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

Abstract

Predicting stress fields in hyperelastic materials with complex microstructures remains challenging for traditional deep learning surrogates, which struggle to capture both sharp stress concentrations and the wide dynamic range of stress magnitudes. Convolutional architectures such as UNet tend to oversmooth high-frequency gradients, while neural operators like DeepONet exhibit spectral bias and underpredict localized extremes. Diffusion models can recover fine-scale structure but often introduce low-frequency amplitude drift, degrading physical scaling. To address these limitations, we propose a hybrid surrogate framework, cDDPM-DeepONet, that decouples stress morphology from magnitude. A conditional denoising diffusion probabilistic model (cDDPM), built on a UNet backbone, generates normalized von Mises stress fields conditioned on geometry and loading. In parallel, a modified DeepONet predicts global scaling parameters (minimum and maximum stress), enabling reconstruction of full-resolution physical stress maps. This separation allows the diffusion model to focus on spatial structure while the operator network corrects global amplitude, mitigating spectral and scaling biases. We evaluate the framework on nonlinear hyperelastic datasets with single and multiple polygonal voids. The proposed model consistently outperforms UNet, DeepONet, and standalone cDDPM baselines by one to two orders of magnitude. Spectral analysis shows strong agreement with finite element solutions across all wavenumbers, preserving both global behavior and localized stress concentrations.

Figures (12)

• Figure 1: Description of the denoising diffusion framework for stress map prediction conditioned by the geometry and loading. The scaled stress maps are learned by the UNet in the conditional DDPM, and the minimum $\&$ maximum von Mises stresses are learned by the DeepONet. A CNN-based encoder is used for feature representation and is used to condition the UNet. The learned scaling and stress maps are combined to get the final stress maps.
• Figure 2: Datasets used for evaluation: The first dataset contains single void and loading values as input to predict the von Mises stress maps. The second dataset comprises geometries with multiple voids, which, along with loading values, are used to estimate the stress.
• Figure 3: Comparison of ground truth FEM stress maps, UNet, cDDPM and cDDPM-DeepONet predictions for single-void (\ref{['fig:DS1_models']}) and multiple-void (\ref{['fig:DS2_models']}) hyperelastic datasets.
• Figure 4: Randomly selected sample from the single-void dataset: Comparison of cDDPM-DeepONet predictions and FEM stress maps. The stress maps in the upper row compare the hybrid model's predictions with the ground-truth FEM results, showing excellent visual correlation and very low absolute error. The line plots illustrate the accurate pixel-wise matching of model outputs and FE along specific horizontal and vertical cross-sections.
• Figure 5: Randomly selected sample from the multiple-void dataset: Comparison of predicted and FEM stress maps. The FEM ground truth and cDDPM-DeepONet hybrid predicted stress maps show strong visual agreement, especially in the regions of high stress concentration induced by the inclusions. The absolute error map confirms this accuracy, showing consistently low error values across the domain. The horizontal and vertical mid-section plots provide quantitative validation, showing that the hybrid model's predicted stress profile closely matches the ground truth across the respective cross-sections.