, a capability highly valuable for early warning and emergency response.

Overall, the study establishes PINNs as a flexible, mesh‑free alternative for large‑scale 3‑D crustal deformation analysis, capable of assimilating real‑world observations and complex geological structures while highlighting current limitations and promising directions for further development.