Point-DeepONet: Predicting Nonlinear Fields on Non-Parametric Geometries under Variable Load Conditions

Nonlinear structural analyses in engineering often require extensive finite element simulations, limiting their applicability in design optimization and real-time control. Conventional deep learning surrogates often struggle with complex, non-parametric three-dimensional (3D) geometries and directionally varying loads. This work presents Point-DeepONet, an operator-learning-based surrogate that integrates PointNet into the DeepONet framework to learn a mapping from non-parametric geometries and variable load conditions to physical response fields. By leveraging PointNet to learn a geometric representation from raw point clouds, our model circumvents the need for manual parameterization. This geometric embedding is then synergistically fused with load conditions within the DeepONet architecture to accurately predict three-dimensional displacement and von Mises stress fields. Trained on a large-scale dataset, Point-DeepONet demonstrates high fidelity, achieving a coefficient of determination (R^2) reaching 0.987 for displacement and 0.923 for von Mises stress. Furthermore, to rigorously validate its generalization capabilities, we conducted additional experiments on unseen, randomly oriented load directions, where the model maintained exceptional accuracy. Compared to nonlinear finite element analyses that require about 19.32 minutes per case, Point-DeepONet provides predictions in mere seconds–approximately 400 times faster–while maintaining excellent scalability. These findings, validated through extensive experiments and ablation studies, highlight the potential of Point-DeepONet to enable rapid, high-fidelity structural analyses for complex engineering workflows.

💡 Research Summary

The paper introduces Point‑DeepONet, a novel operator‑learning surrogate that integrates PointNet into the DeepONet framework to enable fast, high‑fidelity predictions of nonlinear structural response on arbitrary three‑dimensional (3D) geometries under both magnitude‑ and direction‑varying loads. Traditional deep‑learning surrogates either require regular grids, explicit parameterizations of geometry, or can only handle scalar variations in loading, limiting their applicability to real‑world engineering problems where shapes are generated by topology optimization or other non‑parametric processes and where loads may change direction.

Methodology

• Data generation: The authors extend the publicly available DeepJEB dataset (jet‑engine brackets) by performing fully nonlinear static finite‑element analyses (FEA) in Altair OptiStruct. Each bracket is meshed with second‑order tetrahedral elements (≈2 mm element size) and modeled with Ti‑6Al‑4V material using an elastic‑plastic law with linear isotropic hardening (yield = 227.6 MPa, hardening modulus = 355.56 MPa). Load cases include vertical, horizontal, and diagonal forces; additional experiments use randomly rotated load directions to test generalization.
• Network architecture: The branch network consists of a PointNet module that consumes raw point‑cloud coordinates of the geometry and produces a global geometric embedding. A second “auxiliary” branch encodes global physical parameters (load magnitude, direction, boundary conditions). These two embeddings are concatenated and fed into the DeepONet trunk, which receives spatial query points (the original FEM node coordinates) and outputs the desired field values: the three displacement components (uₓ, u_y, u_z) and von Mises stress. This design preserves the classic DeepONet “branch‑trunk” operator approximation while allowing the branch to handle unstructured, non‑parametric inputs.
• Training: The model is trained on a large set of simulated cases (tens of thousands of load‑geometry combinations) using mean‑squared error loss on both displacement and stress fields. No physics‑informed regularization is employed, but the authors perform extensive ablation studies to assess the contribution of each component.

Results

• Accuracy: On a held‑out test set, Point‑DeepONet achieves coefficient of determination R² = 0.987 for displacement and R² = 0.923 for von Mises stress, outperforming baseline CNN‑based surrogates (which suffer from grid interpolation errors) and the previously proposed Geom‑DeepONet (which only handles load magnitude).
• Speed: Inference takes roughly 0.03 s per case, compared with ~19.3 minutes for a full nonlinear FEM run—a speed‑up factor of about 400×.
• Ablation: Removing PointNet (replacing it with a simple MLP) degrades R² to ~0.85, confirming the importance of a dedicated point‑cloud encoder. Excluding load direction from the auxiliary branch reduces stress prediction accuracy dramatically, highlighting the necessity of encoding directional information.
• Scalability: Model performance improves monotonically with larger training datasets, and the computational cost scales linearly with the number of query points, demonstrating suitability for large‑scale design studies.

Comparison with Existing Methods
A concise table in the paper positions Point‑DeepONet against CNNs, PointNet/PointNet++, Graph Neural Networks, PINNs, Fourier Neural Operators, standard DeepONet, and Geom‑DeepONet. Only the proposed method simultaneously satisfies (i) handling of non‑parametric 3D geometries, (ii) full load‑magnitude and direction variability, (iii) high‑resolution inference on the original mesh, and (iv) superior computational efficiency.

Limitations and Future Work

• The current implementation is limited to static, quasi‑static nonlinear problems; extending to dynamic, transient, or coupled multiphysics scenarios will require additional temporal handling (e.g., recurrent or Fourier‑based trunks).
• Material modeling is restricted to isotropic hardening; incorporating anisotropic, visco‑plastic, or composite behaviors is an open challenge.
• While PointNet captures global geometry, it lacks hierarchical local feature extraction; the authors suggest exploring PointNet++ or graph‑based encoders to better represent fine‑scale geometric details.
• Integration of physics‑informed loss terms (PINN‑DeepONet) could further improve extrapolation to unseen load regimes.

Conclusions
Point‑DeepONet represents a significant step forward in surrogate modeling for computational mechanics. By marrying a point‑cloud encoder with the operator‑learning paradigm of DeepONet, the authors deliver a model that can ingest arbitrary 3D shapes without any handcrafted parameterization, incorporate full vectorial load information, and output high‑resolution displacement and stress fields at near‑real‑time speeds. This capability opens the door to rapid design exploration, real‑time structural health monitoring, and uncertainty quantification in complex engineering workflows that were previously infeasible due to the prohibitive cost of repeated nonlinear FEM analyses.