Researchers from Zhejiang University and Zhejiang Advanced CNC Machine Tool Technology Innovation Center, China, have developed a Symmetry-Imbued Hamiltonian Neural Operator (SIHNO) for predicting stress fields in porous metamaterials. The proposed architecture combines rotation-equivariant geometric representations with a Hamiltonian-inspired energy structure to improve prediction accuracy and robustness under different structural orientations.
The framework integrates a Rotation-Steerable Lattice Projector (RSLP), a Hamiltonian-Fueled Propagator (HFP), and a slice-aware convolutional decoder to address two challenges in stress field prediction: orientation bias caused by fixed coordinate representations and non-physical stress leakage near void-solid interfaces.
In comprehensive numerical experiments based on finite-element-generated stress fields, SIHNO outperformed several established neural operator and physics-informed models across multiple evaluation metrics. The model also maintained millisecond-level inference speed, highlighting its potential for rapid stress prediction in additive manufacturing applications.
Porous metamaterials fabricated through additive manufacturing can provide useful combinations of strength, stiffness, and energy absorption, making them attractive for applications such as aerospace, biomedical, and protective structures. For these architected materials, accurate stress field prediction is important for understanding load-bearing paths, identifying local stress concentrations, and assessing structural reliability.
However, stress analysis of complex porous structures can be computationally demanding. Neural operators offer a potential route toward faster surrogate prediction, but existing approaches may struggle when the same lattice structure appears under different in-plane orientations. They may also produce over-smoothed stress fields or non-physical stress leakage near the interfaces between solid and void regions.
In a recent study published in Advanced Manufacturing , Xinyu Lu, Kang Wang, Guodong Yi, Shuyou Zhang, and Jianrong Tan from Zhejiang University and Zhejiang Advanced CNC Machine Tool Technology Innovation Center developed a Symmetry-Imbued Hamiltonian Neural Operator (SIHNO) to address these challenges.
The proposed framework combines two complementary forms of structural prior knowledge. The first is geometric symmetry. Through a Rotation-Steerable Lattice Projector (RSLP), lattice topology images are transformed into rotation-equivariant representations using steerable convolutional kernels. This allows the model to represent structural features more consistently when the same metamaterial is observed at different orientations.
The second is an energy-structured mechanical prior. The Hamiltonian-Fueled Propagator (HFP) introduces a Hamiltonian-inspired latent energy function and uses its gradient to guide feature propagation. Rather than directly imposing pointwise partial differential equation residuals across discontinuous void-solid interfaces, the approach seeks to improve global mechanical coherence through latent-space energy coupling.
These representations are then combined through a slice-aware convolutional decoder, which reconstructs the final stress field while preserving local boundary information and modeling long-range interactions between mechanically related regions.
To evaluate the proposed approach, the researchers constructed a metamaterial dataset from finite-element simulations. The porous structure was sliced along the build direction into 64 × 64 topology images, with corresponding von Mises stress fields used as ground-truth labels. The finite-element analysis was performed using Abaqus for an Aluminum 6061-T6 structure.
Under the standard, unrotated test condition, SIHNO achieved a mean absolute error (MAE) of 0.0438, a relative L2 error of 11.28%, and an R² value of 0.7323, outperforming the other compared models across the reported evaluation metrics.
The researchers also introduced a Void-Region Leakage (VRL) metric to quantify non-physical stress predictions in void regions. SIHNO achieved a VRL of 0.0079, the lowest among the models evaluated, indicating improved behavior around discontinuous void-solid interfaces.
The model was further tested under different in-plane rotations. At rotation angles of 10°, 45°, and 270°, SIHNO achieved MAE values of 0.0634, 0.0953, and 0.0438, respectively. Additional experiments across a continuous 360° rotation range showed that SIHNO maintained a relatively low error envelope across different orientations, demonstrating improved rotational robustness compared with conventional approaches.
An ablation study further examined the contributions of the two main components. Removing either the RSLP or HFP module reduced prediction performance, particularly under non-orthogonal rotations, supporting the combined use of geometric symmetry and Hamiltonian-inspired latent coupling.
Beyond prediction accuracy, computational efficiency is important for potential online monitoring applications. SIHNO contains approximately 1.25 million trainable parameters and requires about 1.91 milliseconds to process a single 64 × 64 topology slice on an NVIDIA 4080 GPU after warm-up.
The study demonstrates that incorporating geometric symmetry and energy-structured inductive biases into neural operator architectures can improve stress field prediction for porous metamaterials. The results provide a potential computational approach for rapid structural analysis and monitoring in additive manufacturing.
The researchers note that further validation using experimental measurements, additional material systems, and more diverse manufacturing conditions will be important to assess the generalization and practical applicability of the proposed framework.
This paper, “Symmetry-imbued Hamiltonian neural operator architecture for stress field prediction of porous metamaterials,” was published in Advanced Manufacturing .
Lu X, Wang K, Yi G, Zhang S, Tan J. Symmetry-imbued Hamiltonian neural operator architecture for stress field prediction of porous metamaterials. Adv. Manuf. 2026(3):0009, https://doi.org/10.55092/am20260009.
Advanced Manufacturing
Experimental study
Not applicable
Symmetry-imbued Hamiltonian neural operator architecture for stress field prediction of porous metamaterials
31-Jul-2026