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Optimize Any Topology: A Foundation Model for Shape- and Resolution-Free Structural Topology Optimization

Amin Heyrani Nobari, Lyle Regenwetter, Cyril Picard, Ligong Han, Faez Ahmed

arXiv:2510.23667v1cs.LGcs.AIcs.CE

TL;DR

Structural topology optimization is computationally expensive and must be restarted when boundary conditions, geometry, or loads change. OAT is a pretrained latent-diffusion framework for minimum-compliance optimization, trained on OpenTO’s 2.2 million optimized structures; it reports lower compliance error than prior models and sub-second inference.

  • Problem

    Conventional topology optimization requires repeated finite-element analysis and must be restarted when boundary conditions, geometry, or loads change, while deep-learning approaches remain narrowly scoped.

    Source: §1 · pp. 1–2

  • Method

    OAT is a pretrained latent-diffusion framework for minimum-compliance topology optimization, using shape- and resolution-independent representations and the 2.2-million-structure OpenTO dataset.

    Source: §1 · p. 2, §4.2 · p. 5

  • Results

    Up to 10x lower compliance error than prior models on established benchmarks, with sub-second inference on a single GPU.

    Source: §1 · p. 3

  • Takeaways & Limitations

    OAT is positioned as a foundational step toward topology-optimization models that generalize across boundary conditions, forces, and resolutions for minimum-compliance linear elasticity.

    Source: §1 · p. 2

    OAT has a notable failure rate on fully random benchmarks, and multi-physics objectives remain future work.

    Source: §6 · pp. 10–11

Abstract

from arXiv · show

Structural topology optimization (TO) is central to engineering design but remains computationally intensive due to complex physics and hard constraints. Existing deep-learning methods are limited to fixed square grids, a few hand-coded boundary conditions, and post-hoc optimization, preventing general deployment. We introduce Optimize Any Topology (OAT), a foundation-model framework that directly predicts minimum-compliance layouts for arbitrary aspect ratios, resolutions, volume fractions, loads, and fixtures. OAT combines a resolution- and shape-agnostic autoencoder with an implicit neural-field decoder and a conditional latent-diffusion model trained on OpenTO, a new corpus of 2.2 million optimized structures covering 2 million unique boundary-condition configurations. On four public benchmarks and two challenging unseen tests, OAT lowers mean compliance up to 90% relative to the best prior models and delivers sub-1 second inference on a single GPU across resolutions from 64 x 64 to 256 x 256 and aspect ratios as high as 10:1. These results establish OAT as a general, fast, and resolution-free framework for physics-aware topology optimization and provide a large-scale dataset to spur further research in generative modeling for inverse design. Code & data can be found at https://github.com/ahnobari/OptimizeAnyTopology.

1 Introduction

Engineering design remains reliant on costly optimization or narrowly scoped models, while topology-optimization predictors struggle to generalize across configurations. OAT combines a large randomized dataset with a single framework for broader structural and geometric coverage, reporting improved benchmark performance and fast inference.

1 Introduction

  • Conventional topology optimization repeatedly runs finite-element analysis and gradient updates, restarting when boundary conditions, geometry, or loads change.The nonlinear, constraint-sensitive physics and large non-convex search space make high precision important.Source: §1 · pp. 1–2
  • Current direct-prediction models struggle with arbitrary boundary conditions and forces, often relying on narrow configuration sets and fixed square grids.The passage cites examples with at most 210 unique setups and only 42 boundary-condition configurations, and describes failures on unseen or more complex settings.Source: §1 · p. 2
  • OpenTO contains 2.2 million optimized structures with randomized boundary conditions, loads, shapes, and resolutions, supporting OAT’s general structural topology-optimization framework.OAT is presented as a foundation-model approach for minimum-compliance structural TO, with scope across arbitrary boundary conditions and resolutions.Source: §1 · p. 2
  • OAT achieves up to 10x lower compliance error than prior models on established benchmarks, even without post-optimization refinement.It also delivers sub-second single-GPU inference and maintains nearly constant inference time from 64 x 64 to 256 x 256 grids.Source: §1 · p. 3

2 Preliminaries

Topology optimization seeks material layouts that optimize physics-based performance under governing equations and material-volume constraints. Diffusion models learn to generate samples by reversing noise, while DDIM offers a faster sampling route used in this work.

2.1 Topology Optimization

  • Minimum-compliance topology optimization distributes material to minimize compliance, or maximize stiffness, under specified forces and boundary conditions.Closed-form solutions are generally unavailable for complex geometries, so the problem is discretized and solved numerically using finite-element analysis.Source: §2.1 · p. 3

2.2 Diffusion Models

  • DDPMs add Gaussian noise over time and train a neural network to predict that noise, then generate samples through learned reverse transitions.The process begins from an approximately Gaussian-noise sample and iteratively moves toward a clean data sample.Source: §2.2 · p. 3
  • This work uses velocity prediction and DDIM’s deterministic, non-Markovian updates to reduce sampling steps while retaining sample quality.DDIM directly predicts an estimate of the clean sample or uses the predicted noise to take larger steps.Source: §2.2 · p. 4

3 Related Works

Prior work motivates compressed latent representations for resolution-flexible diffusion and more general representations of topology-optimization inputs. OAT extends these directions to arbitrary aspect ratios and domain geometries and adopts a point-cloud boundary-condition representation.

Diffusion Models and Resolution Free Variants

  • OAT builds on resolution-free autoencoders and finite-dimensional latent diffusion, extending prior approaches to arbitrary aspect ratios and domain geometries.The related-work discussion contrasts finite-dimensional latent representations with neural-operator approaches that sample all cells or pixels at inference.Source: §3 · p. 4

Representations of TO Problems

  • OAT adopts BPOM, a point-cloud representation of boundary conditions that avoids FEA simulation while mitigating resolution and shape dependence.Earlier image-based representations often require FEA or perform less well without it, and remain resolution- or shape-dependent.Source: §3 · p. 4

4 Methodology

OAT encodes topologies and problem specifications into fixed-size latent representations, then uses conditional latent diffusion to generate designs. Its methodology pairs a shape- and resolution-agnostic autoencoder with OpenTO’s randomized topology data.

4.1 Resolution- and Shape-Agnostic Autoencoder with Neural Fields.

  • The autoencoder maps variable topologies to fixed-dimension latents and uses a neural-field renderer to reconstruct them at arbitrary coordinates and cell sizes.It is trained to minimize L1 reconstruction error; convolutional rendering supports high-frequency detail, and tiled inference supports higher resolutions when memory is limited.Source: §4.1 · pp. 4–5

4.2 Problem Representation

  • OAT encodes domain aspect ratio, cell size, volume fraction, boundary conditions, and forces into a fixed-size problem embedding.Point-cloud boundary conditions and forces use BPOM encoders; MLPs encode volume fraction, cell size, and aspect ratio.Source: §4.2 · pp. 5–6

4.3 Latent Diffusion and Classifier-Free Guidance

  • The diffusion model predicts latent velocity conditioned on the problem specification, while classifier-free guidance adjusts conditioning strength during inference.Problem inputs are omitted for 50% of training samples to support conditional and unconditional denoising.Source: §4.3 · p. 6

4.4 Post Generation Optimizer Refinement.

  • OAT applies 5-10 SIMP steps as a local refinement, compared with 200-500 iterations for full optimization.The authors also report that OAT without direct optimization outperforms other models that require optimization steps.Source: §4.4 · p. 6

4.5 OpenTO Dataset

  • OpenTO addresses prior dataset limits by randomizing domain shapes, resolutions, boundary conditions, and forces, including interior fixtures and loads.It contains 2.194M samples, including 894,000 with labels; its randomized setups include aspect ratios up to 10:1, configurations with as many as 4000 loads, and 5,000 test samples.Source: §4.5 · pp. 6–7
  • On 64 x 64 datasets, OAT without post-generation optimization outperforms NITO + 10, which uses 10 optimization steps.All other models in the table were trained only on 64 x 64 data, while OAT was trained on OpenTO.Source: Table 1 · p. 7

5 Experiments

Experiments compare OAT on established benchmarks and fully randomized OpenTO problems, measuring compliance and volume-fraction errors. OAT outperforms prior models across the reported benchmarks and offers faster inference than iterative optimization.

5 Experiments

  • OAT is trained only on OpenTO, evaluated on existing and randomized benchmarks, and assessed using compliance error and volume-fraction error.The OpenTO setup compares OAT with benchmark-specialized models on their test sets and with a parameter-matched NITO variant on OpenTO.Source: §5 · p. 7

5.1 Comparison To State of The Art

  • 1.74% compliance error versus 2.59% for the best model without direct optimization on the 64 x 64 benchmark; OAT also leads with post-generation optimization.The reported comparison describes lower compliance and volume-fraction errors across all optimized settings.Source: §5.1 · p. 8
  • On 256 x 256, OAT nearly matches costly optimized methods without refinement; on five shapes, it outperforms NITO, while prior methods could not handle mixed shapes.Source: §5.1 · p. 8

5.2 The General Benchmark

  • OAT samples look more realistic than NITO’s noisy, gray outputs, but both models struggle on fully randomized OpenTO problems.Figure 3 also illustrates that slight OAT deviations can misplace material at boundary conditions and cause design failure.Source: §5.2 · p. 8, Figure 3 · p. 9
  • Eight OAT samples yield 20% failure and half the single-sample compliance error; failure rates decline as more candidates are evaluated.The reported results select the best of N generated samples for each problem.Source: Table 5 · p. 9, §5.2 · p. 9
  • Failure means compliance error above 100%; 5 to 10 optimization steps reduce OAT’s failure rate from 39% to 16%.The threshold corresponds to compliance more than double the conventional optimizer’s; the authors note that many failures need only small corrections.Source: §5.2 · pp. 8–9
  • On successful fully random samples, OAT’s compliance error is below 10%, falling to 5% after a few optimization steps.The paper contrasts the 5% result with TopologyGAN’s over 50% error on the limited 64 x 64 dataset.Source: §5.2 · p. 9

5.3 Sampling Efficiency & Inference Speed

  • OAT’s renderer scales as O(N) in the number of pixels or elements, and its GPU inference is faster than optimization, including at low resolution.NITO is faster at 64 x 64 but scales poorly with resolution; OAT uses 20 DDIM steps versus TopoDiff’s 100.Source: §5.3 · p. 10

6 Conclusion & Future Work

OAT is a shape-, aspect-ratio-, and resolution-agnostic foundation-model approach trained on 2.2 million optimized designs. It reports lower mean compliance error than prior deep-learning baselines and sub-second inference, while noting remaining challenges on fully random benchmarks.

6 Conclusion & Future Work

  • 1.7% versus 2.59% mean compliance error on the canonical 64 × 64 benchmark, against prior deep-learning baselines.OAT also reports sub-second inference.Source: §6 · p. 10
  • OAT is agnostic to domain shape, aspect ratio, and resolution, and is trained on 2.2 million optimized designs.The corpus is named OPENTO in the conclusion.Source: §6 · p. 10
  • <10% mean compliance error on a fully general benchmark with random boundary conditions and resolutions.Source: §6 · p. 10
  • OAT's auto-encoder reconstructs heatsink topologies zero-shot without retraining, suggesting extension to other topology-optimization problems and physics.Source: §6 · p. 10
  • OAT retains a notable failure rate on fully random benchmarks; future work targets optimizer-guided reinforcement learning, multiphysics objectives, and few-shot adaptation.Source: §6 · pp. 10–11

Table of Contents for Appendices

The appendix formulates topology optimization as choosing material density to optimize a physics-based objective, then describes the minimum-compliance setting and its numerical solution. It covers static linear elasticity, finite-element discretization, and SIMP material interpolation.

A Topology Optimization and The Minimum Compliance Problem

  • Topology optimization distributes material within a design domain to optimize a physics-based objective; practical problems are discretized and solved numerically with finite elements.Direct binary designs are challenging, and analytical solutions are generally unavailable for complex geometries and material distributions.Source: A Topology Optimization and The… · p. 16

A.1 Minimum Compliance Topology Optimization

  • Minimum-compliance optimization seeks the stiffest material layout under mechanical loads while limiting the total material used.The paper focuses on linear elasticity and iteratively optimizes a discrete topology using finite-element analysis.Source: §A.1 · p. 16, A Topology Optimization and The… · p. 16

A.1.1 The Physics Model: Static Linear Elasticity

  • Static linear elasticity models small deformations in a linearly elastic material, with boundary conditions and density-dependent Young’s modulus determining structural response.For static analyses, the inertial term is set to zero.Source: §A.1.1 · p. 17

A.1.2 Finite Element Discretization and Compliance Calculation

  • Finite-element discretization represents displacement and forces on a mesh, while compliance measures flexibility and is equivalent to maximizing stiffness when minimized.The stiffness matrix depends on the element density variables.Source: §A.1.2 · p. 17

A.1.3 Material Interpolation: The SIMP Model

  • SIMP interpolates element stiffness from density and penalizes intermediate densities, encouraging designs near solid or void.A small positive void modulus prevents a singular stiffness matrix; the data generation uses Emin = 10^-9 with Esolid = 1.0.Source: §A.1.3 · pp. 17–18

A.1.4 The Minimum Compliance Optimization Problem

The minimum-compliance formulation minimizes structural compliance subject to equilibrium and a material-volume limit. Element densities also have lower and upper bounds.

A.1.4 The Minimum Compliance Optimization Problem

  • The objective is to minimize compliance subject to static equilibrium and a maximum total material volume.The formulation is a specific instance of the discretized topology-optimization problem.Source: §A.1.4 · p. 18
  • Each element density is bounded below by ρmin and above by 1; a positive lower bound can keep element stiffness positive or represent minimum manufacturable thickness.The passage gives 10^-9 or 10^-6 as example lower bounds.Source: §A.1.4 · p. 18
  • The material-volume constraint limits the sum of element volumes weighted by their densities to Vmax.The formulation defines ve as the volume of element e and Vmax as the maximum permissible material volume.Source: §A.1.4 · p. 18

B Ablation Studies

The ablations select DDIM sampling with 20 denoising steps and guidance scale 2. The appendix also describes zero-shot reconstruction and randomized dataset domains spanning varied element counts and aspect ratios.

B Ablation Studies

  • DDIM sampling with 20 denoising steps and guidance scale 2 balances CE and VFE, while additional steps bring no benefit and slightly worsen performance.DDIM largely performs better than DDPM despite initially higher absolute VFE; the figure attributes DDPM’s lower CE largely to its higher material use.Source: B Ablation Studies · p. 19

C Out of Distribution Reconstruction

  • OAT’s auto-encoder reconstructs heatsink topologies without additional training, demonstrating zero-shot extension to different physics and topology-optimization problems.Source: Figure 5 · p. 20

D OpenTO Dataset

  • OpenTO randomizes domain discretization, aspect ratio, loads, and boundary conditions to create a diverse topology-optimization dataset.The stated aim is to construct a comprehensive dataset for foundation-model training.Source: D OpenTO Dataset · p. 21

D.1 Domain Definition and Discretization

  • OpenTO samples target element counts from 4096 to 16384 and aspect ratios from a log-normal distribution biased toward values near unity.Rounded domain dimensions are adjusted when needed to keep the element count within the intended range.Source: §D.1 · p. 21

D.2 Load and Constraint Specification

OpenTO randomizes multiple loads and constraints across internal and edge locations, then validates each sampled problem. Invalid instances are discarded and regenerated.

D.2 Load and Constraint Specification

  • OpenTO samples multiple loads and constraints of varied types and locations, including internal and distributed cases rather than limiting them to domain edges.Load and constraint types include internal points, edge and corner points, and distributed placements.Source: §D.2 · pp. 21–22

D.3 Problem Validation and Iteration

  • Generated problems are checked for sufficient constraints to prevent rigid-body motion and for non-trivial solutions; failures trigger regeneration.The validation passage gives coincident load and constraint application points as one possible issue.Source: §D.3 · p. 22

D.4 Algorithm for Data Generation

  • The generation algorithm samples domain dimensions, loads, and constraints, validates the instance, and outputs it only when it passes both checks.Otherwise, the loop repeats with a newly sampled candidate.Source: §D.4 · p. 23

E Further Visualization of Results

The visual examples contrast OAT and NITO on OpenTO, while the implementation details describe OAT’s autoencoder, conditional latent diffusion model, training, and computational resources.

F.1 Autoencoder Architecture

  • OAT’s autoencoder encodes topologies into a latent, decodes them into a feature tensor, and uses a neural-field renderer to reconstruct topologies from features, coordinates, and cell sizes.The renderer uses patch sampling during training and overlapping tiles for high-resolution inference.Source: §F.1 · p. 30

F.2 Autoencoder Training

  • The autoencoder is trained on 2.2 million OpenTO samples, with the renderer trained on sampled 64 × 64 topology patches or full images.Inputs are resized to 256 × 256 for the encoder.Source: §F.2 · p. 30

F.3 Latent Diffusion Architecture

  • The latent diffusion model uses a U-Net to operate on the autoencoder’s 64 × 64, single-channel latent encodings.The model is conditioned on problem specifications.Source: §F.3 · p. 31

Conditioning Mechanism

  • OAT encodes boundary conditions, forces, volume fraction, aspect ratio, and cell size into a problem embedding that conditions the diffusion model’s ResNet and attention blocks.Point-cloud inputs and scalar or low-dimensional conditions are processed separately before their embeddings are concatenated.Source: Conditioning Mechanism · p. 31

F.4 latent Diffusion Training

  • The latent diffusion model trains on latent codes paired with problem specifications, predicts diffusion velocity using mean squared error, and uses DDIM sampling for faster inference.Classifier-free guidance is enabled by randomly dropping problem conditioning during training.Source: §F.4 · p. 31

F.5 Computational Resources

  • Autoencoder training takes two days and diffusion-model training takes four days; both use H100 GPUs, while inference timing is measured on an RTX 4090.Training uses four H100 GPUs overall.Source: §F.5 · p. 32

F.6 Dataset Visualization

OpenTO visualization shows varied force and boundary-condition configurations, and the dataset combines newly generated samples with data from prior work.

F.6 Dataset Visualization

  • OpenTO includes non-repeating complex problem definitions with internal, edge, distributed, and point boundary conditions and forces.The figures provide examples of the dataset and visualize its forces and boundary conditions.Source: §F.6 · p. 32, Figure 16 · p. 33, Figure 17 · p. 34

F.7 Combining Prior Data With OpenTO

  • OpenTO provides labels for 700,000 of 2 million generated samples, 10,000 test samples, and incorporates 194,000 data points from prior work for 2.2 million total samples.The test samples follow the same procedural generation process.Source: §F.7 · p. 43

G Break-Even Analysis

The break-even analysis emphasizes the substantial data-generation cost for OAT-like models and estimates how many uses would offset it.

G Break-Even Analysis

  • About 2.32 million uses are needed to reach break-even under the data-generation-dominated calculation, with data generation costing 168 H100-days versus 16 H100-days for the final training run.The analysis notes that this calculation omits hardware efficiency and the value of faster TO for users with limited compute.Source: G Break-Even Analysis · p. 43
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