A variational physics-informed graph neural network for heterogeneous solid mechanics

A variational, label-free physics-informed graph neural network (PI-GNN) developed by researchers posting to arXiv keeps von Mises error below 3.58% across a stiffness-contrast sweep of E_inc/E_mat from 10^-2 to 10^2, where a strong-form physics-informed neural network (PINN) degrades to 5.58%. The PI-GNN, which carries heterogeneity in a conforming adaptive mesh graph and minimizes discrete total potential energy as a single unweighted objective, also holds displacement error to 0.49% versus 7.66% for the strong-form PINN and halves the sigma_xx error of an energy-based PINN (5.01% versus 10.94%). Training cost exceeds a single finite element solve by more than an order of magnitude, so the authors position the method as a penalty-free interface representation for parametric surrogates and inverse identification rather than a replacement for one-off FE analysis.

arXiv:2609.10983v1 Announce Type: cross Abstract: Stress localization in heterogeneous solids is governed by the bimaterial interface, where the displacement field remains $C^0$-continuous, while in-plane stresses jump due to the stiffness mismatch. Coordinate-based physics-informed neural networks PINNs represent this jump via a prescribed regularization width or a weighted interface penalty, making their accuracy sensitive to how phase-contrast changes are handled. This work presents a variational, label-free physics-informed graph neural network PI-GNN in which the heterogeneity is carried by the discretization rather than by the trial field. The solver operates on a conforming adaptive mesh graph, assigns constitutive behavior per element, and minimizes the discrete total potential energy as a single unweighted objective in which only first derivatives appear. The discrete energy on piecewise-linear elements coincides with the finite element FE Ritz functional. Dirichlet conditions are enforced by construction, with no penalty term, no interface weight, and no prescribed transition width. Using one fixed architecture, optimizer, and loss across small-strain elasticity and finite-strain Neo-Hookean hyperelasticity in two and three dimensions, the von Mises error remains below $3.58\%$ across a stiffness-contrast sweep spanning $ E {\mathrm{inc}}/E {\mathrm{mat}}\in 10^{-2},10^{2} $, where a strong-form PINN degrades to $5.58\%$, and its displacement error reaches $7.66\%$ against $0.49\%$ for the PI-GNN. A trained network halves the $\sigma {xx}$ error of an energy-based PINN $5.01\%$ versus $10.94\%$ . Training cost exceeds a single FE solve by more than an order of magnitude, so the construction is a variationally consistent, penalty-free interface representation for parametric surrogates and inverse identification rather than a replacement for a one-off FE analysis.