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TopologyGAN: Topology Optimization Using Generative Adversarial Networks Based on Physical Fields Over the Initial Domain
Zhenguo Nie, Tong Lin, Haoliang Jiang, Levent Burak Kara
TL;DR
Deep-learning topology optimization has difficulty generalizing from sparse displacement and load boundary conditions to unseen configurations. TopologyGAN augments a cGAN with dense physical fields from the initial domain and combines U-Net and SE-ResNet features in its generator, significantly reducing test errors on previously unseen boundary conditions.
Problem
Deep-learning methods struggle to learn accurate, generalizable mappings from displacement and load boundary conditions to optimal structures, especially for novel boundary conditions.
Method
TopologyGAN augments a conditional generative adversarial network with dense physical fields computed on the original, unoptimized domain and uses a hybrid U-SE-ResNet generator.
Results
TopologyGAN significantly reduces test errors compared with a baseline cGAN on problems involving previously unseen boundary conditions.
Takeaways & Limitations
Initial physical fields and the U-SE-ResNet architecture provide a supported route to improved topology-optimization predictions under previously unseen boundary conditions.
Takeaways & Limitations
TopologyGAN lacks an in-built guarantee of a single connected component or checkerboard avoidance, treats volume fraction as a matching condition rather than a true inequality constraint, and is implemented for 2D problems.
Abstract
from arXiv · showhide
In topology optimization using deep learning, load and boundary conditions represented as vectors or sparse matrices often miss the opportunity to encode a rich view of the design problem, leading to less than ideal generalization results. We propose a new data-driven topology optimization model called TopologyGAN that takes advantage of various physical fields computed on the original, unoptimized material domain, as inputs to the generator of a conditional generative adversarial network (cGAN). Compared to a baseline cGAN, TopologyGAN achieves a nearly $3\times$ reduction in the mean squared error and a $2.5\times$ reduction in the mean absolute error on test problems involving previously unseen boundary conditions. Built on several existing network models, we also introduce a hybrid network called U-SE(Squeeze-and-Excitation)-ResNet for the generator that further increases the overall accuracy. We publicly share our full implementation and trained network.
1 Introduction
Topology optimization seeks optimized structures under material and boundary-condition constraints, but deep-learning methods struggle to generalize from sparse boundary-condition representations to unseen conditions. TopologyGAN addresses this by augmenting conditional generative modeling with dense physical fields computed on the initial domain.
- Motivation: Deep-learning topology optimization must learn accurate mappings from displacement and load boundary conditions to optimal structures, yet generalization to novel conditions remains difficult.Existing approaches often restrict displacement and load configurations because direct mappings are difficult to generalize.
- Method: TopologyGAN uses a conditional generative adversarial network that augments conventional inputs with physical fields computed on the initial, unoptimized material domain.The proposed channels include non-sparse von Mises stress, strain energy, and displacement fields.
- Method: Model-selection studies identify von Mises stress and strain energy density as the most useful fields for augmenting displacement and load boundary conditions.These fields are selected based on their performance in the authors’ model-selection studies.
- Results: TopologyGAN achieves lower test errors than the baseline cGAN’s training error on previously unseen boundary conditions.The comparison evaluates whether initial physical fields improve generalization beyond a cGAN using conventional inputs alone.
- Contributions: The work introduces TopologyGAN, new initial-field input matrices, and the hybrid U-SE-ResNet generator architecture.The authors publicly share the implementation and trained network.
2 Related Work
The paper situates topology optimization within density-based methods, deep learning, GANs, and related network architectures. Prior work seeks faster optimization and mappings from problem conditions to optimized structures, while generalization across boundary conditions remains challenging.
- Topology optimization: Topology optimization selects material distributions that minimize objectives such as compliance or mass under displacement and load constraints.Density-based methods discretize the design domain into finite elements and represent material density between void and full solid.
- Topology optimization: SIMP models element stiffness from material density and penalizes intermediate densities while minimizing compliance.The formulation uses material stiffness, infinitesimal stiffness, and a penalization factor to favor binary outputs.
- Deep learning for topology optimization: Data-driven methods aim to accelerate iterative topology-optimization solvers by learning mappings between problem inputs, constraints, and target designs.Deep neural networks and supervised learning have been applied to engineering design exploration and optimization.
- Related deep generative models: Prior topology-optimization studies use autoencoders, GANs, CNNs, encoder-decoders, super-resolution GANs, and cGANs for generating or representing optimized structures.These works address design alternatives, three-dimensional structures, convergence, low-resolution generation, and compact latent representations.
- Generative adversarial networks: Generative adversarial networks use a generator and discriminator in a minimax game, while cGANs condition the mapping from input x to output y.In topology optimization, cGAN inputs include the full domain, volume fraction, displacement boundary conditions, and external loads; SIMP supplies the ground-truth structure.
- Network architectures: U-Net propagates context across downsampling and upsampling layers, whereas SE-ResNet combines residual skip connections with channel-wise feature recalibration.U-Net also uses reflected-input tiling to provide border context when processing large images.
3 Technical Approach
TopologyGAN augments conditional generative modeling with physical fields computed on the initial domain, rather than mapping sparse boundary-condition inputs directly to optimized structures. Its U-SE-ResNet generator and composite adversarial loss are designed around this formulation.
- Motivation: TopologyGAN addresses the difficulty of generalizing deep-learning topology optimization to previously unseen boundary conditions.Sparse and discrete boundary-condition inputs make the input-to-structure mapping difficult to learn accurately.
- TopologyGAN formulation: TopologyGAN augments problem inputs with initial-domain physical fields, including strain energy, von Mises stress, displacement, and strain fields.These fields are computed on the unoptimized domain and provide non-sparse information alongside the original inputs.
- TopologyGAN formulation: The baseline cGAN directly maps problem inputs x to the output structure y, whereas TopologyGAN maps x through initial fields f before predicting y.During runtime, the initial fields are computed only once.
- TopologyGAN formulation: Initial strain-energy and von Mises-stress maps are hypothesized to correlate more closely with the final structure than the original problem matrices alone.The paper reports that these fields help TopologyGAN achieve higher training and test accuracy than the baseline cGAN.
- Network architecture: The generator predicts the optimized topology from the initial fields and volume fraction, while the discriminator is conditioned on both problem inputs and initial fields.The combined condition is r(x) = [x, f(x)].
- Network architecture: U-SE-ResNet combines a U-Net downsampling-upsampling structure with SE-ResNet modules, while a PatchGAN discriminator models high-frequency features.The generator uses transposed convolutions for upsampling and a sigmoid output for the grayscale structure.
- Loss function: TopologyGAN training combines adversarial loss, generator L2 loss, and absolute volume-fraction error, with empirically selected λ1 = 10,000 and λ2 = 1.The generator minimizes the combined loss while the adversarial discriminator maximizes the adversarial term.
4 Experiments
The experiments compare TopologyGAN and a baseline cGAN on SIMP-generated topology-optimization data, including test boundary conditions withheld during training. Performance is evaluated with image, volume-fraction, and compliance metrics, alongside studies of physical fields and generator architectures.
- Experimental setup: TopologyGAN and the baseline cGAN are trained on the same SIMP-generated data and compared by prediction performance.The experimental design supports a direct comparison between the two models.
- Ablations: The experiments also study which physical fields are useful and compare U-Net, SE-ResNet, and U-SE-ResNet generator structures.These analyses extend the main comparison beyond the baseline cGAN.
- Dataset: The dataset contains 49,078 optimized structures on a 64 × 128 square-element grid, generated with randomized volume fractions, displacement conditions, and boundary load positions.The SIMP filter radius is 1.5.
- Dataset: Four of 42 displacement-boundary-condition scenarios are held out as test conditions unseen during training; the remaining samples are split 80% for training and 20% for validation.The held-out conditions test generalization to previously unseen boundary conditions.
- Dataset: Each sample encodes volume fraction, displacement and x/y load boundary conditions, initial FEM stress, strain, and displacement fields, and a SIMP-generated output structure.The first four images are inputs, while the last image is the output structure.
- Metrics: The evaluation uses MAE, MSE, relative and absolute volume-fraction error, and relative and absolute compliance error.MAE measures average absolute prediction differences, while MSE measures average squared differences.
- Metrics: Ground-truth and predicted 64 × 128 matrices are reshaped into vectors of length N = 8,192 for metric computation.The metrics aggregate errors across data samples and grid elements.
5 Results and Discussions
TopologyGAN outperforms the baseline cGAN, with initial physical fields and the U-SE-ResNet generator improving prediction performance, including on unseen boundary conditions. The study also examines training behavior, field selection, accuracy across data splits, and model limitations.
- Model evaluation: TopologyGAN outperforms the baseline cGAN in both training and test performance.
- Model evaluation: 3× lower MSE and 2.5× lower MAE are achieved by TopologyGAN than by cGAN, with test errors below cGAN’s training errors.
- Model evaluation: TopologyGAN’s generator and whole objective losses decrease during training, while discriminator loss oscillates toward balance.
- Accuracy and performance: After 200 epochs, generated training structures become virtually indistinguishable from ground-truth structures.
- Accuracy and performance: TopologyGAN performs well on training and validation sets, while performance is lower on the unseen test set; volume-fraction error is near zero for most samples.
- Physical-field selection: VF+U+σvm+W performs best on training and validation, whereas VF+σvm+W performs better on the test set.
- Generator architecture: U-SE-ResNet outperforms U-Net and SE-ResNet, combining skip connections, shortcut connections, local information transfer, and adjustable network depth.
- Limitations and future work: TopologyGAN lacks an in-built guarantee of a single connected component or checkerboard avoidance beyond the SIMP-based training data.
6 Conclusions
TopologyGAN augments conventional topology-optimization inputs with dense physical fields computed on the original unoptimized domain and uses a hybrid U-SE-ResNet generator. These choices reduce test errors on problems with previously unseen boundary conditions.
- TopologyGAN augments displacement, load, and target-volume-fraction inputs with dense initial physical fields from the unoptimized domain.
- The U-SE-ResNet generator combines features of U-Net and SE-ResNet.
- The initial fields and U-SE-ResNet significantly reduce test errors on problems involving previously unseen boundary conditions.