cDDPM-DeepONet: Hybrid Surrogate for Stress Analysis

• The paper introduces a hybrid surrogate framework that decouples stress morphology from amplitude to significantly improve von Mises stress prediction accuracy.
• It employs a cDDPM with a UNet backbone to generate normalized stress fields and a modified DeepONet to predict global stress extrema.
• The framework mitigates oversmoothing, spectral bias, and amplitude drift, achieving remarkable error reductions compared to standalone methods.

cDDPM-DeepONet is a hybrid surrogate framework for full-field von Mises stress prediction in 2D hyperelastic materials with complex void geometries under tensile loading. It combines a conditional denoising diffusion probabilistic model (cDDPM), built on a UNet backbone, with a modified DeepONet that predicts global stress scaling parameters. The defining idea is to decouple stress morphology from stress magnitude: the cDDPM generates a normalized stress field conditioned on geometry and loading, while the DeepONet predicts the minimum and maximum von Mises stress used to reconstruct the physical field. In the reported formulation, this decomposition is designed to address three specific deficiencies of standalone surrogates: oversmoothing in UNet, spectral bias in DeepONet, and low-frequency amplitude drift in diffusion models (Kota et al., 18 Mar 2026).

1. Concept and operator-learning rationale

DeepONet belongs to the class of neural operator models that learn mappings between function spaces rather than merely vector-to-vector regressions. In standard form, a branch network encodes an input function and a trunk network encodes the evaluation coordinate, with the final output obtained through a feature interaction that represents the learned operator. This branch/trunk decomposition is the basic operator-learning mechanism used across contemporary DeepONet applications in uncertainty propagation, dynamical systems, combustion chemistry, and predictive control (Garg et al., 2022, Jong et al., 23 May 2025).

Within that broader setting, cDDPM-DeepONet uses DeepONet in a deliberately reduced role. The model does not ask the operator network to synthesize the full stress field. Instead, the diffusion model is assigned the task of generating the normalized spatial stress morphology, while the DeepONet predicts only two global scalars per sample: the minimum and maximum von Mises stress. This design reflects the paper’s diagnosis that full-field neural operators tend to underpredict localized extremes because of spectral bias, whereas diffusion models can recover fine-scale structure but may mis-scale the absolute stress magnitude (Kota et al., 18 Mar 2026).

The resulting division of labor is central to the identity of cDDPM-DeepONet. It is neither a conventional full-field DeepONet nor a standalone conditional diffusion model. It is a hybrid operator-generative surrogate in which morphology and amplitude are learned by different modules and recombined only at the reconstruction stage.

2. Hybrid architecture

The cDDPM module is a conditional denoising diffusion probabilistic model built on a UNet backbone. Its conditioning input is a two-channel image consisting of a geometry mask and a load image or loading-magnitude representation. A CNN-based encoder maps this physical input to an embedding vector, $\boldsymbol{\zeta}_{emb} = g(\mathbf{y})$, and this embedding is fused with the timestep embedding through a bilinear transformation and injected at each resolution level of the UNet. The UNet includes residual blocks, a global attention mechanism, and multiresolution feature extraction, so the diffusion pathway is explicitly structured to resolve both local stress concentrations and long-range spatial dependencies (Kota et al., 18 Mar 2026).

The DeepONet pathway is modified in two ways. First, its target is reduced from a dense field to a two-dimensional output, $[\hat{\sigma}_{vM,\min}, \hat{\sigma}_{vM,\max}]$. Second, because the physical input $U \in \mathbb{R}^{N \times N \times 2}$ is high-dimensional, the branch network is implemented as a UNet-based encoder rather than as a simple multilayer perceptron. The trunk network is implemented as an MLP and acts as a basis encoder; after inner-product fusion of branch and trunk features, a final projection layer outputs the predicted extrema (Kota et al., 18 Mar 2026).

A common misconception is that the hybrid simply places DeepONet and diffusion in parallel as interchangeable predictors. In the reported design, they are not interchangeable. The cDDPM predicts a normalized field, whereas the DeepONet predicts only the global scaling parameters required for denormalization. The framework therefore depends on a strict asymmetry between spatial synthesis and amplitude calibration.

3. Mathematical formulation

The diffusion component operates on a normalized von Mises stress field rather than on the physical stress field directly. The forward diffusion process progressively corrupts the clean sample, and the reverse process is learned through conditional noise prediction. The paper uses a cosine noise schedule and trains the cDDPM with the standard noise-prediction mean-squared-error objective. Performance is reported to improve with increasing diffusion horizon $T$, with diminishing returns beyond about $T = 100$ (Kota et al., 18 Mar 2026).

The DeepONet component is formulated as a reduced operator

$$\mathcal{F} : \mathcal{A} \rightarrow \left[ \min_{s \in \Omega} \phi(U)(s),\; \max_{s \in \Omega} \phi(U)(s) \right].$$

Its training objective is an $L^1$ loss on the extrema,

$$\mathcal{L}_{DeepONet} = \left\| \hat{\mathbf{z}} - \mathbf{z} \right\|_1,$$

where $\mathbf{z}$ contains the true minimum and maximum stress values (Kota et al., 18 Mar 2026).

The final reconstruction is the defining equation of the hybrid:

$$\hat{\sigma}_{vM} = \hat{\sigma}_{vM,\min} + \hat{\sigma}^{norm}_{vM}\left(\hat{\sigma}_{vM,\max} - \hat{\sigma}_{vM,\min}\right).$$

Here $[\hat{\sigma}_{vM,\min}, \hat{\sigma}_{vM,\max}]$0 is the normalized field produced by the cDDPM, and the predicted extrema come from the DeepONet. The paper interprets this denormalization rule as the mechanism by which the framework mitigates two separate failure modes: spectral bias in operator learning and amplitude drift in diffusion generation (Kota et al., 18 Mar 2026).

The material model in the benchmark problems is compressible Neo-Hookean, with strain-energy density

$[\hat{\sigma}_{vM,\min}, \hat{\sigma}_{vM,\max}]$1

where $[\hat{\sigma}_{vM,\min}, \hat{\sigma}_{vM,\max}]$2. The output of interest throughout is the full-field von Mises stress (Kota et al., 18 Mar 2026).

4. Data generation, physical setting, and training protocol

The datasets are generated from 2D finite-element simulations in FEniCS on square domains containing polygonal voids under uniaxial tensile loading. Two datasets are reported. The single-void dataset contains one polygonal void with number of sides $[\hat{\sigma}_{vM,\min}, \hat{\sigma}_{vM,\max}]$3, with randomized size, orientation, and location. The loading magnitude is sampled from $[\hat{\sigma}_{vM,\min}, \hat{\sigma}_{vM,\max}]$4, and the dataset contains 19,900 training samples and 3,317 testing samples. The multiple-void dataset uses several polygonal voids with shapes ranging from triangles to heptagons, orientations $[\hat{\sigma}_{vM,\min}, \hat{\sigma}_{vM,\max}]$5, $[\hat{\sigma}_{vM,\min}, \hat{\sigma}_{vM,\max}]$6, and $[\hat{\sigma}_{vM,\min}, \hat{\sigma}_{vM,\max}]$7, varying size and position, and load magnitude sampled from $[\hat{\sigma}_{vM,\min}, \hat{\sigma}_{vM,\max}]$8. Its size is 20,000 training samples and 3,000 testing samples (Kota et al., 18 Mar 2026).

The void geometry is encoded by a binary mask $[\hat{\sigma}_{vM,\min}, \hat{\sigma}_{vM,\max}]$9, with $U \in \mathbb{R}^{N \times N \times 2}$0 in material and $U \in \mathbb{R}^{N \times N \times 2}$1 in void, and the effective Young’s modulus is written as

$U \in \mathbb{R}^{N \times N \times 2}$2

The boundary conditions are fixed-bottom, top tensile loading, and traction-free vertical sides (Kota et al., 18 Mar 2026).

The paper also reports ensemble statistics that quantify the variability of the stress fields. For the single-void dataset, the mean of means is 404.335, the standard deviation of means is 180.43, the mean of maxima is 3364.41, and the standard deviation of maxima is 2842.95. For the multiple-void dataset, the corresponding values are 394.56, 162.32, 3047.62, and 1795.37. This suggests that the benchmark is not merely geometrically diverse but also strongly heterogeneous in global stress level and peak-stress behavior (Kota et al., 18 Mar 2026).

Training uses the standard diffusion noise-prediction MSE for the cDDPM and an independent $U \in \mathbb{R}^{N \times N \times 2}$3 loss on extrema for the DeepONet. The extracted description states that field-level comparisons in the results section use $U \in \mathbb{R}^{N \times N \times 2}$4 for the surrogates, while the diffusion objective itself remains the standard MSE on noise (Kota et al., 18 Mar 2026).

5. Accuracy, error structure, and spectral fidelity

The paper evaluates MAE, RMSE, Relative MAE (RelMAE), Peak Absolute Error (PAE), Localized Stress Gradient (LSG), and Peak-to-Valley (PV) error. These metrics are defined from the true and predicted von Mises stress fields; for example,

$U \in \mathbb{R}^{N \times N \times 2}$5

$U \in \mathbb{R}^{N \times N \times 2}$6

$U \in \mathbb{R}^{N \times N \times 2}$7

and

$U \in \mathbb{R}^{N \times N \times 2}$8

where $U \in \mathbb{R}^{N \times N \times 2}$9 (Kota et al., 18 Mar 2026).

Dataset / Model	MAE	RMSE
Single-void / UNet	194.85	350.65
Single-void / cDDPM	28.36	84.32
Single-void / cDDPM-DeepONet	4.12	5.91
Multiple-void / UNet	166.99	235.25
Multiple-void / cDDPM	41.56	74.03
Multiple-void / cDDPM-DeepONet	23.52	45.16

The remaining metrics show the same ordering. On the single-void dataset, cDDPM-DeepONet achieves RelMAE 0.0017, PAE 47.79, LSG 0.73, and PV 47.78, compared with 0.0147, 1809.21, 41.95, and 1398.73 for UNet, and 0.0282, 269.07, 11.30, and 338.45 for standalone cDDPM. On the multiple-void dataset, the hybrid achieves RelMAE 0.0021, PAE 5.83, LSG 3.67, and PV 38.19, compared with 0.1086, 838.51, 39.04, and 826.24 for UNet, and 0.0169, 338.50, 13.07, and 266.92 for cDDPM. The paper states that, relative to UNet, the hybrid reduces MAE by 99.29% on the single-void dataset and by 86.77% on the multiple-void dataset; relative to cDDPM, the reductions are 84.59% and 46.84%, respectively (Kota et al., 18 Mar 2026).

Spectral analysis is a second major validation axis. The paper computes 2D Fourier magnitude spectra, 1D isotropic energy spectra $T$0, and an area-between-curves summary metric

$T$1

Lower $T$2 indicates better agreement with finite-element spectra across all wavenumbers. Reported values are 5.2572 for UNet, 4.5372 for cDDPM, and 4.4198 for cDDPM-DeepONet on the single-void dataset, and 5.3313, 4.5451, and 4.4166 on the multiple-void dataset. The paper interprets these results as evidence that UNet behaves as a low-pass surrogate, cDDPM improves high-frequency recovery but exhibits amplitude mismatch, and cDDPM-DeepONet best preserves both low-frequency global structure and high-frequency stress concentrations (Kota et al., 18 Mar 2026).

6. Position within the DeepONet literature, misconceptions, and limitations

In the broader DeepONet literature, operator networks have been used to map stochastic forcing histories to dynamical responses, to advance stiff combustion chemistry over short time windows, to perform one-shot multi-step prediction for model predictive control, and to handle unaligned observation grids through decoder-based fusion (Garg et al., 2022, Kumar et al., 2023, Jong et al., 23 May 2025, Chen et al., 2023). cDDPM-DeepONet differs from these variants in a specific way: it does not primarily extend DeepONet’s operator expressivity toward larger spatiotemporal outputs. Instead, it contracts the operator target to two global observables and delegates the high-dimensional field generation to a conditional diffusion model.

This makes cDDPM-DeepONet a notable example of modular operator learning. A related tendency appears in model-free physics-informed DeepONet, where the conventional residual-based “physics” term is replaced by a learned surrogate of short-term dependence extracted from sparse data (Sun et al., 22 Feb 2026). The shared pattern is architectural hybridization: DeepONet is strengthened not by scaling it into a universal full-field predictor for every subtask, but by pairing it with an auxiliary structured model that handles the regime in which DeepONet alone is weak.

Several misconceptions can therefore be excluded. cDDPM-DeepONet is not a physics-informed method in the residual-enforcement sense; the extracted description explicitly states that the paper does not include physics constraints inside the learning objective beyond supervised training. It is also not a general-purpose full-field DeepONet; the operator network predicts only min/max stress. Nor is it simply a diffusion model with an auxiliary regressor attached, because the denormalization step is the mechanism by which the physical stress scale is restored (Kota et al., 18 Mar 2026).

The reported limitations are equally specific. Validation is restricted to 2D hyperelastic problems with polygonal void geometries under uniaxial tension and to compressible Neo-Hookean materials. The DeepONet amplitude model is limited to two descriptors, minimum and maximum stress, which may be insufficient if a problem requires richer amplitude characterization. The paper lists future directions including 3D representative volume elements, rate-dependent materials, history-dependent materials, stochastic microstructures, multiscale constitutive modeling, and physics-constrained or solver-integrated diffusion processes (Kota et al., 18 Mar 2026).