Split-Node Physics-Encoded Finite-Element Network for Forward and Inverse Modeling of Fault-Slip-Induced Discontinuous Deformation

Modeling crustal deformation induced by fault slip is a fundamental problem in structural geology and seismology. However, the challenges of data sparsity and spatial discontinuity impose significant limitations on conventional forward and inverse methods, often resulting in low computational efficiency and limited accuracy. Although AI-based approaches such as Physics-Informed Neural Networks (PINNs) and Physics-Encoded Finite Element Networks (PEFEN) offer new solutions for sparse-data problems governed by physical laws, their underlying assumption of spatial continuity conflicts with the inherent displacement discontinuities of fault-slip fields. To address this limitation, we propose a novel method—the Split-Node Physics-Encoded Finite Element Network (SN-PEFEN)—which integrates the node-splitting mechanism into the PEFEN framework. By explicitly encoding spatial discontinuities into the nodal topology during mesh preprocessing, SN-PEFEN not only overcomes the theoretical limitations of existing PEFEN models in handling discontinuous fields but also maintains the physical consistency. We apply SN-PEFEN to perform forward and inverse modeling of deformation fields induced by complex fault slip in both 2D and 3D heterogeneous media. For a model with over one million degrees of freedom, the forward simulation achieves over 40× speedup compared to traditional FEM (~1,800s vs. 42s), while maintaining comparable accuracy. In inverse modeling, the solution converges within only 100 iterations, with a total runtime of approximately 2,000 s, demonstrating high computational efficiency. This method establishes a new high-efficiency paradigm for analyzing complex discontinuous deformation in geomechanics, offering promising applications in multi-fault system analysis and fault-slip inversion. Furthermore, SN-PEFEN facilitates rapid, physics-based assessments for emergency seismic response and disaster management, while laying the groundwork for next-generation data-driven regional earthquake early warning systems.