Source-linked AI summary
Geom-DeepONet: A Point-cloud-based Deep Operator Network for Field Predictions on 3D Parameterized Geometries
Junyan He, Seid Koric, Diab Abueidda, Ali Najafi, Iwona Jasiuk
TL;DR
Repeated simulations on changing 3D geometries can be costly, while available neural architectures have limited support for varying 3D shapes. The paper proposes Geom-DeepONet to predict fields on parameterized 3D geometries. It reports greater accuracy than PointNet and vanilla DeepONet, and better generalization than vanilla DeepONet on unseen dissimilar designs.
Problem
Repeated high-fidelity simulations across changing 3D geometries can be impractical, and the paper identifies a gap in neural architectures for field prediction on varying 3D shapes.
Source: §1 · pp. 2, 4Method
Geom-DeepONet combines DeepONet with SDF and SIREN representations to predict fields on parameterized 3D geometries.
Source: §1 · p. 4Results
Geom-DeepONet was more accurate than PointNet and vanilla DeepONet, and generalized better than vanilla DeepONet to unseen, dissimilar designs.
Source: §4 · pp. 21–22Takeaways & Limitations
The authors present the model as a tool for preliminary design evaluations that can provide directional insights and help filter undesirable designs.
Source: §4 · p. 22The model relies on design-parameter descriptions of input geometries, which may not be feasible or available for complex components and assemblies.
Source: §4 · p. 22
Abstract
from arXiv · showhide
Modern digital engineering design process commonly involves expensive repeated simulations on varying three-dimensional (3D) geometries. The efficient prediction capability of neural networks (NNs) makes them a suitable surrogate to provide design insights. Nevertheless, few available NNs can handle solution prediction on varying 3D shapes. We present a novel deep operator network (DeepONet) variant called Geom-DeepONet, which encodes parameterized 3D geometries and predicts full-field solutions on an arbitrary number of nodes. To the best of the authors' knowledge, this is the first attempt in the literature and is our primary novelty. In addition to expressing shapes using mesh coordinates, the signed distance function for each node is evaluated and used to augment the inputs to the trunk network of the Geom-DeepONet, thereby capturing both explicit and implicit representations of the 3D shapes. The powerful geometric encoding capability of a sinusoidal representation network (SIREN) is also exploited by replacing the classical feedforward neural networks in the trunk with SIREN. Additional data fusion between the branch and trunk networks is introduced by an element-wise product. A numerical benchmark was conducted to compare Geom-DeepONet to PointNet and vanilla DeepONet, where results show that our architecture trains fast with a small memory footprint and yields the most accurate results among the three with less than 2 MPa stress error. Results show a much lower generalization error of our architecture on unseen dissimilar designs than vanilla DeepONet. Once trained, the model can predict vector solutions, and speed can be over $10^5$ times faster than implicit finite element simulations for large meshes.
1. Introduction
The paper addresses the gap in neural architectures for field prediction on varying 3D geometries by proposing Geom-DeepONet, a DeepONet variant for parameterized shapes. It combines operator learning with geometric representations to predict fields on such geometries.
1. Introduction
- Repeated high-fidelity simulations over many design scenarios or geometries can become impractical, motivating surrogate models for changing 3D domains.The paper identifies design exploration, digital twins, and optimization among tasks that may require extensive evaluations.Source: §1 · p. 2
- Existing geometry-learning approaches have limits: CNNs poorly fit irregular meshes, point-cloud methods struggle to capture edge or pairwise relations, and GNNs can incur training costs and generalization limits.The cited discussion notes that coordinate transformations can help CNNs but do not handle complex geometries with many edges or holes.Source: §1 · pp. 2–3
- The paper draws on SDFs to represent signed distance to a shape boundary and SIREN's sinusoidal activations to encode complex, variable inputs.SDF sign indicates inside versus outside, while magnitude gives distance to the nearest surface.Source: §1 · p. 3
- DeepONet maps parameters to solution fields and can infer new fields without retraining, but prior work had not addressed varying 3D computational domains.The paper also notes that existing DeepONets generally encode constant coordinate information in the trunk.Source: §1 · pp. 3–4
- Geom-DeepONet is proposed for field prediction on parameterized 3D geometries, a setting the authors identify as not previously addressed by DeepONet architectures.The paper describes this as its primary novelty.Source: §1 · p. 4
2. Methods
The methods benchmark PointNet and vanilla DeepONet alongside Geom-DeepONet, designed for field prediction on parameterized geometries with varying mesh sizes. Two geometry families provide the training and evaluation data.
2.1. PointNet and vanilla DeepONet for parameterized geometries
- The benchmark compares a 5,000-point PointNet with vanilla DeepONet, whose flexible node dimension supports prediction on arbitrary node counts.PointNet receives point coordinates and the applied load; vanilla DeepONet resamples nodes and fields to a fixed quantity during training so differently sized meshes can be batched.Source: §2.1 · pp. 4–5
2.2. Geom-DeepONet based on SDF and SIREN
- Geom-DeepONet uses intermediate branch–trunk fusion, applies SIREN layers in the second portion of its trunk, and supports vector-field outputs on variable node counts.Training nodes are resampled to a fixed length for batching; the benchmark model has 25,568 trainable parameters, while the four-component version has 38,298.Source: §2.2 · p. 7
2.3. Training data generation
- Two geometry families serve distinct purposes: the circular-hole beam benchmarks three networks, while the more variable cuboid with an ellipsoidal void tests geometry changes and vector prediction.Both families use parameterized designs with mesh discretizations that vary across geometries.Source: §2.3 · p. 8
2.3.1. Parameterized beam with a circular hole
- The beam dataset contains 3,000 designs parameterized by length, thickness, hole radius, and pressure; quasi-static, small-deformation FE simulations provide nodal von Mises stress labels.The beams use quadratic hexahedral elements and an elastic-plastic material model.Source: §2.3.1 · p. 8
2.3.2. Cuboid with a randomly oriented ellipsoidal void
- The cuboid dataset contains 2,500 designs parameterized by eight geometric variables, with tensile strain also varied; the predicted fields are von Mises stress and displacement components.Each cuboid has a randomly oriented ellipsoidal void and is meshed with quadratic tetrahedral elements.Source: §2.3.2 · p. 9
3. Results and discussion
The results evaluate stress prediction on the beam dataset and stress plus displacement prediction on the cuboid-with-void dataset, using relative error and mean absolute error. Across these tests, Geom-DeepONet is compared with baseline models, dissimilar-geometry splits, larger model sizes, and different meshes.
3.1. Effect of number of resample points
- N=2000 was optimal for beam prediction error, while N=5000 balanced accuracy, variability, and training time; the authors note that the optimal N is problem-dependent.Prediction errors on all mesh nodes were only slightly larger than on the resampled points, indicating generalization from the subset to full meshes.Source: §3.1 · pp. 11–12, Figure 8 · p. 11
3.2. Comparison with PointNet and vanilla DeepONet
- Geom-DeepONet achieved stress MAEs below 2 MPa on resampled points and all mesh nodes, over 50% lower prediction error, under-10-minute training, and 1.2 GB training memory.PointNet training took over 10 times as long and required almost 38 GB of memory.Source: §3.2 · p. 13, Table 3 · p. 13
- Geom-DeepONet handled variable node counts at prediction and was more accurate than vanilla DeepONet near the beam’s circular hole, where vanilla predictions often erred.PointNet predictions were limited to 5000 resampled points, whereas the DeepONets predicted on all mesh nodes.Source: Figure 10 · p. 13, Figure 11 · p. 14, Figure 12 · p. 14
3.3. Generalization in the parameter space
- Under the similarity split, vanilla DeepONet test error rose to around 12 MPa, while Geom-DeepONet remained below 2.5 MPa; its error also grew less with design dissimilarity.The similarity split trained on the 80% most similar geometries and tested on the 20% most dissimilar.Source: §3.3 · pp. 15–16, Figure 13 · p. 16
3.4. Extension to vector predictions: Cuboid with ellipsoidal void
- The cuboid-with-void experiment extended Geom-DeepONet from stress prediction to four vector components: von Mises stress and the X, Y, and Z displacement components.The baseline model had 38,298 trainable parameters; 95.24% of the meshes had more degrees of freedom than that parameter count.Source: §3.4 · p. 16, §3.4.1 · p. 16
3.4.1. Effect of Geom-DeepONet model size
- Increasing model size lowered stress MAE by 1.1 MPa and displacement-vector relative error by 1.2%, while training time rose 21%.At the 90th percentile, stress MAE fell from below 9 MPa for the baseline to below 7 MPa for the larger model; neither model captured the worst case well, and the larger model’s error rose more slowly with dissimilarity.Source: Table 4 · p. 17, §3.4.1 · pp. 18–19
3.4.2. Prediction on different meshes
- On refined meshes, Geom-DeepONet predictions were more than 10^4 times faster than finite element simulations; fitted time scaling was O(n^0.83) for prediction versus O(n^1.36) for simulation.Stress and displacement MAEs remained similar across meshes of different densities and element connectivity.Source: §3.4.2 · pp. 19, 21, Figure 19 · p. 20, Table 5 · p. 20
- Geom-DeepONet smoothed sharp stress transitions in the finite element fields, a limitation that may matter near cracks, defects, and sharp corners with steep stress gradients.The paper identifies this as spectral bias in the predictions.Source: §3.4.2 · p. 21
4. Conclusions, limitations, and future work
Geom-DeepONet extends DeepONet to parameterized 3D geometries and arbitrary-node field prediction using SDF and SIREN. The paper reports stronger accuracy than PointNet and vanilla DeepONet, lower generalization error than vanilla DeepONet, and faster prediction than finite element simulations, while noting limitations in geometry parameterization and high-frequency learning.
4. Conclusions, limitations, and future work
- It augments trunk inputs with node-level signed distance functions, replaces dense trunk layers with SIREN, and fuses branch and trunk representations.These choices are used for field prediction on parameterized 3D geometries.Source: §4 · p. 21
- Geom-DeepONet was more accurate than PointNet and vanilla DeepONet, trained more efficiently than PointNet, and generalized better than vanilla DeepONet on unseen dissimilar designs.The training-efficiency comparison was made using identical hardware.Source: §4 · pp. 21–22
- The architecture predicts on meshes with arbitrary node counts and was demonstrated on vector fields including stress and three displacement components.The vector-field demonstration used a cuboid with random elliptical voids.Source: §4 · pp. 21–22
- For models with a large number of degrees of freedom, neural-network prediction can be over 10^5 times faster than finite element simulations, while the proposed network has a relatively small memory footprint in training and prediction.Source: §4 · p. 22
- The authors identify design-parameter-based geometry descriptions as a limitation when parameterizations are infeasible or unavailable for complex components and assemblies.They propose exploring implicit and parameter-free geometry encoding in future work.Source: §4 · p. 22
- The model exhibits spectral bias: high-frequency solution components are not learned effectively, which may matter where geometry creates sharp stress gradients.The paper gives cracks, defects, and sharp corners as examples of such geometric features.Source: §4 · p. 22