A variational physics-informed graph neural network for heterogeneous solid mechanics
Abstract
Stress localization in heterogeneous solids is governed by the bimaterial interface, where the displacement field remains C0-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 (Einc/Emat∈[10−2,102]), 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 (σxx) 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.
Related
- INI-VPINN: A Variational Physics-Informed Neural Network with Implicit Neumann and Interface Handling for Multi-Material Domains with Geometric Singularities
- A well-balanced shock-capturing variational physics-informed neural method for shallow-water flow with topography
- Comparison of a Parametric Physics-Informed Neural Network and a Tensorial Reduced-Order Model for the Shallow-Water Dam-Break Problem
- Solving Fredholm Integral Equations with Physics-Informed Neural Networks: Single-and double-network approaches
- QD-PINN:A Physics-Informed Neural Network Approach to Bayesian Quickest Detection in High-Dimensional Diffusions
BibTeX
@misc{yadav2026variational,
title = {{A variational physics-informed graph neural network for heterogeneous solid mechanics}},
author = {Aashay Rajan Yadav and Amiya Prakash Das and Ratna Kumar Annabattula},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.10983},
eprint = {2609.10983},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10983},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A variational physics-informed graph neural network for heterogeneous solid mechanics
AU - Aashay Rajan Yadav
AU - Amiya Prakash Das
AU - Ratna Kumar Annabattula
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.10983
UR - https://arxiv.org/abs/2609.10983
AB - 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_inc/E_mat∈[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 (σ_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.
ER -