FEM-aligned weak-form physics-informed neural networks with Laplace–Beltrami geometry encoding for linear elastic surrogate modeling

Reliable prediction of subsurface deformation is central to geo-engineering applications such as geological CO 2 storage and underground hydrogen storage. We propose a finite-element–aligned physics-informed neural network (FEM–PINN) for small-strain linear elasticity. Unlike conventional strong-form PINNs that minimize PDE residuals at collocation points, the FEM–PINN minimizes the total potential energy—a variational (weak-form) objective—that mirrors the structure, numerical quadrature, and post-processing of classical finite element methods (FEM). Geometry is encoded using a compact Laplace–Beltrami eigenbasis computed on the analysis mesh, while training minimizes the total potential energy evaluated consistently at FEM Gauss points. Essential boundary conditions are enforced through smooth architectural gating complemented by light penalty regularization. The framework outputs FEM-grade quantities, including nodal displacements, Gauss-point strains and stresses, von Mises stress, and total potential energy, enabling direct one-to-one comparison with a finite-element baseline. Three benchmarks are examined. A uniaxial compression test is used as a verification case, yielding relative L 2 displacement errors of approximately 7% and energy gaps of 0.2–3.5% across all cases. A second, large-scale prismatic block with free lateral boundaries and a fixed base demonstrates robustness under boundary conditions representative of geomechanical compression scenarios, with displacement errors in the range of 6%–8% and near-unity parity with FEM across all displacement components. A third, quasi-two-dimensional domain with mixed directional boundary conditions is verified, yielding relative L 2 displacement errors of approximately 8%. Stress and strain fields are recovered consistently at Gauss points. A sensitivity study over the eigenmode budget m confirms that LBO encoding is essential for variational consistency: the coordinate-only case ( m = 0 ) yields an energy gap of 137%, whereas m = 24 achieves 0.33%, the best among all configurations tested. Displacement errors vary non-monotonically with m , consistent with the non-convex optimisation landscape at a fixed training budget. By aligning weak-form training, geometry representation, and numerical integration with FEM practice, the proposed FEM–PINN provides a physically interpretable and practical surrogate for geomechanics workflows and offers a natural pathway toward hydro–mechanical coupling via augmented variational formulations.