OpenTO: 2D Topology Optimization Dataset

• OpenTO is a large corpus of optimized 2D structural layouts generated via SIMP-based minimum-compliance topology optimization, offering diverse design challenges.
• It features mixed resolutions, arbitrary aspect ratios, and randomized boundary as well as interior fixtures and loads, enabling resolution- and shape-agnostic learning.
• OpenTO underpins the OAT framework, contributing to significant compliance error reductions and sub-second inference times through latent diffusion techniques.

OpenTO is a large, procedurally generated corpus of optimized 2D structural layouts introduced to address the generalizability limitations of prior deep-learning datasets for topology optimization. It was designed to support resolution- and shape-agnostic learning under arbitrary boundary conditions and forces, with broad variation in aspect ratio, discretization, volume fraction, fixtures, and loads, including interior point fixtures and interior forces (Nobari et al., 26 Oct 2025). Within the same work, OpenTO serves as the data foundation for OAT, a single model intended to generalize across unseen aspect ratios, resolutions, and boundary/load configurations in minimum-compliance topology optimization under linear elasticity (Nobari et al., 26 Oct 2025).

1. Origin and problem setting

OpenTO was introduced to overcome what the source paper calls the “generalizability challenge” in deep-learning-based topology optimization (Nobari et al., 26 Oct 2025). Earlier datasets were typically small and narrow: they often used fixed square domains, commonly $64 \times 64$, a small number of hand-coded boundary-condition configurations, single loads, and loads or fixtures applied only on boundaries. The paper states that some prior settings used as few as 42 boundary-condition configurations, which made learned models resolution- and shape-dependent and caused failure on unseen boundary conditions (Nobari et al., 26 Oct 2025).

The corpus was therefore designed to change the regime of supervision rather than merely increase sample count. Its defining design choices are mixed resolutions, arbitrary rectangular aspect ratios within a single dataset, randomized fixtures and loads, interior as well as boundary applications, many simultaneous loads per problem, and broad coverage of target volume fractions and discretizations (Nobari et al., 26 Oct 2025). This suggests that OpenTO is best understood not simply as a benchmark repository, but as infrastructure for training models whose latent representations and decoders are not tied to a single grid or to a small catalog of canonical mechanics problems.

The underlying optimization problem throughout the corpus is minimum-compliance topology optimization under linear elasticity with a volume constraint on total material. OpenTO’s layouts are therefore not arbitrary synthetic patterns; they are converged solutions generated by a standard SIMP-style iterative solver with finite-element analysis at each step and volume-constraint enforcement (Nobari et al., 26 Oct 2025).

2. Scale, composition, and coverage

OpenTO contains 2.194 million samples of optimized topologies and approximately 2 million unique boundary-condition configurations obtained by fully random sampling of fixture locations and directions; interior fixtures are explicitly included (Nobari et al., 26 Oct 2025). The dataset also incorporates 194,000 samples from the prior NITO dataset, bringing legacy benchmarks into the same corpus (Nobari et al., 26 Oct 2025).

A central distinction in the corpus is between labeled and unlabeled samples. The paper states that 894,000 samples include full problem specifications paired with optimized solutions, while the remaining samples provide topologies only (Nobari et al., 26 Oct 2025). The labeled subset is used for conditional generation, whereas the full corpus is used to pretrain a resolution-free autoencoder.

Component	Quantity	Role
Total samples	2.194 million	Full OpenTO corpus
Unique BC configurations	approximately 2 million	Coverage of randomized fixtures and loads
Labeled samples	894,000	Conditional training with full specifications
Added NITO samples	194,000	Unified legacy benchmark integration
Held-out test problems	5,000	Fully randomized generalization benchmark

The geometric and loading coverage is unusually broad for the literature described in the paper. Rectangular domains span narrow to square shapes, with aspect ratios up to $10\!:\!1$, and pixel or cell sizes range from $1/64$ to $1/1024$ (Nobari et al., 26 Oct 2025). Resolutions include canonical $64 \times 64$ and $256 \times 256$ cases, but also other rectangular grids. Each problem records a target volume fraction $\mathrm{VF}$, domain aspect ratio $a$, cell size $s$, and sets of forces and fixtures (Nobari et al., 26 Oct 2025).

Force configurations are fully randomized in location, direction, and magnitude, and may include up to 4000 loads in a single problem, with distributed loads covered (Nobari et al., 26 Oct 2025). Fixture sets consist of point fixtures with binary features indicating fixed degrees of freedom, and both boundary and interior fixtures are possible (Nobari et al., 26 Oct 2025). The paper emphasizes that inputs are distributed broadly and randomly across the corpus to ensure coverage rather than concentration in a few easy regimes.

3. Physical formulation and generation pipeline

OpenTO’s ground-truth layouts are generated from the standard density-based formulation of topology optimization. The equilibrium equation is

$K(\rho)u = f,$

where $10\!:\!1$0 is the displacement field, $10\!:\!1$1 the global force vector, and $10\!:\!1$2 the stiffness matrix assembled from element moduli interpolated by $10\!:\!1$3 (Nobari et al., 26 Oct 2025). Compliance is written as

$10\!:\!1$4

and the elementwise material interpolation follows the SIMP law

$10\!:\!1$5

with $10\!:\!1$6, base Young’s modulus $10\!:\!1$7, a small positive void modulus $10\!:\!1$8, and penalization exponent $10\!:\!1$9 (Nobari et al., 26 Oct 2025). The volume-fraction constraint is

$1/64$0

where $1/64$1 is the number of elements and $1/64$2 is the target volume fraction (Nobari et al., 26 Oct 2025).

The paper also gives the corresponding continuous statement on a domain $1/64$3 with Dirichlet conditions:

$1/64$4

together with the constrained optimization problem

$1/64$5

All physics assumes linear elasticity, and the resulting samples are stated to satisfy equilibrium and volume-fraction constraints (Nobari et al., 26 Oct 2025).

The source paper characterizes the generator as a standard SIMP-style iterative solver with FEA at each step and convergence under the volume constraint (Nobari et al., 26 Oct 2025). It notes that regularization typical of SIMP, such as density filtering or projection to mitigate checkerboarding and control length scales, may be applied, though exact parameterization is not the focus of the dataset description (Nobari et al., 26 Oct 2025). A typical SIMP run in the paper’s speed comparison takes on the order of 150 iterations to converge for $1/64$6 to $1/64$7 problems, which is presented as context for the cost of producing ground truth (Nobari et al., 26 Oct 2025).

4. Data representation and conditioning variables

OpenTO stores topology fields $1/64$8 on rectangular grids at native resolution and aspect ratio, so individual samples are variable-sized arrays rather than being forced into a single fixed raster (Nobari et al., 26 Oct 2025). This representation is central to the dataset’s intended use in resolution-free autoencoding and implicit neural-field decoding.

For the 894,000 labeled samples, each problem specification includes four groups of conditioning information: the aspect ratio $1/64$9 and cell or pixel size $1/1024$0; the target volume fraction $1/1024$1; the force set $1/1024$2, consisting of point coordinates plus 2D load vectors; and the fixture set $1/1024$3, consisting of point coordinates plus binary degree-of-freedom fix indicators (Nobari et al., 26 Oct 2025). Because the force and fixture sets are variable-length, the associated OAT model encodes them with BPOM, an order-invariant MLP, while separate small MLPs encode $1/1024$4, $1/1024$5, and $1/1024$6 (Nobari et al., 26 Oct 2025).

The train/validation/test organization is asymmetric. The paper states that 5,000 fully randomized problems are held out for testing, while the rest of the corpus constitutes the training pool for pretraining and conditional generation (Nobari et al., 26 Oct 2025). The unlabeled 1.3M+ layouts are used to pretrain the autoencoder, and the labeled 894k subset trains the conditional latent diffusion model (Nobari et al., 26 Oct 2025).

This design implies that OpenTO separates two learning problems: learning a resolution- and shape-agnostic latent geometry prior from all layouts, and learning conditional generation from fully specified mechanics problems using the labeled subset. The paper does not frame this as a formal hierarchy, but the split is structurally aligned with that interpretation.

5. Relationship to OAT and benchmark role

OpenTO is inseparable from the development of OAT, “Optimize Any Topology,” which the paper presents as a foundation-model framework for shape- and resolution-free structural topology optimization (Nobari et al., 26 Oct 2025). In that framework, a resolution- and shape-agnostic autoencoder provides fixed-dimensional latents for variable-resolution inputs, an implicit neural-field renderer decodes layouts at arbitrary resolutions, and a conditional latent diffusion model predicts optimized structures from encoded problem specifications (Nobari et al., 26 Oct 2025).

The conditioning embedding used by OAT is given as

$1/1024$7

The autoencoder is trained with an $1/1024$8 reconstruction loss,

$1/1024$9

and the latent diffusion model uses a velocity-parameterized objective with classifier-free guidance (Nobari et al., 26 Oct 2025). During training, conditions are dropped for 50% of samples to enable guidance (Nobari et al., 26 Oct 2025).

The paper attributes several capabilities of OAT directly to OpenTO’s breadth. The unlabeled portion supports learning resolution- and shape-agnostic latents and high-fidelity reconstructions across resolutions from $64 \times 64$0 to $64 \times 64$1 and aspect ratios up to $64 \times 64$2, while the labeled portion supplies the diversity of boundary conditions and loads required for robust permutation-invariant conditioning on unseen fixtures, loads, and domain geometries (Nobari et al., 26 Oct 2025).

OpenTO also serves as an explicit generalization benchmark. Its held-out test split contains 5,000 fully randomized problems, and the paper reports that even the best prior shape- and resolution-dependent models struggle on these randomized settings (Nobari et al., 26 Oct 2025). That role is methodologically important: the dataset is not only training data, but also a stress test against overfitting to conventional square-grid, boundary-loaded benchmark families.

6. Reported impact, scope limits, and future directions

In the reported experiments built on OpenTO, OAT achieves up to an order-of-magnitude, approximately 90%, reduction in mean compliance error relative to the best prior models on canonical benchmarks, while adhering more tightly to volume constraints (Nobari et al., 26 Oct 2025). The paper also reports sub-1 second inference on a single GPU with nearly constant runtime from $64 \times 64$3 to $64 \times 64$4, which it attributes to latent diffusion combined with the implicit renderer (Nobari et al., 26 Oct 2025). These results are claims about OAT rather than the dataset alone, but they function as evidence that OpenTO’s scale and diversity are sufficient to support foundation-model training for physics-aware inverse design.

The scope of OpenTO remains deliberately narrow in physical terms. It targets 2D minimum-compliance topology optimization under linear elasticity and does not include stress or buckling constraints, multi-physics settings such as thermoelasticity, or 3D problems (Nobari et al., 26 Oct 2025). The paper further notes that generative models trained on such data can still fail through small misplacements near fixtures or loads, producing high compliance despite apparently plausible layouts (Nobari et al., 26 Oct 2025). It reports that OAT’s failure rate on fully random problems drops substantially with a few SIMP refinement steps, for example 10 iterations, which indicates that learned prediction and classical refinement remain complementary (Nobari et al., 26 Oct 2025).

The future directions proposed in the source are dataset-centered as much as model-centered. These include extension to 3D and multi-physics settings, larger labeled subsets with finer-grained metadata, incorporation of manufacturing constraints such as length-scale and overhang restrictions, robust-design variants, and exploration of reinforcement learning or preference optimization to reduce failure rates and align generation more closely with optimizer feedback (Nobari et al., 26 Oct 2025). A plausible implication is that OpenTO is intended not as a terminal benchmark, but as the first large-scale substrate for a broader family of learned topology optimization systems.

OpenTO is released open-source through the OptimizeAnyTopology repository, with the paper directing readers to the repository’s LICENSE file for exact terms (Nobari et al., 26 Oct 2025). Its significance lies in the combination of scale, randomized mechanics specifications, native variable-resolution representation, and explicit benchmarking for out-of-distribution generalization—features that collectively reposition dataset design as a primary determinant of what learned topology optimization systems can generalize to in practice (Nobari et al., 26 Oct 2025).