Robin-Neumann Coupling of PINN and FEM Solvers: A Steklov-Poincaré View, with Application to Fluid-Structure Interaction with Contact

2026-06-12

AI Summary

This work proposes a hybrid computational framework that integrates meshfree physics-informed neural networks (PINNs) with conventional finite element methods (FEM) to address fluid–structure interaction problems involving geometric topological changes, such as contact. By modeling the interface coupling through interactions of Steklov–Poincaré operators, the study establishes, for the first time, a rigorous theoretical foundation for PINN–FEM coupling and introduces a closed-form, non-iterative expression for interfacial impedance. Leveraging a Robin–Neumann domain decomposition strategy, the authors prove the contractivity of the coupling scheme under PINN training perturbations and devise Fourier modal probes to diagnose the resolvable spectral content of the PINN. Numerical experiments demonstrate prediction errors in contraction rates below 7% for 1D/2D Poisson coupling problems and achieve only 0.4% error in contact reaction forces for Stokes flow interacting with a rigid disk—handling topological changes solely by excluding collision points, without remeshing or cut-cell techniques.

Abstract

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.

Problem

• PINN-FEM coupling
• Fluid-Structure Interaction
• Contact
• Steklov-Poincaré operator
• Domain decomposition

Innovation

• Steklov-Poincaré operator
• Robin-Neumann coupling
• Physics-informed neural networks
• Domain decomposition
• Fourier-mode interface probe