A Hybrid Conditional Diffusion-DeepONet Framework for High-Fidelity Stress Prediction in Hyperelastic Materials

Accurate prediction of stress and strain fields in hierarchical composite microstructures is critical for physics-informed material design, yet conventional finite element method (FEM) simulations are computationally prohibitive at scale, requiring minutes to days per evaluation. In this work, we propose a hybrid UNet-Transformer architecture that predicts complex mechanical field distributions directly from composite microstructure geometry images, serving as an efficient surrogate for FEM across ten distinct stress and strain field types spanning diverse two-phase composite configurations including square, hexagonal, and triangular tessellations, multiple boundary conditions, and high-resolution geometries. Results demonstrate that the proposed architecture achieves strong predictive performance across the majority of subdatasets, with peak accuracy on periodic tessellation geometries reaching R2=0.9991, SSIM=0.9936, and MAE=0.0050 on the boundary condition subdataset and the triangular tessellation subdataset respectively. Across six of the eight evaluated subdatasets, MAE remains below 0.05 on the normalized [0,1] pixel scale. Encoder attention analysis via Grad-CAM and Grad-CAM++ confirms that the model develops physically meaningful internal representations, localizing attention at mechanically critical regions including phase boundaries, ligament junctions, and indenter contact zones without explicit structural supervision. Performance degrades on irregular square-grid geometries with sparse soft-phase inclusions, with the S11 normal stress subdataset yielding R2=0.7735 and SSIM=0.7126, consistent with the known limitation of smooth-loss image translation models in reproducing sharp stress discontinuities.

We propose a physics-informed machine learning framework called P-DivGNN to reconstruct local stress fields at the micro-scale, in the context of multi-scale simulation given a periodic micro-structure mesh and mean, macro-scale, stress values. This method is based in representing a periodic micro-structure as a graph, combined with a message passing graph neural network. We are able to retrieve local stress field distributions, providing average stress values produced by a mean field reduced order model (ROM) or Finite Element (FE) simulation at the macro-scale. The prediction of local stress fields are of utmost importance considering fracture analysis or the definition of local fatigue criteria. Our model incorporates physical constraints during training to constraint local stress field equilibrium state and employs a periodic graph representation to enforce periodic boundary conditions. The benefits of the proposed physics-informed GNN are evaluated considering linear and non linear hyperelastic responses applied to varying geometries. In the non-linear hyperelastic case, the proposed method achieves significant computational speed-ups compared to FE simulation, making it particularly attractive for large-scale applications.

The discovery of constitutive laws from experimentally accessible measurements is a central problem in nonlinear computational mechanics. Many data-driven constitutive identification approaches rely either on paired strain-stress data or on full-field displacement measurements, both of which are difficult to obtain in realistic three-dimensional settings. We present a differentiable finite element framework for the discovery of hyperelastic material laws from partial observations, including boundary-only displacement measurements and global reaction forces. The method embeds the nonlinear finite element equilibrium problem directly into the learning loop, so that candidate strain-energy densities are assessed through the deformation fields and reactions they induce. This formulation enforces mechanical equilibrium as a constraint and allows the loss function to be evaluated only at observed locations. To ensure physical admissibility and promote numerical solvability throughout training, the constitutive response is represented by Hyperelastic Neural Networks, a structure-preserving neural class that enforces residual energy and stress-free conditions, frame indifference, isotropic material symmetry, polyconvexity, coercivity, and controlled volumetric growth by construction. The resulting PDE-constrained learning problem is solved using a quasi-Newton strategy combined with continuation and solver-aware backtracking. Numerical experiments in two- and three-dimensional finite elasticity demonstrate accurate recovery of hyperelastic isotropic responses from boundary-only data, robustness to measurement noise, and generalization across geometries, loading conditions, and boundary conditions.