Physics-informed neural networks (PINNs) are meshless and carry moving geometry and topology change through resampling of collocation points; the finite-element method (FEM) is the workhorse for boundary-fitted discretisations. Coupling the two across a shared interface promises the best of both, yet existing PINN-FEM schemes are validated only empirically. We put the coupling on a domain-decomposition footing: viewing each solver as a Steklov-Poincaré (trace-to-flux) operator, we transfer the classical Dirichlet-Neumann (DN) divergence diagnosis and its Robin-Neumann (RN) cure, including a closed-form, sweep-free interface impedance, and prove a PINN-specific contraction theorem: a trained network realises only a perturbed Steklov operator with a per-step training residual, and RN still contracts, with no shared-eigenbasis hypothesis, to a floor set by the achieved training loss. Because a PINN has no stiffness matrix, we introduce a Fourier-mode interface probe that recovers the network's resolvable Steklov eigenvalues to within 0.5% and doubles as a diagnostic of the network's spectral cap. The theory predicts measured PINN-FEM contraction rates to within 7% on 1D and 2D Poisson couplings, and a two-slab analogue of the large-added-mass regime shows RN's per-mode impedance matching winning decisively where tuned scalar relaxation saturates. We demonstrate the framework on a Stokes/rigid-disc problem with Alart-Curnier contact: the meshless PINN fluid absorbs the topology change at contact by collocation exclusion alone, no remeshing and no cut cells, and the static-equilibrium contact reaction matches the submerged weight to 0.4% under mesh refinement. We quantify remaining limitations: the warm-started PINN drifts off the Stokes manifold over long horizons, and matched FEM-FEM benchmarks attribute pre-impact squeeze-film signatures to PINN under-resolution.

翻译：物理信息神经网络（PINN）具有无网格特性，可通过重新采样配点实现移动几何形状和拓扑变化；而有限元法（FEM）则是边界贴合离散化的核心工具。在共享界面上耦合二者有望兼具两者优势，但现有PINN-FEM方案仅经过经验验证。我们为这种耦合建立了区域分解理论基础：将每个求解器视为Steklov-Poincaré（迹-通量）算子，将经典的Dirichlet-Neumann（DN）发散诊断及其Robin-Neumann（RN）修复方法（包括闭式、免扫掠的界面阻抗）迁移至该框架，并证明了针对PINN的收缩定理：训练后的网络仅实现带有每步训练残差的扰动Steklov算子，而RN方法在无需共享特征基假设的条件下，仍能收敛至由训练损失决定的下界。由于PINN缺乏刚度矩阵，我们引入了一种傅里叶模态界面探测技术，能以0.5%的误差恢复网络的可分辨Steklov特征值，同时兼具网络谱上限的诊断功能。该理论预测PINN-FEM收缩率的实测偏差在1D和2D泊松耦合中不超过7%；在大附加质量效应的双片层类比中，RN的逐模态阻抗匹配方法在调谐标量松弛法饱和后仍能取得决定性优势。我们通过含Alart-Curnier接触的Stokes/刚性圆盘问题验证了该框架：无网格PINN流体通过配点排除单独吸收接触时的拓扑变化（无需重新网格划分或切割单元），且静力平衡接触反作用力在网格细化后与浸没重量吻合度达0.4%。我们量化了现有局限性：热启动的PINN在长时程中会偏离Stokes流形，而匹配的FEM-FEM基准测试将碰撞前挤压膜特征归因于PINN分辨率不足。