Researchers at the Institute for Fluid Power Drives and Systems at RWTH Aachen University have demonstrated that physics-informed graph neural networks and physics-informed neural networks can accurately calculate the nonlinear deformation of hyperelastic materials under mechanical loads without relying on labeled training data. In a study published in Advanced Modeling and Simulation in Engineering Sciences, investigators Faras Brumand-Poor, Matthias Trautmann, and Katharina Schmitz introduced a computational framework that pairs deep energy minimization with automated Bayesian hyperparameter optimization. By comparing model predictions directly against high-fidelity finite element reference solutions generated in Abaqus, the team established that both deep learning architectures reliably capture complex displacement fields in isotropic, nonlinear solids across both structured and unstructured geometric meshes.
Predicting mechanical deformations in solid bodies subject to external forces represents a classical challenge in continuum mechanics and tribology. In elastohydrodynamic lubrication simulations, such as those performed using the dynamic description of sealings tool developed at RWTH Aachen University to calculate friction in reciprocating seals, traditional engineering workflows rely heavily on iterative numerical procedures including the finite element method and the finite difference method. These conventional tools resolve complex fluid-structure interactions by coupling hydrodynamic factors, such as lubricant cavitation and fluid pressure distributions governed by the Reynolds equation, with solid counterbody mechanical deformations. However, because the resulting structural deflections simultaneously modify the geometry of the sealing gap and reconfigure fluid flow, standard iterative solvers incur substantial computational demands when executing repeated forward evaluations.
To overcome these computational bottlenecks, the authors formulated a physics-driven learning system grounded in nonlinear continuum mechanics and hyperelasticity. The mechanical behavior of the investigated solid bodies was governed by the Mooney-Rivlin constitutive material law, which characterizes isotropic rubber-like materials using a strain energy density function. In this formulation, mechanical strain energy is computed from the deformation gradient tensor and the invariants of the right Cauchy-Green deformation tensor, alongside empirical material parameters and a dedicated compressibility coefficient. Rather than relying on classical linear elasticity assumptions where stress and strain follow proportional relationships, the framework calculates the first Piola-Kirchhoff stress tensor by deriving the strain energy density function with respect to the deformation gradient, capturing finite strains and nonlinear geometric responses under surface traction.
The engineering team addressed the governing equations through the weak formulation of static equilibrium, implementing the Deep Energy Method. Under this variational principle, static equilibrium is achieved by finding a displacement field that minimizes the total potential energy of the physical system, defined as the difference between internal strain energy and external work performed by applied boundary tractions. Body forces, such as gravity, were set to zero for all investigated configurations. A central theoretical advantage of minimizing potential energy via the weak form over solving the strong differential equilibrium equations is that it eliminates the need to compute second-order spatial derivatives of the displacement field, reducing required mathematical operations to first-order derivatives with respect to initial reference coordinates and bypassing explicit stress tensor evaluations.
To evaluate alternative deep learning predictors within this variational framework, the authors implemented both multilayer perceptron physics-informed neural networks and message-passing physics-informed graph neural networks. The graph neural network approach models the finite element mesh as an undirected graph comprising material nodes and connecting bidirectional edges, adopting an encoder-processor-decoder topology inspired by MeshGraphNets. Learnable multilayer perceptrons first encode undeformed coordinates and relative spatial edge distances into latent feature vectors. A sequence of graph network blocks subsequently executes message generation along edges, aggregates incoming messages using permutation-invariant functions, and updates nodal feature representations across successive message-passing steps before a decoder maps latent states back into physical displacement predictions.
A critical technical challenge encountered when stacking multiple message-passing layers in deep graph networks is the over-smoothing phenomenon, where repeated information aggregation across expanding spatial neighborhoods causes nodal feature vectors to converge exponentially toward homogeneous values. To preserve feature distinctiveness and ensure stable gradient backpropagation, the authors incorporated residual connections across every graph network block, adding the prior block’s latent features directly to newly computed node and edge updates. Systematic evaluations of mathematical activation functions demonstrated that the hyperbolic tangent function produced the most reliable convergence across network architectures, outperforming sigmoid, exponential linear unit, and Gaussian error linear unit activations, while linear activations were implemented in final decoding layers to prevent non-physical displacement cutoffs.
Boundary conditions were rigorously handled by embedding kinematic constraints directly into network predictions. In boundary-value problems, enforcing Dirichlet constraints through additive penalty terms within the training loss converts unconstrained energy minimization into a constrained optimization problem, which frequently triggers severe numerical instability and convergence failure in the Deep Energy Method. To circumvent this vulnerability, the researchers adopted a hard enforcement strategy utilizing geometric distance functions. Raw nodal displacement outputs from the neural predictors were transformed through distance functions that identically satisfy zero-displacement clamping boundaries by mathematical construction, allowing downstream optimization to proceed entirely unconstrained while preserving exact boundary adherence.
Evaluating internal strain energy and external surface work integrals across two-dimensional continuous domains required numerical integration schemes adapted to mesh discretization. The authors analyzed four discrete approximation methods: the Monte Carlo technique, the Trapezoidal rule, the Elementwise-Mean method, and Gauss quadrature. Due to comparatively higher discretization errors, Monte Carlo sampling and the Trapezoidal rule were set aside in favor of higher-order schemes. For structured meshes, two-by-two Gauss quadrature was utilized across quadrilateral elements, whereas specialized triangular Gauss quadrature schemes and the Elementwise-Mean method were implemented for irregular geometric domains. Furthermore, surface traction integrals governing external work along one-dimensional Neumann boundaries were evaluated using one-dimensional Gauss quadrature formulas.
Training physics-informed models under the Deep Energy Method presents inherent optimization difficulties because the objective loss function measures the energetic difference between internal strain and external work. In equilibrium states where external loading induces large structural displacements, external work can mathematically exceed internal strain energy, yielding valid negative potential energy values that rule out standard absolute or squared norm regularizations. To prevent uncontrolled network divergence driven by high-frequency displacement oscillations, the researchers instituted an automated two-level optimization scheme. Network weights and biases were updated using the Adam optimizer with exponential learning rate decay, while a surrogate Gaussian process optimizer executed Bayesian hyperparameter optimization over decay rates, latent layer widths, network depths, and graph block counts to minimize the mean root squared error relative to reference finite element solutions.
The comparative accuracy of the optimized architectures was evaluated across four distinct benchmark scenarios: a slender Bending Beam, a Rectangle subjected to pure lateral Shear, a Rectangle subjected to vertical Stretch, and a Half Circle with a curved boundary. In the Bending Beam case, a structured quadrilateral mesh representing a 0.1-meter tall beam clamped at its base was loaded with a 10-Pascal horizontal surface traction at its tip. The physics-informed graph neural network achieved a mean root squared error of 1.08226e-4 meters, closely matching the standard neural network’s 1.1045e-4 meters. However, the graph network demonstrated superior fidelity in regions experiencing the largest deflections, yielding a maximum root squared error of 1.7411e-4 meters compared to 5.2622e-4 meters for the standard network, with cross-sectional relative errors remaining below 0.34 percent versus 1.39 percent.
In the Rectangle Shear and Rectangle Stretch configurations, both machine learning solvers accurately mirrored finite element reference behaviors, although spatial error distributions differed systematically between models. Under 150 Pascals of shear traction, the standard network achieved a slightly lower overall mean root squared error of 7.0647e-5 meters compared to 7.3565e-5 meters for the graph network. The graph network resolved interior displacements with higher precision, keeping relative errors across horizontal sections mostly below 1.0 percent, but exhibited larger deviations along exterior boundaries. Under 1000 Pascals of tensile stretching, where the Elementwise-Mean integration method was deployed to suppress non-physical asymmetry, the standard network again attained a slightly superior domain-wide mean root squared error of 4.67885e-5 meters against 5.57745e-5 meters for the graph network.
The unstructured Half Circle test case, comprising 996 nodes arranged in a quad-dominated mesh with triangular elements under a negative 1000-Pascal vertical traction, highlighted clear structural trade-offs between the two predictive approaches. While the graph neural network maintained superior displacement tracking through the central core of the geometry, its prediction error increased significantly toward the curved traction boundary, reaching relative errors up to 12 percent compared to 8 percent for the standard network. The researchers attributed this boundary degradation to topological constraints inherent in message-passing operations, where perimeter nodes possess fewer neighboring connections and spatially restricted receptive fields. Computational resource demands also varied sharply: the graph network required up to 332,551 trainable parameters compared to 2,842 in standard networks, necessitating dedicated graphics processing acceleration to complete training.
Subject of Research: Physics-informed deep graph learning frameworks for hyperelastic material deformation modeling
Article Title: Advancing deformation calculation: a physics-informed deep graph learning framework for hyperelastic materials
Article References: Brumand-Poor, F., Trautmann, M., & Schmitz, K. (2025). Advancing deformation calculation: a physics-informed deep graph learning framework for hyperelastic materials. Advanced Modeling and Simulation in Engineering Sciences, 12(1), Article 20. https://doi.org/10.1186/s40323-025-00304-7
Image Credits: AI Generated
DOI: 10.1186/s40323-025-00304-7
Keywords: physics-informed machine learning, deep energy method, graph neural networks, physics-informed neural networks, hyperelastic materials, Mooney-Rivlin model, Bayesian optimization, elastohydrodynamic lubrication, finite element method, displacement field, Dirichlet boundary conditions, Gauss quadrature, bending beam, rectangle shear, half circle scenario, message-passing neural networks, over-smoothing, strain energy density
News Source: Katie Riggs. (November 13, 2026). Physics-Informed Graph Networks Accurately Predict Nonlinear Hyperelastic Material Deformations. Scienmag.
