Optimize Any Topology (OAT) Framework

• Optimize Any Topology (OAT) is a foundation-model framework for 2D linear-elastic topology optimization that predicts minimum-compliance layouts across arbitrary aspect ratios, resolutions, and load configurations.
• It combines a shape- and resolution-agnostic autoencoder, an implicit neural-field decoder, and a conditional latent-diffusion model trained on the extensive OpenTO dataset for robust design generation.
• Empirical results show OAT significantly reduces compliance errors, achieving up to 90% improvement over prior models with sub-second inference times, despite limitations on handling more complex physics.

Optimize Any Topology (OAT) is a foundation-model framework for structural topology optimization that directly predicts minimum-compliance layouts for arbitrary aspect ratios, resolutions, volume fractions, loads, and fixtures. It combines a resolution- and shape-agnostic autoencoder, an implicit neural-field decoder, and a conditional latent-diffusion model trained on OpenTO, a corpus of 2.2 million optimized structures covering 2 million unique boundary-condition configurations. Within the domain considered—2D linear-elastic minimum-compliance design—OAT is intended to replace repeated instance-wise optimization with a single pre-trained model that is shape- and resolution-free, physics-aware through data, and fast at inference (Nobari et al., 26 Oct 2025).

1. Definition, scope, and optimization setting

OAT addresses minimum-compliance structural topology optimization under linear elasticity. In continuous form, the displacement field $u$ and density field $\rho$ satisfy

$$\mathcal{L}\bigl(\rho(x), u(x)\bigr) = f(x) \quad \text{in } \Omega, \qquad u(x) = g(x) \quad \text{on } \mathcal{D} \subset \overline{\Omega},$$

with a volume constraint

$\int_{\Omega} \rho(x) \, dx \le V_{\max}.$

The corresponding topology optimization problem is

$$\min_{\rho(\cdot)} C(\rho) \quad\text{s.t.}\quad \mathcal{L}(\rho,u) = f,\; u = g \ \text{on }\mathcal{D},\; \int_{\Omega} \rho(x)\,dx \le V_{\max},$$

where $C(\rho)$ is the structural compliance. In the discrete finite-element setting used for data generation, compliance is represented as $C = \mathbf{F}^\top \mathbf{U}$, and the training targets are SIMP-optimized minimum-compliance layouts (Nobari et al., 26 Oct 2025).

The design objective is therefore not to solve arbitrary topology optimization problems in the broadest possible sense, but to learn the mapping

$\hat{P} \mapsto \rho^*(\cdot;\hat{P}),$

where $\hat{P}$ encodes the design problem and $\rho^*$ is the optimized minimum-compliance solution. The paper defines $\rho$0 through boundary conditions, forces, volume fraction, cell size, and aspect ratio; OAT is consequently “shape- and resolution-free” only within this structural design class, not across arbitrary physics or objectives (Nobari et al., 26 Oct 2025).

The model is positioned against prior topology-optimization learning systems that are typically limited to fixed square grids, a small number of hand-designed boundary conditions, pixel encodings tied to finite-element preprocessing, and frequent reliance on post-hoc optimization. OAT is designed specifically to avoid those restrictions by using a continuous decoder and point-set conditioning rather than grid-tied PDE fields (Nobari et al., 26 Oct 2025).

2. Architectural composition

OAT consists of three coupled components: a resolution- and shape-agnostic autoencoder, an implicit neural-field renderer, and a conditional latent-diffusion model. The architecture has roughly 730M parameters, with about 40M in the autoencoder and the remainder in the latent-diffusion network (Nobari et al., 26 Oct 2025).

The autoencoder operates on optimized topologies $\rho$1. Each topology is padded and resized to a fixed $\rho$2 canvas before encoding: $\rho$3 The encoder $\rho$4 maps the input topology to a fixed-size latent representation $\rho$5, independent of the original aspect ratio or resolution. The decoder does not directly reconstruct a full image. Instead, it produces a feature tensor,

$\rho$6

which is then passed to an implicit renderer (Nobari et al., 26 Oct 2025).

The renderer is a neural field: $\rho$7 where $\rho$8 denotes spatial coordinates and $\rho$9 denotes cell or pixel size. This representation allows reconstruction on arbitrary target grids because the topology is sampled from a continuous field rather than emitted at one fixed raster resolution. The renderer is local and convolutional rather than a purely independent per-point MLP, which the paper associates with better capture of high-frequency structural detail and sharper boundaries (Nobari et al., 26 Oct 2025).

Diffusion is performed in latent space rather than pixel space. Given a clean latent $$\mathcal{L}\bigl(\rho(x), u(x)\bigr) = f(x) \quad \text{in } \Omega, \qquad u(x) = g(x) \quad \text{on } \mathcal{D} \subset \overline{\Omega},$$0, the forward process is

$$\mathcal{L}\bigl(\rho(x), u(x)\bigr) = f(x) \quad \text{in } \Omega, \qquad u(x) = g(x) \quad \text{on } \mathcal{D} \subset \overline{\Omega},$$1

OAT uses a velocity parameterization,

$$\mathcal{L}\bigl(\rho(x), u(x)\bigr) = f(x) \quad \text{in } \Omega, \qquad u(x) = g(x) \quad \text{on } \mathcal{D} \subset \overline{\Omega},$$2

and trains a conditional UNet-like model $$\mathcal{L}\bigl(\rho(x), u(x)\bigr) = f(x) \quad \text{in } \Omega, \qquad u(x) = g(x) \quad \text{on } \mathcal{D} \subset \overline{\Omega},$$3 to predict $$\mathcal{L}\bigl(\rho(x), u(x)\bigr) = f(x) \quad \text{in } \Omega, \qquad u(x) = g(x) \quad \text{on } \mathcal{D} \subset \overline{\Omega},$$4. The latent-diffusion loss is

$$\mathcal{L}\bigl(\rho(x), u(x)\bigr) = f(x) \quad \text{in } \Omega, \qquad u(x) = g(x) \quad \text{on } \mathcal{D} \subset \overline{\Omega},$$5

At inference, OAT samples a latent with DDIM steps and decodes it through $$\mathcal{L}\bigl(\rho(x), u(x)\bigr) = f(x) \quad \text{in } \Omega, \qquad u(x) = g(x) \quad \text{on } \mathcal{D} \subset \overline{\Omega},$$6 and $$\mathcal{L}\bigl(\rho(x), u(x)\bigr) = f(x) \quad \text{in } \Omega, \qquad u(x) = g(x) \quad \text{on } \mathcal{D} \subset \overline{\Omega},$$7 to obtain a topology at the requested resolution (Nobari et al., 26 Oct 2025).

3. Conditioning, invariance, and problem representation

A central design feature of OAT is its representation of boundary conditions and loads as point sets rather than as fixed-grid fields. Each topology optimization instance is described by

$$\mathcal{L}\bigl(\rho(x), u(x)\bigr) = f(x) \quad \text{in } \Omega, \qquad u(x) = g(x) \quad \text{on } \mathcal{D} \subset \overline{\Omega},$$8

where $$\mathcal{L}\bigl(\rho(x), u(x)\bigr) = f(x) \quad \text{in } \Omega, \qquad u(x) = g(x) \quad \text{on } \mathcal{D} \subset \overline{\Omega},$$9 is the point set of support locations and directional constraints, $\int_{\Omega} \rho(x) \, dx \le V_{\max}.$0 is the point set of load locations and associated 2D force vectors, $\int_{\Omega} \rho(x) \, dx \le V_{\max}.$1 is the target volume fraction, $\int_{\Omega} \rho(x) \, dx \le V_{\max}.$2 encodes cell size, and $\int_{\Omega} \rho(x) \, dx \le V_{\max}.$3 is the domain aspect ratio (Nobari et al., 26 Oct 2025).

OAT maps this structured specification to a fixed-length conditioning vector

$\int_{\Omega} \rho(x) \, dx \le V_{\max}.$4

The encoders $\int_{\Omega} \rho(x) \, dx \le V_{\max}.$5 and $\int_{\Omega} \rho(x) \, dx \le V_{\max}.$6 are order-invariant point-set encoders introduced to avoid dependence on the order or cardinality of supports and loads. Scalar quantities such as target volume fraction, cell size, and aspect ratio are embedded with separate MLPs and concatenated (Nobari et al., 26 Oct 2025).

This conditioning scheme replaces the field-based conditioning used by some earlier diffusion methods. A closely related precursor, “Diffusing the Optimal Topology,” already framed an “Optimize Any Topology” capability as the ability to map loads, boundary conditions, and volume fractions to performant layouts within a 2D linear-elastic setting, but its field-conditioned TopoDiff variants relied on physics fields or kernel relaxations plus optional SIMP refinement rather than a shape- and resolution-free foundation-model architecture (Giannone et al., 2023). OAT generalizes that ambition to arbitrary rectangular aspect ratios and resolutions within one learned latent space (Nobari et al., 26 Oct 2025).

Classifier-free guidance is used during diffusion. During training, conditioning is dropped with probability 0.5, enabling interpolation between conditional and unconditional predictions at inference: $\int_{\Omega} \rho(x) \, dx \le V_{\max}.$7 This provides tunable conditioning strength during DDIM sampling (Nobari et al., 26 Oct 2025).

4. OpenTO dataset and training regime

OAT is trained on OpenTO, a large-scale topology-optimization corpus designed to support a general, reusable model rather than a benchmark-specific predictor. OpenTO contains 2.194M optimized structures, with 894k samples carrying full problem specifications used for conditional modeling and 1.3M additional topologies used for autoencoder training. The dataset covers roughly 2M unique boundary-condition configurations (Nobari et al., 26 Oct 2025).

The data span rectangular domains with aspect ratios from approximately $\int_{\Omega} \rho(x) \, dx \le V_{\max}.$8 to $\int_{\Omega} \rho(x) \, dx \le V_{\max}.$9, and cell sizes from $$\min_{\rho(\cdot)} C(\rho) \quad\text{s.t.}\quad \mathcal{L}(\rho,u) = f,\; u = g \ \text{on }\mathcal{D},\; \int_{\Omega} \rho(x)\,dx \le V_{\max},$$0 to $$\min_{\rho(\cdot)} C(\rho) \quad\text{s.t.}\quad \mathcal{L}(\rho,u) = f,\; u = g \ \text{on }\mathcal{D},\; \int_{\Omega} \rho(x)\,dx \le V_{\max},$$1 of the domain size. Boundary conditions include interior supports and varied combinations of constrained directions. Loads include interior and multiple forces, with some configurations containing up to about 4000 distributed loads. Volume fractions are also randomized over typical engineering regimes. Ground-truth solutions are produced with a SIMP-based minimum-compliance solver (Nobari et al., 26 Oct 2025).

The autoencoder is trained on all 2.194M topologies with an $$\min_{\rho(\cdot)} C(\rho) \quad\text{s.t.}\quad \mathcal{L}(\rho,u) = f,\; u = g \ \text{on }\mathcal{D},\; \int_{\Omega} \rho(x)\,dx \le V_{\max},$$2 reconstruction loss,

$$\min_{\rho(\cdot)} C(\rho) \quad\text{s.t.}\quad \mathcal{L}(\rho,u) = f,\; u = g \ \text{on }\mathcal{D},\; \int_{\Omega} \rho(x)\,dx \le V_{\max},$$3

using the encode–decode–render pipeline $$\min_{\rho(\cdot)} C(\rho) \quad\text{s.t.}\quad \mathcal{L}(\rho,u) = f,\; u = g \ \text{on }\mathcal{D},\; \int_{\Omega} \rho(x)\,dx \le V_{\max},$$4, $$\min_{\rho(\cdot)} C(\rho) \quad\text{s.t.}\quad \mathcal{L}(\rho,u) = f,\; u = g \ \text{on }\mathcal{D},\; \int_{\Omega} \rho(x)\,dx \le V_{\max},$$5, and $$\min_{\rho(\cdot)} C(\rho) \quad\text{s.t.}\quad \mathcal{L}(\rho,u) = f,\; u = g \ \text{on }\mathcal{D},\; \int_{\Omega} \rho(x)\,dx \le V_{\max},$$6. The latent-diffusion model is then trained on the 894k labeled samples using the velocity objective above. The reported training budget for the latent-diffusion stage is 50 epochs and 349,219 optimization steps (Nobari et al., 26 Oct 2025).

The OpenTO corpus functions as more than a training dataset. It also defines a generalization benchmark of 5000 fully random test problems with unseen configurations. This is the principal evaluation setting for claims about OAT’s shape-, resolution-, and boundary-condition generality (Nobari et al., 26 Oct 2025).

5. Empirical performance and scaling behavior

OAT is evaluated on four public benchmarks and on two unseen tests, with compliance error (CE), median CE, volume fraction error (VFE), failure rate, and inference time as primary metrics. The paper states that 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 $$\min_{\rho(\cdot)} C(\rho) \quad\text{s.t.}\quad \mathcal{L}(\rho,u) = f,\; u = g \ \text{on }\mathcal{D},\; \int_{\Omega} \rho(x)\,dx \le V_{\max},$$7 to $$\min_{\rho(\cdot)} C(\rho) \quad\text{s.t.}\quad \mathcal{L}(\rho,u) = f,\; u = g \ \text{on }\mathcal{D},\; \int_{\Omega} \rho(x)\,dx \le V_{\max},$$8 and aspect ratios as high as $$\min_{\rho(\cdot)} C(\rho) \quad\text{s.t.}\quad \mathcal{L}(\rho,u) = f,\; u = g \ \text{on }\mathcal{D},\; \int_{\Omega} \rho(x)\,dx \le V_{\max},$$9 (Nobari et al., 26 Oct 2025).

On the canonical $C(\rho)$0 benchmark with 42 hand-defined boundary-condition configurations, OAT achieves mean CE $C(\rho)$1, median CE $C(\rho)$2, and VFE $C(\rho)$3 without refinement. The best prior model reported in the same table, TopoDiff with guidance, attains CE $C(\rho)$4. With 5 or 10 SIMP refinement steps, OAT reaches CE $C(\rho)$5 and $C(\rho)$6, respectively, outperforming prior refined baselines on that benchmark (Nobari et al., 26 Oct 2025).

On the $C(\rho)$7 benchmark, OAT achieves CE $C(\rho)$8, median CE $C(\rho)$9, and VFE $C = \mathbf{F}^\top \mathbf{U}$0 without refinement, compared with $C = \mathbf{F}^\top \mathbf{U}$1 CE for NITO and $C = \mathbf{F}^\top \mathbf{U}$2 for TopoDiff. On the mixed five-shape benchmark, OAT records CE $C = \mathbf{F}^\top \mathbf{U}$3 without refinement and $C = \mathbf{F}^\top \mathbf{U}$4 after 10 SIMP iterations, again matching or surpassing specialized alternatives (Nobari et al., 26 Oct 2025).

The OpenTO general benchmark is more demanding because it includes arbitrary shapes, resolutions, and unseen random boundary conditions. There, OAT obtains CE* $C = \mathbf{F}^\top \mathbf{U}$5, median CE* $C = \mathbf{F}^\top \mathbf{U}$6, VFE* $C = \mathbf{F}^\top \mathbf{U}$7, and failure rate $C = \mathbf{F}^\top \mathbf{U}$8 without refinement; with 10 refinement steps, CE* drops to $C = \mathbf{F}^\top \mathbf{U}$9, median CE* to $\hat{P} \mapsto \rho^*(\cdot;\hat{P}),$0, VFE* to $\hat{P} \mapsto \rho^*(\cdot;\hat{P}),$1, and failure rate to $\hat{P} \mapsto \rho^*(\cdot;\hat{P}),$2 (Nobari et al., 26 Oct 2025).

The paper also reports a multi-sample selection effect. Because OAT is generative, multiple candidates can be produced and evaluated. On the general benchmark, best-of-$\hat{P} \mapsto \rho^*(\cdot;\hat{P}),$3 selection improves performance steadily: best-of-2 gives CE* $\hat{P} \mapsto \rho^*(\cdot;\hat{P}),$4 and failure $\hat{P} \mapsto \rho^*(\cdot;\hat{P}),$5; best-of-64 gives CE* $\hat{P} \mapsto \rho^*(\cdot;\hat{P}),$6 and failure $\hat{P} \mapsto \rho^*(\cdot;\hat{P}),$7 (Nobari et al., 26 Oct 2025).

The following table summarizes the benchmark structure and representative OAT outcomes already reported in the paper.

Benchmark	OAT setting	Reported outcome
64×64, 42 B.C.s	No refinement	CE $\hat{P} \mapsto \rho^*(\cdot;\hat{P}),$8, median CE $\hat{P} \mapsto \rho^*(\cdot;\hat{P}),$9, VFE $\hat{P}$0
64×64, 42 B.C.s	+10 SIMP	CE $\hat{P}$1, median CE $\hat{P}$2
256×256, 42 B.C.s	No refinement	CE $\hat{P}$3, median CE $\hat{P}$4, VFE $\hat{P}$5
Five-shape benchmark	+10 SIMP	CE $\hat{P}$6, median CE $\hat{P}$7
OpenTO general test	No refinement	CE* $\hat{P}$8, failure $\hat{P}$9
OpenTO general test	Best-of-64	CE* $\rho^*$0, failure $\rho^*$1

A further distinguishing result is scaling. Reported inference time for OAT is about $\rho^*$2 s at $\rho^*$3 and about $\rho^*$4 s at $\rho^*$5, effectively constant over that range. By contrast, TopoDiff, DOM, SIMP, and large NITO variants all show substantially worse scaling with resolution. This behavior is attributed to latent-space diffusion and coordinate-based rendering rather than pixel-space generation (Nobari et al., 26 Oct 2025).

6. Downstream uses, comparative context, and limitations

OAT is explicitly presented as a foundation model rather than merely a benchmark-specific topology generator. A downstream example is TopoEdit, which treats a pre-trained OAT latent-diffusion model as a topology foundation model for post-optimization editing. TopoEdit uses OAT’s neural-field autoencoder as an encoder–decoder pair, OAT’s latent diffusion model as a conditional prior over optimized structures, and OAT’s conditioning interface to update boundary conditions and volume fractions during editing. In that setting, OAT is not retrained; its frozen latent space is repurposed for drag-based warping, shell–infill lattice replacement, and no-design-region enforcement (Chen et al., 25 Feb 2026).

This downstream use suggests that OAT’s latent space is structured enough to support partial-noise editing while preserving instance identity and enabling global structural recovery. TopoEdit reports that edited candidates can be generated in sub-second diffusion time per sample and that OAT-based latent editing better preserves mechanical performance than direct density-space edits in the tested cases (Chen et al., 25 Feb 2026). A plausible implication is that OAT’s learned representation organizes optimized layouts by mechanically meaningful structure rather than only by raster similarity.

Within the broader topology-optimization landscape, OAT differs from modular optimization frameworks such as SOPTX, which provide a multi-backend, automatic-differentiation-enabled computational infrastructure for topology optimization rather than a pre-trained generative model. SOPTX is closer to an extensible solver architecture, whereas OAT is a learned generative prior for minimum-compliance layouts (He et al., 5 May 2025). OAT also differs from feature-mapping and constructive-solid-geometry approaches such as TreeTOp, which optimize parameterized geometry directly through differentiable Boolean trees rather than learning a broad conditional distribution over optimized structures (Padhy et al., 2024).

The limitations identified for OAT are substantial and define its current scope. The model is trained only for 2D minimum-compliance topology optimization under linear elasticity. It does not address stress-constrained, buckling, thermal, fluid, nonlinear, or multi-objective topology optimization. It does not provide hard guarantees of volume or compliance satisfaction; those properties are learned statistically from data and can be improved, but not guaranteed, by optional SIMP refinement. Failure cases remain, particularly on the general benchmark, and commonly involve missing material at supports or load points that produces catastrophic compliance increases (Nobari et al., 26 Oct 2025).

The paper frames several extensions as open directions: reinforcement-learning or optimizer-guided training to reduce failures, few-shot or task-specific adaptation to new physics, broader objectives such as heat-sink optimization, and full 3D extensions. In that sense, OAT is best understood not as a universal topology optimizer, but as the first large-scale foundation-model formulation for shape- and resolution-free structural minimum-compliance design, together with a dataset and representation scheme intended to support subsequent work in generative inverse design (Nobari et al., 26 Oct 2025).