Optimize Any Topology: A Foundation Model for Shape- and Resolution-Free Structural Topology Optimization
Authors: Amin Heyrani Nobari
Lyle Regenwetter
Cyril Picard
Ligong Han
Faez Ahmed
Venue: NeurIPS 2025 poster ()
Keywords: Topology Optimization
Diffusion Models
Deep Generative Models
Primary area: machine_learning_for_sciences
TL;DR: OAT introduces the first step towards foundation models for topology optimization by combining a neural field auto-encoder and latent diffusion models with large scale training on a new general dataset of 2M optimized topologies called OpenTO.

Abstract: 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 × 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.

Optimize Any Topology: A Foundation Model for Shape- and Resolution-Free Structural Topology Optimization
Amin Heyrani Nobari
Lyle Regenwetter
Massachusetts Institute of Technology Cambridge, MA, 02139 Aomobari@mit..o edu
Massachusetts Institute of Technology Cambridge, MA, 02139 regenwet@mit. edu
Cyril Picard
Ligong Han Red Hat AI, MIT-IBM Watson Al Lab Cambridge, MA, 02139 ligong. han@ibm com
Massachusetts Institute of Technology Cambridge, MA, 02139 cyrilpsmit.edui
Faez Ahmed
Massachusetts Institute of Technology Cambridge, MA, 02139 faez@mit. edu
Abstract
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, few hand-coded bound- ary conditions, and post-hoc optimization, preventing general deployment. We introduce Optimize Any Topology (OAT), a foundation-modes framework that directly predicts minimum-compliance layouts for arbitrary aspect ratios, reso- lutions, volume fractions, loads, and fixtures. OAT combines a resolution- and shape-agnostic autoencoder with an implicit neural-fiee decoder and a conditional lafent-diffusion model trained OpenTO, new corpus of 2.2 million optimized structures covering million unique houndary-condinng public benchmarks and two challenging unseen tests, OAT lowers mean compliance up to 90% relative to the best prior models and delivers sub-1 I second inference on a single GPU across resolutions from 64 x 64 to 256 256 and aspect ratios as high as 10:1. These results establish OAT general, fast, and resolution-free frame- work for physics-awand topology optimization and provide large-scale dataset spur further research in generative modeling for inverse design. Code & data can found at
configurations. On four
Introduction
Foundation models-large models trained on broad data that can be adapted to many downstream tasks-have transformed modern AI. They underpin advances in vision, language, and multimodal reasoning, powering systems such large language models (LLMs) and image generators that drive progress across domains, including scientific breakthroughs in protein folding [21] and drug discovery [16]. contrast, engineering design remains dominated by inverse problems, where the goal is to infer a design that meets stated constraints while maximizing performance objectives. Designing a bridge, for example, may require allocating fixed mass of steel so the finished span withstands specified loads with required stiffness. Because such performance maps highly the field continues to rely direct optimization [6]
nonlinear, inteennntensvve, and constraint-sensiteee or narrowly scoped deep-learning surrogates and generative models [38]. Limited datasets and the need for high precision have so far hindered the emergence foundation models for engineering design [38].
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Structural topology optimization (TO) exemplifies these challenges [49] and presents good testbed for developing engineering foundation models. TO seeks the optimal material distribution within design domain to maximize physics-based performance metrics such as stiffness strength. Solving TO conventionally involves repeated Finite Element Analysis (FEA) and gradient-based updates until convergence [3, 41, 50]. This process is computationally expensive and must be entirely restarted when boundary conditions, geometry, or load cases change, limiting its practicality in interactive or large-scale design settings. Solving the underlying PDEs on arbitrary shapes and boundary conditions also yields huge, non-convex search space, where tiny design changes can trigger failure; high precision is therefore essential.
To alleviate these costs, deep learning methods have emerged to accelerate or replace op- timization [38, 49]. While comprehensive reviews detail this expansive field [49, 62]. our focus on the evolution of models that directly predict near-optimal topologies from problem specifications (boundary conditions, forces, volume fraction) [13, 14, 20, 30, 33- 35]. By directly predicting optimal topolo- gies, these aim to substitute the entire opti- mization process with a rapid predictive frame- work. These typically employ Deep Genera- tive Models (DGMs) such as Variational Au- toencoders (VAEs), Generative Adversarial Net- works (GANs) [2, 26, 32, 37, 48, 61, 63], and more recently Denoising Diffusion Probabilistic Models (DDPMs) [13, 14, -0 frame TO as a conditional generation task. Despite numerous innovations, these direct prediction approaches have faced significant criticism by a few TO researchers for their limitations [34, 35, 62].
Food in Both and Ω Design Domain
F
Optmize Phycics-Basoe Objective Jutpp's P)
Flbodd in Y
d d.eundary Concitios Foroas
Optimieed Distibution
Torgel Vilume Phyyiis
Vinax
Alpis), s(x))
p(z)
Figure 1: In TO, the objective is to distribute material in a domain (a density field p(x)) to obtain optimal physics- based performance. Above, we show an example of TO for maximum stiffness. given boundary conditions of material supports and forces applicd.
The most significant issue, termed the generalizability challenge, is characterized by the inability of current models to manage arbitrary boundary conditions and forces [62]. Current direct prediction methods are trained on finite, often small, sets of predefined problem configurations (c.g.. a maximum of 210 unique setups [35]). Consequently, they fail to generalize to unseen or more complex settings, fundamentally limiting their practical utility. Key limitations include: 1) Restricted domain geometry, often confined to fixed square grids [13, 14, 30, 33], rendering many models limited proofs-of- concept. 2) Simplistic problem representations. frequently adapting FEA fields as images for standard vision models (e.g., CNNs) [34]. While recent works [34, 64] show promise in addressing some of these domain and representation issues, more critical limitation persists: 3) Poor generalizability. Existing models are trained on narrow, controlled problem sets (c.g., only 42 boundary condition configurations [14, 30, 34)) and fail on unseen conditions, considered major hurdle by critics [62].
We introduce the Optimize Any Topology (OAT) framework, a pre- pre-trained latent diffusion model for general structural TO focused on minimum compliance. OAT represents the first attempt to simultaneously address the representation and shape/resolution limitations noted in prior work, while also tackling the generalizability challenge. It is positioned as a foundational step towards TO models capable of handling any arbitrary boundary conditions and domain shape/resolution. Our foundational scope refers specifically to the model's 's ability to generalize across arbitrary boundary conditions, forces, and resolutions-ccaaalllii research. We emphasize that OAT is focused on the ceniimmm-comliannee problem under linear clasticity and does not encompass all forms of physics-driven topology optimization. Thus, rather than claiming a universal foundation model for physics-ban based design, we present OAT as the first step toward such general frameworks: a foundation model that captures the minimum compliance problem in its most general structural and geometric form.
that have not been demonstrated in prior topology optimization
The primary contributions of this work are as follows: 1) Open'T Dataset: We introduce Open'TO, the largest and first general-puryge -purpoee topology optimization dataset, comprising 2.2 million optimized structures with fully randomized boundary conditions, loads, shapes, and resolutions. OpenTO establishes scalable foundation for data-driven research in physics-based structural design. 2) OAT Framework: We propose Optimize Any Topology (OAT), the first deep learning framework capable of addressing the general topology optimization problem-handling arbitrary boundary conditions and resolutions within single model. 3) State-of-the-An Performance: Through extensive experiments, we demonstrate that OAT achieves up to 10x lower compliance error than prior models on established benchmarks, outperforming all cxisting architectures even without post-optimization refinement. It also performs better than existing methods on the foundational dataset of Open 4) Scalable Inference: We show that OAT achieves sub-second inference on single GPU and scales efficiently to higher resolutions, maintaining nearly constant inference time from 64 x 64 to 256 x 256
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grids-where prior methods experience severe slowdowns- thus enabling near real-time topology generation across diverse design problems.
Preliminaries
This section introduces the preliminaries of topology optimization and then examines the challenges and criticisms leveled towards deep learning approaches for TO.
2.1 Topology Optimization
Structural Topology Optimization (TO) is the process of determining the optimal distribution of a limited amount of material within a defined design space to maximize (or minimize) specific physics- based performance metrics. Let 52 C IRd represent a bounded design domain with boundary = 00.
The system's 's physical behavior is described by a state field u Q - IR" (e.g., displacement), which is the solution to a governing partial differential equation (PDE). For simplicity, considering only Dirichlet boundary conditions, this PDE takes the form:
[((()), u(x)) f(x in S2
w(x) g(x) on P 57
(I)
Here, p(x) Q + [0, 1] is the material density at each point I, acting as the design variable we seek to optimize. is a differential operator representing the underlying physics (e.g., clasticity), f(x) represents applied forces sources, and g(x) prescribes values for u(x) on a portion of the domain D (which can be on the boundary I' or within !!.. The general continuous TO problem is to find the material density distribution p(x: that minimizes a performance objective J (u(p), p). such as structural compliance (inverse of stiffness) or thermal resistance. This is subject to the governing PDE (Eq. 6) and constraint on the total material volume Vmax:
min Ju((p) p) s.t. ((p,u) u) = f(x), u(x) = g(x) on D. pds S Vmax- p(-)
This process and the overall problem are depicted in Figure 1. Analytical solutions to the PDE in Eq. are generally unobtainable for complex geometries and material distributions. These practical difficulties necessitate numerical approach to TO. This is usually done by discretizing the domain into a mesh and solving the problem using Finite Element Analysis (FEA). In this paper, we focus on a common type of topology optimization problem, the minimum compliance TO. The goal in minimum compliance TO is to distribute material to minimize compliance (i.e., maximize stiffness) given some boundary conditions and forces (Figure 1). We detail the underlying physics and optimization scheme for minimum compliance optimization, optimizers such as SIMP [3, 41] are generalizable, meaning that the domain can have any arbitrary shape, loads may be applied at any node of the FEA mesh and in any direction/magniuuae and the boundary conditions may also be applied in any location and for any direction. As such, a general framework should, in theory, be able to handle these generalities. This, however, proves to be rather difficult in deep learning frameworks built for TO.
Appendix A. Despite the cost and complexities of conventional
2.2 Diffusion Models
Denoising Diffusion Probabilistic Models (DDPMs) [18] are a class of generative models that learn to synthesize data by reversing noise corruption process. Given data sample zn from distribution q(zu), a fixed forward diffusion process gradually adds Gaussian noise over T discrete time steps. This process can be characterized by noise, and a is predefined noise schedule that decreases from ug approximately distributed as (0 I). Typically, neural network, ca f), is trained to reverse this process by predicting the noise component added at time step f to the noisy sample x,, The standard training objective, derived from a variational lower bound on the data log-likelhhoo, simplifies to minimizing the mean squared error between the true and predicted noise:
Varat + vI dge, where σ-r =s O, such that xy is
(N 7) is random
LDDEM = Exo... [[|c [ t)]
(x,,
(3)
Once trained, samples are generated by iteratively applying the learned reverse transition po ) starting from xy ~ (0, I). Alternative parameterizations for the reverse process exist. For instance, the model can be trained to predict "velocity" term vo (x,, !), such as Varze v1 04 which can offer benefits in training stability, uniform signal to noise ratio and sample quality [22, 44].
The objective then becomes ((0) Ezo.,, [| t) o| =|]2].
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For more efficient sampling, Denoising Diffusion Implicit Models (DDIMs) [53] introduce non- Markovian reverse process that allows for deterministic generation and significantly fewer sampling steps. DDIMs achieve this by deriving an update rule that directly predicts an estimate of xn (or uses co) to take larger, deterministic steps towards the clean data sample. In this way, DDIMs enable fewer sampling steps while retaining quality - crucial in the TO context, where the primary benefit of deep learning is solution speed. In our work, we use the "velocity" term formulation and DDIM rather than the vanilla DDPM objective.
3 Related Works
We now discuss prior works closely related our framework, building upon the established TO background and our paper's motivation.
Difftusion Models and Resolution Free Variants. Originating with Sohl-Dickstein et al. [52],
diffusion models [54-56] are now state-of-the-art for conditional and unconditional image synthe- GANs [15] and utility for inverse problems [57], an important feature for TO. Their iterative nature means slow inference, a drawback partially ad- dressed by faster sampling 124, 29, 45]. Nevertheless, pixel-space diffusion remains intensive for high resolutions, limiting prior TO works [34]; latent space compression offers a path to mitigate these issues. Our work aligns with Latent Diffusion Models (LDMs) [40], which perform diffusion in compressed latent space typically learned via autoencoders [51, 59]. Critical to this paradigm is the nature of the latent representation upon which the diffusion model operates. Most conventional autoencoders work on excd-resolutioonshaae images/domains, unsuitable for TO's need for arbitrary aspect ratios and resolutions, thus motivating resolution-
sis [10, 19, 31, 42, 43]. offering stable training
latent representations. Recent efforts extend diffusion to arbitrary resolutions [5, 8, 12, 23, 27]. One such approach performs diffusion in diiiitt-ddmesiinaal continuous spaces using neural operators [5, 12, 23, 25, 27]. offering training scalability but slow high-resolution inference as all cells/pixels must be sampled in inference. Alter- natively, resolution-free autoencoders with inite-dimensional latents [8] preserve LDM efficiency, often using Implicit Neural Representations (INRs) for decoding arbitrary resolutions. Given INRs' promise in TO [34, 35), we are inspired by Image Neural Field Diffusion (INFD) [8]. INFD introduces Convolutional Local Image Functions (CLIFs) for a high-quality, resolution- frree autoencoder with finite latents, relaxing INR's 's rixel-independence. Our approach extends this idea beyond variable resolution to arbitrary aspect ratios and domain geometries, integrating conditional latent diffusion model tailored for physics-based topology optimization.
Representations of TO Problems. Many prior DL TO works used image-based boundary con- ditions and force representations [14, 20, 30, 33]. These often require FEA simulations, limiting FEA-free inference, or offer lower performance if FEA-free [14], and invariably suffer from res- olution/shape dependence. Nobari et al. [34] proposed the Boundary Point Order-invariant MLP (BPOM), point-cloud boundary condition representation that matches FEA-based input performance without requiring FEA simulation (See Figure 2) and mitigating the resolution/shape dependence issucs. We adopt BPOM for its generalizability and efficiency.
Methodology
Our method, OAT, a Latent Diffusion Model (LDM) [40], first employs autoencoder to map topologies to fixed-dimension latents (decoded by a neural field for arbitrary resolution), then con- ditional diffusion model generates topologies from these latents based on TÔ problem specifications (Figure 2; Appendix for implementation details).
4.1 Resolution- and Shape-Agnostic Autoencoder with Neural Fields.
To handle the variability of mixed resolutions and aspect ratios (Fig. 2), we employ an autoencoder with a neural field renderer, inspired by INFD [8]. The encoder maps topology T to latent 2 ETT;; decoder D converts 2 to a feature tensor á = D(2); and renderer R, extending the CLIF concept [8], reconstructs / R((,, c, s) using o, coordinates c, and cell/pixel sizes
Scalable Training and Inference. Inspired by VQ-VAE [60] in LDMs [40], our autoencoder
pads and resizes variable topologies to fixed 256 x 256 inputs for encoder E (Fig. 2). Decoder D
then generates fixed resolution feature tensor c. A key challenge in TO is high-frequency changes
with small changes in boundary conditions. To address this, we employ a convolutional renderer R (unlike point-wise models [7, 34]) for better high- figh-frequency details and scale consistency [8].

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Open'Q Dataset
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Variable Shape & Resolution
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Petrestruttd Pasch
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Fandi Sandled fe Natiee fics usin
Refined Topolocy
BPOM
Siuree
Stepe Generated Topology
TO Problem
Ω Design Domain
Boundary Condition
Point Cloud
Shoundrry
BPDM
Prablem Embedding
P
Vinax
MLP
Ratio
MLP MLF
Boundary Conditions
Forces
Diffuson
D
R
Figure 2: OAT generative framework overview. resolution-free autoencoder encodes variable Open'T dataset topologies into fixed-resolution latent space. Latent diffusion models (LDMs) then conditionally generate topologies. Problem specifications, including forces and boundary conditions represented as point clouds, processed by BPOM models and MI.Ps to form fixed-size embedding to condition the LDM.
R's local operator (inputs: nearest pixel, coordinates, cell sizes; Fig. 2) requires connected patch sampling given its convolutional nature. Thus, scalable training is possible with random patches from non-padding areas (Fig. 2). High-resolution inference also possible by stenciled, overlapping padded tiles (padding > R's receptive field) when memory is limited.
Training Objective. This neural field design yields resolution and shape-agnostic autoencoder that gives us fixed-resolution latent representations. We train this autoencoder to minimize reconstruction error. For ground truth patch or topology Tor (with pixel positions e and cell sizes 8) and its reconstruction = R(ő, c, 8), the objective is to minimize L, norm, which has proven more effective for reconstruction in prior works [7, 8]: CAE
T -Trlll..
4.2 Problem Representation
In topology optimization, key challenge is to represent the underlying topology optimization problem, which includes domain shape, boundary conditions, and loads. This is difficult, as the number and position of these problem settings change in every configuration. We represent the TO problem (Eq. 8) by encoding its primary components using different resolution and indeeededeee encoders. The problem domain, S2 (rectangular grid with regular cells in our dataset), described by its aspect ratio E R² and its resolution via cell/pixel size (Fig. 2). The target material volume, Vmnx. is represented
scalar volume fraction VF Vmax/ Vna which indicates the ratio allowed material to total volume of the domain Vg. The more complex components, forces, and boundary conditions are handled as point clouds. Boundary conditions, which fix specific degrees of freedom, are represented as a point set Shoundary with binary features indicating directional fixture (Fig. 2). Similarly, applicd forces are point set Storee with 2D vector features denoting load magnitude and direction (Fig. 2). As these sets order invariant, MLP (BPOM) approach [34] to learn the embeddings of Shoundary and Storee The complete latent problem embedding, P. is concatenation of five embeddings from distinct neural networks:
employ the Boundary Point Order-invariant
P = (BPCa
(Soundary)) (S (BOO,((Ss..,,, ( MLP (VVV)) MLL MLP (a)), (a)),,
(4)
where BPOM, and BPOM, embed boundary conditions and forces, respectively, while Multi-Layer Perceptrons (MLPs) embed the volume fraction (VF), cell size (s), and aspect ratio (a). This construction yields a fixed-size embedding P capable of representing TO problems with diverse domain shapes, resolutions, and arbitrary boundary conditions and forces. For brevity, we denote this entire encoding process as P LP()), where P encompasses all raw problem inputs.
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4.3 Latent Diffusion and Classifier-Free Guidance
Having trained the autoencoder, obtain latent representations ET) for each topology T. Furthermore, the TO problem is encoded into a latent problem embedding = LP(P). We then train our diffusion model to predict the 'velocity' conditioned on the problem definition, P, and train with the following modified diffusion objective:
LUMM Ez E(T) [10-~0 20 (2,,t) P) 1)
(5)
where v. vaze - v1- azz, To enable classifier-free guidance [17], the problem inputs P are omitted for 50% of training samples, allowing the model to learn both conditional and unconditional denoising. This allows us to control the influence of conditioning during inference as described by Ho and Salimans [17]. Classifier-free guidance can be applied in our approach by adjusting the denoising velocity. Normally, in conditional denoising, one would use the predicted velocity at time step I, n0(=,, P), which predicts the velocity for the current noisy sample 2 given problem embedding P. When applying classifier- free guidance we can also compute the un-conditional denoising, namely "0(2,, Ø), and rather than just using (z L P), we measure the shift in velocity direction as result of conditioning, (2,, t P) (2 @) and amplify this shift by an over-relaxation
factor, w, leading to final denoising velocity ūe (z( PP) = vo(zi, Ø) + w (10(2),
P) 0o 2)), which can be used in place of vo(z,, P), mimicking what Ho and Salimans [17] propose for classifier-free guidance.
4.4 Post Generation Optimizer Refinement.
Despite the ever-increasing quality of DGM-generated solutions and other deep learning methods for TO, we observe that these models often lack precision in many cases and can lead to failed designs. To address this, we follow the few-step direct optimization approach commonly used in literature [14, 34]. A few optimization steps (5-10) of SIMP (small compared to 200-500 iterations for full optimization) are equivalent to a local search operation and are applied to topologies synthesized by OAT to improve the accuracy of the final samples. We also incorporate this into our framework; however, as we discuss in our experiments, even without any direct optimization, OAT outperforms other models that require optimization steps.
4.5 Open'O Dataset
Existing datasets in DL for TO face four challenges that we have overcome in creating the OpenTO
dataset: 1) Most existing datasets are limited one specific domain shape (square) and one resolution
(64 x 64), with the most expansive dataset including five different shapes and resolutions [34]: 2) Most existing works are limited to a small set of pre-defined boundary conditions (a maximum of 42 in 2D dataset [34]); 3) Existing datasets typically feature only single applied force; 4) All existing datasets only apply forces and boundary conditions at the borders of the design space (boundary of domain), and lack interior forcing and fixtures.
These limitations restrict the development of foundation models and are key factor behind the generalizability challenges. Thus, we see this major gap and introduce the largest open-source Topology Optimization (OpenTO) dataset for general purpose topology optimization with 194M samples. To enable a foundational scale of data, OpenTO comprises procedurally-creat creeted random TO problems, running a SIMP solver, and obtaining optimal topologies from the solver. The data generation is done such that OpenTO overcomes the aforementioned limitations of prior works. The main features of what makes Open'TO effective are: 1) Design domains in Open'O are fully random in both resolution and shape with aspect ratios ranging from very narrow (10 to 1), to square to 1), and pixel/cell size ranging from 1/64 to 1/1024, covering the exhaustive range of rectangular domain shapes and resolution in most practical real-world applications of TO in 2D; 2) OpenTO includes randomly sampled boundary conditions, including interior boundary condition point, with every sample having unique boundary condition configuration; 3) Open'O includes fully random forces including interior forces, and configurations with many 4000 loads (see Appendix D on distributed forces) in one configuration. The exhaustive nature of OpenTO makes it the first dataset to tackle the general TO problem and a starting point for the development of foundation models for TO. Open'TO includes 5,000 test samples, with a fully randomized configuration not in the main training data for testing the performance of models in a general problem setting. The full details of the procedural data generation can be found in Appendix D.

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Finally, OpenTO additionally includes the prior datasets developed in limited settings (194,000 samples from Nobari et al. [34]). Overall, Open'O includes 2.194M samples of topologies, with 894,000 samples being labeled (with defined P) and the rest only including the topology. We use all samples to train our autoencoder, and only use the 894,000 samples with labels to train our LDM.
5 Experiments
In this section, we perform experiments to showcase OAT's increased performance compared to prior works trained on the smaller, limited datasets. This constitutes the first truly general TO benchmark on fully randomized problems. Before detailing the experiments, we will first briefly describe the evaluation metries we use measure the performance of each model.
Evaluation Metrics: As we described in Section 2.1, 1, TO involves both a compliance objective and volume fraction constraint. Thus, we measure the performance of models with respect to these two requirements of the TO problem. First, to measure how well the models perform in compliance minimization, we measure compliance error (CE), which is calculated by subtracting the compliance of a generated sample from the compliance of the SIMP-optimized solution for the corresponding problem, then normalizing by the SIMP-optimized compliance value. Since the mean compliance error tends to be dominated by just a few outlier samples, we also report the median compliance error. We also report mean volume fraction error (VFE), which quantifies the absolute error between the generated topology's 's actual volume fraction and the target volume fraction for the given problem. This benchmarking is standard practice for DGMs in TO [14, 30, 33, 34).
Experimental Setup. Our experiments benchmark OAT both on existing benchmarks with limited boundary con- ditions and on our new TO test set with generalized problem configurations. In all experiments, exclusively on the Open'TO dataset, and never on any manchmrrk-speific data. On existing benchmarks, we compare against the specialized existing models trained specifically for these benchmarks and report performance metrics as reported by the original authors. On the Open'TO benchmark, only one prior work, NITO [34], has the architectural generalizability to be trained on this data. Thus, we train OAT with a combined 729. 99.94M (672.00M for LDM and 57.90 17.94M for AE) parameters, training the diffusion model for 50 epochs (349,219 steps). For a fair baseline, train variant of NITO for the same number of training steps. scaled to 732M parameters to approx- imately match OAT. For each method, we then sample solutions for each problem and test performance with and without direct optimization. In all experiments, OAT sam- pling is done with 20 DDIM deterministic denoising steps with classifier-fre free guidance scale of 2.0 (as informed by our ablation studies presented in Appendix B. We sam- ple 10 sets of solutions for each problem and report the average across runs.
Table 1: Quantitative evaluation on 64 x 64 datasets. All models were only trained on 64 x 64 data, except OAT, which is trained on the OpenTO dataset. w/ G: using a classifier and regression guidance. CE: Compliance Error. VFE: Volume Fraction Error. + 5 and 10 indicate and 10 steps of post-generation direct optimization. OAT, without any post- generation optimization, outperforms NITO 10, the SOTA model that also uses 10 steps of optimization.
train OAT
Model
CE&
CE
% Med
VFE %
TopologyGAN [33]
48.51
2.06
11.87
cDDPM [14]
60.79
TopoDift [34]
3.23
0.45
1.14
TopoDiff w/ G [34]
2.59
0.49
1.18
DOM w/o TA [14]
13.61
DOM w/ TA [14]
4.44
0.74
0.74
NITO [34]
8.13
0.47
1.40
OAT (Ours)
1.74
0.32
0.25
NITO 5 [34]
0.30
0.12
0.40
'TopoDif'' + [34]
3.55
0.42
0.67
Topa w/ 5 [34]
2.24
0.44
0.69
OAT (Ours)
0.22
NITO 10 [34]
0.17
0.071
0.25
TooooDiff 10 [34]
1.38
TopoDiff' w/ G+ 10 [34]
1.05
0.32
0.45
OAT (Ours) + 10
0.0503
0.014
0.27
5.1 Comparison To State of The Art
Prior works are typically limited to specific domain shapes and resolutions. We first compare the performance of OAT with prior state-of-the-art on their own test sets [30, 33, 34, 34].
The 64% x 64 Benchmark. The most common benehmark that previous state-of-the-ar models [14, 33, 33, 34] have trained on and benchmarked against is TO problems with a single force, and only 42 distinct boundary condition configurations defined on a 64 64 domain. We evaluate OAT on this benchmark with no specific training fine-tuning. We also test post-generation optimization, which some prior works [14, 34] explore. Table shows the results
the 64 64 benchmark. We can that OAT has the lowest compliance error amongst models without direct optimization, with only 1.74% error compared to 2.59% from the best model (Topo with classifier guidance), showing a 32% reduction in compliance error. Notably, OAT achieves this while maintaining stricter adherence to the target volume fraction, demonstrating superior efficiency in material usage. Table shows that when post-gencration optimization is applied, OAT continues outperform all competing methods,
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Figure 3: Left: Samples from ground truth (top), NITO (middle), and OAT (bottom) random Open'O benchmark; compliance errors shown below. NITO outputs noisy with many gray pixels, unsuitable for real- world use. Right: Example highlighting sensitivity in TO; slight deviations by OAT led to material misplacement at boundary conditions, causing design failure. This highlights the precision requirements of TO.
achieving the lowest compliance and volume-fraction errors across all settings. This represents the first instance in topology optimization where a foundation moode--odl-bbaee based approach not only generalizes across problem configurations but also surpasses all specialized, task task-specific models trained on restricted datasets. This establishes OAT major milestone in advancing generalizable, data-driven topology optimization.
The 256 x 256 Benchmark. Nobari al. [34], their neural field-based method, expand the prior mentioned data by solving the same problems at a much higher resolution of 256 x 256 resolution (still the same boundary conditions and single load) and test their method and the highest performing prior work, Difo [30], on this data by retraining a larger TopoDiff on this resolution. We report the results of our model and the reported values for NITO and TopoDiff [34], in Table 2. The trends here are similar the 64 x 64 data, with an even larger gap in performance, with specialized models unable beat the general framework of OAT in this harder high resolution benchmark. However, we do see that NITO, which predicts blurry gray samples [34], allows more freedom for direct optimization, thus yielding marginally better compliance eπrors with few-step optimization. However, note that OAT can perform nearly par with these costly direct optimization methods without using any direct optimization.
Table 2: Quantitative evaluation on 256 x 256 datasets. Same settings as 64 x 64.
Extended five shape benchmark. Aside from extending the dataset to higher resolution, Nobari al. [34] also expands the data by introducing three additional domain shapes apart from 64 x 64 and 256 256. The authors add data for 64x48, 64x32, and 64x 16 domains, still with the same boundary condition configuration adapted to these new shapes. The authors then train their model on all of this data and report the performance on this benchmark. Unlike prior works, which could not handle this kind of mixed shapc/resolution data, our approach, like NITO [34], can be adapted to this data easily. Table 3 shows how a NITO model only trained on this dataset lags behind OAT, we saw in prior benchmarks.
Model
CE %
CE % Med
VFE %
Topo [34]
16.62
0.59
2.92
NITO [34]
9.178
0.96
1.52
OAT (Ours)
1.51
0.51
-0.17
NITO [34]
0.25
0.09
0.34
OAT (Ours) + 5
0.54
0.37
-0.12
NITO 10 [34]
0.033
0.012
0.130
OAT (Ours) 0.081
0.16
-0.0502
Table 3: Quantitative evaluation on the com- bined 5 shape datasets, NITO results are only available with direct optimization the orig- inal work [34].
5.2
The General Benchmark
The only prior work that can handle the OpenTO dataset is the NITO [34] model, which is non-generative neural field model. As such, we train NITO on Open'O and report the performance of the model our fully random benchmark. Notably, even though we train variant of NITO with the same number of parameters, we observe very poor visual quality in samples generated by NITO (Figure 3). The authors of the original work had also observed non-sharp boundaries and suggested that this issue [34] may be due to the deterministic genn-genrratie) nature of their model. OAT learns this more complex benchmark significantly better. Figure 3 shows that the quality of samples generated by OAT is more realistic. Nonetheless, Op presents a challenging benchmark that both OAT and NITO struggle with.
Model
CCE
CE
Med VFE
NITO [34]
0.56
0.13
0.39
OAT (Ours)
1.94
0.89
0.103
OAT (Ours) +
0.82
0.12
0.35
OAT (Ours)
0.23
0.091
0.28

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Table 4: Quantitative evaluation on the fully general henchmark of TO problems. It can he seen that OAT outperforms NITO-L, which is the only existing model that could be trained on OpenTO. Values are reported for the non-fainlee non-failure samples.
Model
CE %
CE* % Med VFB* %
Failure Rate %
NITO
13.14
6.43
5.3861
60.01
OAT (Ours)
8.17
3.75
-0.12
39.22
NITO +
5.10
1.78
1.75
23.88
OAT (Ours) +
6.29
2.86
0.68
22.11
NITO 10
3.56
1.33
0.83
16.39
OAT (Ours)= 10
4.31
2.24
0.57
15.90
Table 5: Inference time scaling results of generating multiple samples and reporting results for the best of N samples generated by OAT on the fully random benchmark.
Best of CE %
CE*
Med
VFE* %
Failure Rate
7.40
3.02
-0.0021
32.95
5,50
2.65
0.041
27.50
8
4.14
2.41
0.096
21.50
16
1.97
1.58
0.14
16.15
32
1.76
1.55
0.15
12.92
64
0.4439
1.31
0.19
11.51
Failure Study. Given the precise nature of the topology optimization problem, small changes in material distribution can cause catastrophic failures in the outcome. This often occurs when the model fails to place material on boundary conditions loads precisely. The example in Figure 3 shows how failed topology looks very similar to the ground truth topology but has poor physical performance. Since DGMs often focus on distribution matching but fail the problem, such failures are common in DL-based TO. In this benchmark, we define failure as having compliance error greater than 100%, meaning that the generated topology has a compli- ance more than double that of the conventional optimizer. Despite OAT frequently incurring such failures, its failure rate is still significantly lower than NITO's. s, as shown Table 4. Given that the topologies in most failure cases only need small corrections, 5 to 10 steps of optimization reduce OAT's failure rate from 39% to only 16%. This confirms that these imprecisions a handful of optimization steps. Further improvements to failure rate may be addressed by larger models and datasets, as well as more advanced paradigms emerging in foundation model training. such as reinforcement learning techniques [36, 47] utilizing invalid data [39].
Table 6: Average inference time at 64 x 64 and 256 x 256 resolutions on an RTX 4090 GPU (10-run average). Some timings include refinement iterations (marked w/ Ref). NITO-S NITO-L are original and larger variants [34]: OAT timings steps. Parentheses show increase factor when elements scale by 16x. SIMP times are reported using an intel 14-9008 CPU (RTX4090 for SIMP-GPU), All SIMP infer- ence speeds are measured for 150 steps of optimization (average needed to converge).
to capture the precise requirements
20 DDIM
be healed with only
Model
Parameters (M)
Inference Time
TepoDitf
1.86
TopoDiff w/
4.79
DXM
121
0.82
SIMP
3.45
SIMP GPU
9.13
NITO-S
22
0.005
Generalized Performance. In addition to the failure rate, we also report the compliance and volume fraction er- rors for each method with and without direct optimization,
NTTO-S w/ Ref NITO-L
22
0.14
732
0.05
NITO
Ref
732
0.18
OAT
0.508
measured only on the successful samples. OAT signifi- cantly outperforms NITO in both of these physics- performance metrics. Notably, OAT has a compliance error of less than 10% the fully random general data,
Ref
0.637
Resolution
TopoDiff
553
10.81 (5.812x)
TopoDilf w/
1092
22.04 (4.601 x)
DOM
553
7,82 (9.557 x)
which reduces to only 5% after few steps of optimization. Comparing this to the first models for deep learning TO,
SIMP GPU
SIMP
69.45 (20.13x)
(1.30(19880x)
NITO-S
22
0.16 (32.00>)
such as TopologyGAN [33], which had over 50% error on the limited 64 x 64 dataset, 5% error on fully random data marks significant step forward for the community and heralds a potential paradigm shift toward foundation models.
NTTO-S w/ Ref
NITO-
22
2,88 (20.57
NITO-L
(Ours)
(1.01)))
OAT (Ours) w/ Ref
730
3.28 (5.129x)
Inference Time Scaling Study. The high failure rate observed in Table may seem stark upon first glance; however, we believe that the generative nature of OAT provides clear path to alleviate this issue. With generative models like OAT, one can generate vast numbers of candidates in short amount of time, considering batch inference of diffusion models scales better than sampling one

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problem at a time. In Table 5, we generate multiple candidates for each problem and measure the performance of the best amongst all samples. These results clearly show how the failure rate goes down rapidly as more samples are evaluated, with as few as eight samples yielding only 20% failure and a CE that is half that of the single sample run in Table 4. We believe this shows how powerful foundational generative frameworks like OAT can be for physical AL. Moreover, this favorable scaling with more samples clearly shows how emerging approaches around reinforcement learning [36, 46] in generative models can be employed here to further improve the performance of OAT.
5.3 Sampling Efficiency & Inference Speed
Inference speed is one of the key advantages of deep learning over iterative optimization. It's important to mention that a high-level analysis of computational cost in this case would result in an O(N) complexity for OAT, where N is the number of pixels/elements for given sample. This is because the non-constant cost of OAT is in the neural field renderer, which would have complexity of ((N). This is compared to the expensive FEA simulations at each step of the optimization, yielding an ((N3) complexity (Solving a linear system of equations). However, given that FEA matrices in this setting are highly sparse and the meshes in our work are structured, specialized multi-grid solvers can, in theory, reach an ((N) efficiency [l. 28, 58]. This high-level analysis is however, fails to realize the inference efficiency of deep learning models compared to conventional optimization, given that it does not take into account the highly parallelized inference of deep learning models on the GPU compared to the iterative nature of linear system solvers. This makes inference time overall better measure of performance in practical consumer hardware. Thus, it is important to analyze the inference time of different approaches. Table 6 shows the results of inference speed for different models. NITO, the only model with comparable generalizability. is faster at 64 64 resolution, but requires the whole network for each sample point over the entire field, meaning that scales poorly with resolution. In contrast, the latent diffusion model, whose autoencoder only has 40M parameters, can be sampled at high resolution much more efficiently. NITO heavily relies on post-sampling optimization, which often contributes the majority of inference cost. Even ignoring this optimization case, OAT scales significantly faster than NITO and generates samples much faster at higher resolutions, despite having the same number of parameters. OAT even outperforms similar diffusion models in inference speed. For example, compared to Difoss 100 denoising steps, OAT's s DDIM sampling requires only 20 denoising steps, significantly accelerating inference. Most importantly, unlike some prior diffusion models, OAT remains faster than optimization even at low resolution, where direct optimization is relatively fast. Overall, OAT is significantly faster than SIMP, making it an attractive option for fast design space exploration.
It is important note that some critics of generative models for TO point the break-even analysis of computational cost [62]. We estimate this and discuss this concern in more detail in Appendix G.
6 Conclusion & Future Work
We have introduced Optimize Any Topology (OAT), the first foundation-model approach to structural topology optimization that is agnostic to domain shape, aspect ratio, and resolution. Trained on the new OPENTO corpus comprising 2.2 million optimized designs, OAT couples resolution-free latent auto-encoder with a conditional diffusion prior and achieves much lower mean compliance error than prior deep-learning baselines on the canonical 64 x 64 benchmark (1.7 % versus 2.59 %) while delivering sub-second inference. Moreover, OAT is easily extendable to different TO problems and physics, such as heatsink optimization, given OAT's auto-encoder has great zero-shot capability for reconstructing such topologies without re-training (see Appendix C). This clearly solidifies the case for OAT as a foundational framework for TO. On fully general benchmark with random boundary conditions and resolutions, OAT attains < 10 % mean compliance error. These results demonstrate that large-scale pre-training combined with resolution-fis -ree latent diffusion is a viable path towards real-time, physics-awar design exploration. Despite the promising performance of OAT, significant challenges remain. First, we note that OAT faces notable failure rate on fully random benchmarks, which future work will focus on addressing through directions such as reinforcement learning [36, 46] based on optimizer guidance in diffusion models [1]]. Future work should also focus on addressing multi-physics objectives, and develop few-shot fine-tuning approaches to quickly adapt OAT or other foundational frameworks to new physics such as strets-constrained and buckling TO problems.
7 Acknowledgements
We would like to acknowledge Akash Srivastava and the MIT-IBM Watson Al Labs for helpful insights and guidance in the early stages of developing our methods. We would also like to acknowledge the crucial guidance from Professor Josephine Carstensen in developing the solver used to generate the data and perform benchmarks in this work. PI Ahmed acknowledges support from the National Science Foundation under CAREER Award No. 2443429.
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An exploration
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Table of Contents for Appendices
A Topology Optimization and The Minimum Compliance Problem
21
A.I Minimum Compliance Topology Optimization
22
A.1.1
The Physics Model: Static Linear Elasticity
22
A.1.2 Finite Element Discretization and Compliance Calculation
22
A.1.3
Material Interpolation: The SIMP Model
23
A.1.4
The Minimum Compliance Optimization Problem
23
B Ablation Studies
24
C Out of Distribution Reconstruction
25
D Open'OO Dataset
26
D.1 Domain Definition and Discretization
26
D.2 Load and Constraint Specification
26
D.3 Problem Validation and Iteration
27
D.4 Algorithm for Data Generation
28
E Further Visualization of Results
29
Implementation Details
34
F.1
Autoencoder Architecture
34
F2
Autoencoder Training
34
F3
Latent Diffusion Architecture
35
F44
latent Diffusion Training
35
F55
Computational Resources
36
F6
Dataset Visualization
36
F.7
Combining Prior Data With Open
46
G Break-Even Analysis
46
H Code & Data
46
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A Topology Optimization and The Minimum Compliance Problem
Structural Topology Optimization (TO) is the process of determining the optimal distribution of a limited amount of material within a defined design space to maximize (or minimize) specific physics- based performance metrics. Let Ω C IRd represent a bounded design domain with boundary I -= 0Q. The system's 's physical behavior is described by a state field Ω - "R (e.g., displacement). which is the solution to a governing partial differential equation (PDE). For simplicity, considering only Dirichlet boundary conditions, this PDE takes the form:
[(p(x), u(x))= f(z in S2
u(x) g(x) on D DCD 5
(6)
Here, p(x): Ω - [0, is the material density at each point æ, acting as the design variable we seek to optimize. L is a differential operator representing the underlying physics (e.g., elasticity), f() represents applied forces or sources, and g(x) prescribes values for u(z) on a portion of the domain DD (which can be on the boundary P or within Q). The general continuous TO problem is to find the material density distribution p(x) that minimizes a performance objective J (u(p), p). such as structural compliance (inverse of stiffness) or thermal resistance. This is subject to the governing PDE (Eq. 6) and constraint on the total material volume Vmax:
min [u(p), P) s.t. u)p f(x), u(x) = g(x) on D, J. pdx S Vaax- p(-)
This process and the overall problem are depicted in Figure 1. While the ideal outcome a distinct structure W C Ω with clear boundaries (binary densities), optimizing for such a structure directly is challenging. Furthermore, analytical -closed-form) solutions to the PDE in Eq. 6 are generally unobtainable for complex geometries and material distributions. These practical difficulties necessitate a numerical approach TO. This is usually done by discretizing the domain into a mesh with finite elements/cells, and solving the problem this discretized space using Finite Element Analysis (FEA) [9].
So far, we described the general topology optimization problem in continuous space; however, as we alluded to, this problem is often not approached in the continuous form. To make the problem tractable. the continuous domain f2 is discretized into finite element mesh Tr, composed of N small elements Th = [Ke J Within this framework, the continuous material density p(z) is approximated by a vector element densities Ps [PI,.... pN]] where each Pe E [0, 1]. Similarly, the state field (x) becomes vector u, [ul..... uM] I representing values M discrete points or "degrees of freedom" (DoFs) of the mesh (e.g.. 2D displacements at nodes). Applied forces f(x) and prescribed boundary values g(z) are also discretized into vectors fr. and gh. where the boundary conditions are applied at specific DoFs Dr. The local element operator (e.g., stiffness matrix) Ke is derived from L for each element [9]. The numerical TO problem then becomes finding the optimal discrete densities Ph:
K K(pn) un Sn, N
min Jn (un(p)) s.t. eV Pl
Ve S Vmaxx
(8)
U4 = Js: E Dr,
where Jn is the discretized objective function, K(pa) is the global system matrix (e.g., global stiffness matrix) assembled from element contributions. which depends on the material densities Ph. and v. is the volume of element e. The first constraint, K ((pn)ux = Sn, is the discretized form of the governing PDE (Eq. 6). Although the goal is often a binary design (p. E {0. 1}. indicating presence or absence of material), directly sol this as a mixed-integer non-linear program is computationally prohibitive for realistic problem sizes. Therefore, the optimization typically allows continuous densities Pe E [0. 1] and employs penalization techniques to encourage solutions that are nearly binary. A widely used method is Solid Isotropic Material with Penalization (SIMP) [3, 41]. In SIMP, effective element densities are related design variables &e E [0, 1] by Pe = d% where p >> 1 is penalization factor that makes intermediate density values (between 0 and 1) structurally inefficient, thus favoring &e values close to 0 or 1.
In our work, we focus on the minimum compliance problem, which involves solving the linear clasticity problem and using an FEA solver and iteratively optimizing the topology in discrete space. We detail this in the section below.
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A.1 Minimum Compliance Topology Optimization
This section details a prominent application of Topology Optimization (TO): minimum compliance structural optimization, often referred to as stiffness optimization. The primary objective to determine the material layout that results in the stiffest possible structure under applied mechanical loads, subject to a constraint on the total amount of material used. We will first describe the underlying physics model for this class of problems.
A.1.1 The Physics Model: Static Linear Elasticity
In most structural topology optimization problems, including minimum compliance, the behavior of structure under load is modeled using linear elasticity. This model assumes that the structure composed of a material that exhibits linear elastic behavior (i.e., stress is proportional to strain via Hooke's 's Law) and that deformations are small.
The governing partial differential equation (PDE) for linear elasticity in its general dynamic form is the Navier-Cauchy equation:
2(1+v) V'u+ (a+t)l1-20) 2v)
E
Ev
E
+
V(V u) + foody = Pphys U, 2(1
(9)
where u(x, 1) is the displacement vector field (corresponding to u(x) in Eq. 6, where m would be the number of spatial dimensions, e.g., mn 2 or m = 3), ú is its second derivative with respect to time (acceleration), foody(2) (z) is the body force vector field (e.g., gravitational loads, corresponding to f(x)) in Eq. 6 if C is defined as the internal force operator), of material stiffness), is the Poisson's ratio (characterizing transverse contraction/expansion)) and Pphys is the physical mass density of the material.
the Young's modulus (a measure
For static or quasi-static analyses, which are typical in minimum compliance problems, we are interested in the steady-state deformation under load. This allows us set the inertial term Pphys ü to The governing PDE for static linear elasticity then becomes:
E' 2(1 v)
Ev
E 2(1 ( +)
V'u+ (1+
V(V u) + faody 0.
(10)
2v)
This equation, along with appropriate boundary conditions (such as prescribed displacements g(x) on D as in Eq. 6), defines the structural response. In the context of TO, the Young's ss modulus E is not uniform throughout the domain Q; instead, it varies spatially depending the material density distribution p(x), which is the design variable.
A.1.2 Finite Element Discretization and Compliance Calculation
As discussed in the general TO framework (leading to Eq. 8), analytical solutions to Eq. 10 are generally intractable for complex domains and material distributions. Therefore, the Finite Element Method (FEM) is employed. The continuous domain Ω is discretized into a mesh Tr, of N elements, and the continuous displacement field u(z) is approximated by vector of M discrete nodal displace- ments Un RRM, The body forces foody (z) and any applied surface tractions are discretized into global force vector fn E RR Dirichlet boundary conditions (supports) are enforced by prescribing values for certain components of un.
The FEM discretization of Eq. 10 results in system of linear algebraic equations:
K(pa)u: fn:
(11)
where ((px) RMXMM is the global stiffness matrix. This matrix depends on the vector of element pN]] T these densities determine the material properties
design variables (densities) pa (specifically, Young's modulus) within each element.
The compliance C' of the structure is measure of its overall flexibility; minimizing compliance is equivalent to maximizing stiffness, For the discrete system, compliance is calculated the work done by the external forces:
Clun, fm) = fh un.
(12)
Using the cquilibrium condition fr. K K(p))un compliance can also be written as C = uff K ((p))u
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A.1.3 Material Interpolation: The SIMP Model
To enable optimization, the material properties of each clement must be related to the design variables Pe- The new paper's Pe E (0, 1] are the design variables for each element e. common approach is the Solid Isotropic Material with Penalization (SIMP) method. In this scheme, the Young's 's modulus Eg of an element is interpolated based its normalized density Pe:
(pe) Ermin + (f((Foooi Fmmin):
(13)
where Esotid is the Young's s modulus of the solid material, Emin is small positive Young's 's modulus assigned to void regions (e.g., Pe x 0) to prevent the stiffness matrix K (ps) from becoming singular, and p is the penalization factor, typically p 2 This penalization makes intermediate densities (e.g., Pr. = 0.5) structurally inefficient, meaning they contribute less to stiffness than their "cost" in material. This encourages the optimization process to yield designs with densities close to O (void) or I (solid). The element stiffness matrices K2 are computed using (ee)) and then assembled into the global stiffness matrix (ph). Note that the choice of Emin purely to avoid singular values and is selected to be 10 9 in data generation compared to the much larger value of Esolid 1.0. This choice is informed by prior literature recommending such value as good balance between stable optimization and accuracy of simulations [50]. Since in our implementation we set Emin = 10 the choice of Pmin 0 is used, although numerically any design element in mesh which receives zero density will effectively have a Young's modulus of 10-9 which given our choice Egoii = 1 would be equivalent to setting Emin and Pmmin
10-9.
A.1.4 The Minimum Compliance Optimization Problem
The goal of minimum compliance topology optimization is to find the distribution of material densities Ph that minimizes the compliance C (Eq. 12), subject to the static equilibrium constraint (Eqq. 11) and constraint on the total volume of material used. This is specific instance of the general discretized TO problem (Eq. 8). The formulation is:
minimize Ph subject to:
C(un, fn) = fh un
K(pp)un = fn
(Static Equilibrium)
(14)
Σ VoPe S Vmmx
(Volume Constraint)
Prnin S S
for e = 1,..., N (Density Bounds),
where v. is the volume of element e, Vmax is the maximum permissible total volume of the material (as defined in Eq. 7 and Eq. 8), and Pemin is small positive lower bound for the element densities (e.g.. 10-9 or 10 () to ensure Eg is always positive if Emsin = 0 is chosen, or to represent a minimum manufacturable material thickness. This formulation widely used in structural optimization. The aim of approaches like Neural Network based Inverse Topology Optimization (NITO), as alluded to in your previous work, would be to predict the optimal densities Ps directly, given the problem definition (loads fh. boundary conditions represented in K and fn, and the target volume fraction V Vinx/ Σ Ue).
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B
Ablation Studies
To determine the appropriate denoising steps and guidance scale, and the denoising strategy, perform ablation studies by sampling the model with different parameters and testing the performance on the OpenTO benchmark mentioned in the experiments. Figure shows the results of our ablation study on sampling parameters. study the effects of denoising steps, we run both DDPM and DDIM denoising with guidance scale of 2.0 with 5, 10. 20, and 40 denoising steps. We observe that DDIM sampling largely achieves better performance despite initially having a larger absolute VFE. Thus, we determine DDIM sampling to be more effective, especially for more efficient lower denoising step counts, crucial for computational efficiency. From here, we can clearly sampling strikes the perfect balance of CE and VFE 20 denoising steps, and any more comes with no benefits and, fact, slightly worse performance, Finally, to determine the optimal guidance scale, we run DDIM sampling for guidance scales of 1, 2, 4, 6, 8, and 10 using the 20 denoising steps we found to be most effective. As seen in Figure 4, guidance scale of strikes good balance between VFE and CE, which informs our final choice of DDIM denoising with 20 steps and guidance scale of 2.
in Figure 4 that DDIM
CE vs Dencising Steps
VFE vs Denoising Steps
VFE PM
0.4
0.3
8.5
Denolsing Steps
Derobing Steps
CE Guidance Scale
VFE Guidance Scale
18
1.5
1.0
VIE
12
0.5
8-
20
Guidence Scale
Guidance Scade
Figure 4: Top: Results of ablation studies on denoising steps. Observe that despite CE heing lower in DDPM sampling, this is largely due to DDPM sampling using significantly more material, as evidenced by the much higher VFE values. Bottom: Ablation on the guidance scale shows that CE largely stays the same for the guidance
while VFE optimal at guidance scale 2.0.
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C Out of Distribution Reconstruction
To further demonstrate potential downline adaptations of OAT foundational framework, reconstruction using our auto-encoder completely out-of-distributioo data, involving optimized topologies for heat sinks introduced by Bernard al. [4]. This involves entirely different physics and optimization in comparison to the data we train OAT on. Despite this, we see an Intersection over Union (IoU) of 0.94 on these samples. The reconstruction quality is visualized in Figure 5. These results show how OAT, as a pre-trained foundation model, can easily be extended to different TO problems involving different physics and topologies, even without retraining the entire framework.
Ground Truth Reconstruction
Ground Truth Reconstruction
Ground Truth Reconstruction
Ground Truth Reconstruction
Figure Heatsink optimized topologies reconstructed using OAT's auto encoder without any additional training.
This demonstrates OAT's zero-shot capability extend different physies and TO problems with relative
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D
OpenTO Dataset
Here we describe the details of the Open'TO dataset and the data generation specifics, and at the end, provide visual examples of the topologies and boundary conditions in the problem.
The generation of each topology optimization problem within the OpenTO dataset follows a structured, randomized pipeline designed produce diverse set of problems. This diversity manifests in the domain discretization (resolution), aspect ratio, and the configuration of applied loads and boundary conditions. The overarching aim is to construct a comprehensive dataset suitable for foundation model training for TO.
D.1 Domain Definition and Discretization
The first step involves defining the physical domain of the topology optimization problem. This is characterized by its resolution (total number elements) and its aspect ratio.
1. Target Element Count (EC): The total number of discrete elements in the design domain is sampled from uniform distribution. Let EC be the target element count, then
EC U12 2
This range ensures a variety of problem sizes, from 4096 to 16384 elements.
2. Aspect Ratio (AR): The aspect ratio of the design domain is sampled from standard log-normal distribution. Let AR be the aspect ratio, then
AR LogNormal 1)
where the parameters (0, 1) represent the mean and standard deviation of the natural loga- rithm of the variable, respectively. This choice allows for wide range of domain shapes, biased towards aspect ratios closer to unity but permitting more elongated or flattened domains. The bias towards square domains is mainly because these problems are more common and physically more stable, yielding fewer failed simulations in the optimization process.
3. Problem Dimensions (Nz, Ny): Given the target element count EC and the sampled aspect ratio AR, the dimensions of the problem domain, N₂ (number of elements in the x-direction) and Ny (number of elements in the y-direction), are determined. A Assuming AR = N₂/N, we have N₂ = AR Ny. Since the total number of elements is EC ze N₂ Ng. we can substitute to get EC = (AR N2) N, = AR N Thus,
EC
N ~
AR
N₂ ~ AR N The values for N₂ and Ny are then rounded to the nearest integers, ensuring that their product N₂- N, is close to the target EC. Furthermore. if this leads to an element count outside the intended distribution, one side is randomly adjusted to ensure this does not occur.
D.2 Load and Constraint Specification
Once the domain is defined, loads and boundary constraints are applied. This involves determining the number of loads and constraints, their types, and their specific locations and orientations. Unlike prior datasets, which hand-select a finite number of boundary conditions and apply a single load on the boundary, TO samples boundary conditions randomly and applies multiple random loads of different kinds. Furthermore, unlike prior works, which limit forces and boundary conditions to the edges of the domain, we sample internal boundary conditions and loads for full generality.
OpenTO also includes loads, which we refer as distributed loads. which are typically characterized formally as Neumann boundary conditions when loads are a result of stress/pressus
applied at the boundaries, and body forces when an internal load is applied to part of the domain interior (often seen in electromagnetic forces or gravitational body forces). Below are details of how loads and boundary conditions are sampled in OpenTO:
Number of Loads (NL): The number of applied loads sampled from a geometric distribution with parameter p = 0.3, shifted by +1 to ensure at least one load. Let NL be the number of loads, then
NL Geom(0.3) +
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2. Number of Constraints (NC): The number of boundary constraints (fixed degrees of freedom) is sampled from a geometric distribution with parameter p = 0.2, shifted by +2 to ensure least two constraints. Let NC be the number of constraints, then
NC Geom(0.2) + 2
This ensures baseline level of constraint necessary for valid mechanical problem. If fewer constraints present, the FEA problem (Appendix A) will become singular and thus not solvable.
3.
Load and Constraint Types and Placement: For each of the L loads and NC constraints, a type is selected from predefined discrete probability distribution. The available types and their selection probabilities are:
Internal point force (single force in the interior of the domain): 50%
Edge point (a point on one of the four edges, not corner): 10%
Cornrr point (one of the four corners): 10%
Distributed load/constraint (equal magnitude applied across many nodes of line
segment edge) on partial edge: 10%
Distributed load/constraint on full edge: 10%
Internally distributed load/constraint (distributed on a line or area of random ellipse):
10%
The specific placement of load or constraint is then determined randomly based on the selected type. For example:
Internal point: A random coordinate (ar, y) within the domain interior.
Edge point: An edge (top, bottom, left, or right) is chosen uniformly at random, and then random position along that edge is selected.
Corner point: One of the four domain corners ((0,0), (N,, 0), (0, Ny), (Ng, N₂) in elemental coordinates, or corresponding nodal coordinates) is chosen uniformly at random.
Distributed on partial edge: An edge is chosen, and then random sub-segment of that edge is selected.
Distributed full edge: An edge is chosen.
Internally distributed: A region (e.g., rectangular or circular patch or line) is randomly defined within the interior of the domain. The specific shapes and parameters for these internal distributions are further randomized. For comprehensive details on the implementation of these variations, readers are directed to the project's codebase.
Loads are typically defined by a vector (e.g., magnitude and direction), which can also be randomized.
4. Constraint Direction: Each of the NC' constraints is assigned a direction application. The degrees of freedom (DOFs) to be constrained are chosen based on the following probabilities:
Constraint in the lateral (e.g., x-direction) direction only: 30%
Constraint in the vertical (e.g., y-direction) direction only: 30%
Constraint in both lateral and vertical directions: 40%
D.3 Problem Validation and Iteration
critical final step in the generation pipeline the validation of the constructed problem.
1. Validity Check: The problem, now fully defined by its domain, loads, and constraints, is assessed to ensure it is:
The applied boundary conditions must sufficient to prevent rigid body motion (translation and rotation) of the structure under the applied loads. This is typically checked by ensuring the global stiffness matrix is nonsingular.
Fully constrained:
Not trivially solved: The problem should not be over-constrained to the point of having an obvious or degenerate solution. For instance. applying a load and a fixed constraint at the exact same degree of freedom might lead to issues or trivial (zero vector) forcing terms.
2. Regeneration on Failure: the generated problem fails the validity check (e.g., due to coincident load and constraint application points, insufficient constraints, or other prob- lematic configurations), the entire problem instance is discarded. The generation process,
27

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starting from the sampling of EC, is then re- re-initiated to produce a new, statistically inde- pendent problem candidate. This iterative loop continues until valid problem instance is successfully generated.
This rigorous, multi-stage randomized procedure ensures the creation of a diverse and challenging set of topology optimization problems, forming the basis of the OpenIO dataset.
D.4 Algorithm for Data Generation
Below a summary of the data generation process in algorithmic form.
Algorithm OO Problem Instance Generation
1: Initialize: ProblemIs Valid +- false
2: while Problemls Valid is false do
3: // Step 1: Domain Definition and Discretization
4: Sample target element count EC ~ 12 2
5: Sample aspect ratio AR LogNormal 1)
6: Calculate Ny, Ny.float V /EC/AR
7:
Calculate Na.float < AR AR.Ny.flood Ny.float
8: Set N f (Dound(NN,..ooo
9:
Set N₂ < round(N) (NN.
10:
if N₂
Ny < then
continue // Restart generation if dimensions are invalid end if
12:
13:
// Step 2: Load and Constraint Specification
14:
Sample number of loads NL ~ Geom((.3) +
Sample number constraints N ~ Geom .). Initialize empty lists: LoadsList, ConstraintsList
16:
17:
for = 1 to N L do
18:
Sample load type Leyne from (InternalPt (0.5), EdgePt (0.1), CornerPt (0.1), PartialEdge (0.1), FullEdge (0.1), Internal (0.1))
Determine load placement Eplace based on Ltype and domain (Na, Ny) (randomized selec- tion)
20:
Determine load magnitude and direction Liee (randomized)
21:
Add (Ityyes Tsplace: Ireee) to LoadsL.st
22:
end for
for j= 1 NC do
24:
Sample constraint type Ctype from (InternalPt (0.5), EdgePt (0.1), CornerPt (0.1), Par-
tialEdge (0.1), FullEdge (0.1), InternalDist (0.1)) Determine constraint placement Oplace based on Otype and domain (Na, N2) (randomized selection)
25:
26:
Sample constraint direction Cair from (Lateral (0.3), Vertical (0.3), Both (0.4))
Add (Ceyper Cplace: Cáir) to ConstraintsList
end for
29:
// Step 3: Problem Validation
30:
Perform check for sufficient constraints (prevent rigid body motion).
31:
Perform check for non-trivial solution (e.g., avoid coincident loads/constraints at same DOF if problematic).
if problem fully constrained AND not trivially solved then
33:
ProblemIs Valid +- true
34:
else
35:
Problemis Valid <- false // Problem discarded, loop will reiterate end if
36:
37:
end while
Output: Generated problem instance (Domain: N₂, Ny: Loads: LoadsList; Constraints: Con- straintsList) =0
28

---

E Further Visualization of Results
Here we visualize more samples from OAT and NITO the fully general Open'TO benchmark. The figures below clearly highlight how NITO fails generate high-quality samples and generates largely invalid, blurry topologies that are not mostly binary as ideally would be in TO. NOTE: Samples with CE higher than 100% are considered failed and do not enter CE computation the reported results.
Ground Truth
NITO
5.76
144665554
OAT
Figure 6: Samples generated on the Open'O benchmark by NITO and OAT. Compliance error for each generated sample is shown underneath.
Ground Truth
NITO
C2:63766 B
77.35
7197952
OAT
CE 16:M
Figure Samples generated OpenTO benchmark by NITO and OAT. Compliance enor each generated sample shown underneath.
29

---

NITO
CE 1555343.94
1955.92
CE7577
OAT
CE 4.22
160927.L
1826
CE 10T
02:1111
Figure 8: Samples generated the OpenTOO benchmark by NITO and OAT. Compliance error for each generated sample is shown underneath.
Ground Truth
NITO
18:
0.76
OAT
CE 04544
Figure 9: Samples generated on the Open'TO benchmark by NITO and OAT. Compliance emor for each generated sample is shown underneath.
30

---

Ground Truth
NITO
CE 1127665
CE 18192716
CE7
45023.26
03320
OAT
LE: 19130.45
58285380
129
515.18
Figure 10: Samples generated on the OpenTO benchmark by NITO and OAT. Compliance error for each generated sample is shown underneath.
Ground Truth
NITO
CE 071
0E:41.01
OAT
135
CE: 385507.42
CE 0.16
2232148.55
CE: 9.89.8
Figure 11: Samples generated on the OpenTO henchmark by NITO and OAT. Compliance error for each generated sample is shown underneath.
31

---

Truth
33:335
CE 139282
0005
"CE 51709.18
OAT
CE: 30:01
72.50
D2: 1.80
Samples generated on the OpenTO benchmark by NITO and OAT. Compliance error for each generated sample is shown underneath.
Figure 12:
Ground Truth
NITO
0.000
19822333
CE: 8.01
3760117.9
OAT
CE: (77077
Figure 13: Samples generated on the OpenTO benchmark by NITO and OAT. Compliance error for each generated sample is shown underneath.
32

---

Ground Truth
NITO
12952911
CE 1238042.07
C 2.00
OAT
CE 175
853052.75
Samples generated on the OpenTO benchmark by NITO and OAT. Compliance error for each generated sample is shown underneath.
Figure 14:
Ground Truth
NITO
100.64
OAT
DE 5.05
CE 11756.66
CE 0.39
Figure 15: Samples generated on the OpenTO henchmark by NITO and OAT. Compliance error for each generated sample is shown underneath.
33

---

Implementation Details
In this section, we describe the full details of the training and architecture of OAT. First, we discuss the autoencoder training, then will go over the LDM training.
F1 Autoencoder Architecture
The autoencoder comprises an encoder, a decoder, and a neural field renderer.
Encoder
The encoder maps an input topology T to a latent representation 2 = E(T).
Input Processing: Topologies are padded and resized to a fixed 256 256 resolution with input channel.
Core Structure: A Convolutional Neural Network (CNN) with initial convolution is followed by three downsampling levels, each employing ResNet-style blocks.
Middle Section: Features are further processed by middle section containing additional ResNet-style blocks and an attention mechanism.
Output: Final convolutional layers produce 1-channel latent tensor of size 64 x 64.
Decoder
The decoder reconstructs a feature tensor ø (=)) from the latent z.
Input:
The latent tensor is processed by initial convolutional layers.
Core Structure: Symmetrical the encoder, features middle section with ResNet-style blocks and an attention mechanism, followed by three upsampling levels composed of ResNet-style blocks.
.
Output: A fixed-resolution feature tensor (256 256) & with 128 channels.
Neural Field Renderer The renderer reconstructs the topology T = R(ő, c, 8) from 6, pixel coordinates c, and cell sizes s.
A convolutional renderer, inspired by Convolutional Local Image Functions (CLIF), is employed. It processes the decoded features, coordinates, and cell sizes, with optional positional encoding for the latter two.
Architecture:
Network: The renderer consists
sequence of convolutional and ResNet-style blocks, terminating in an output convolution with a Tanh activation function.
Scalability: Training utilizes patch sampling. High-resolution inference employs stenciled, overlapping tiles.
Full details of the architecture can be found our publicly available code.
F.2
Autoencoder Training
The autoencoder is trained on the TO dataset (2.2 million pre-training samples).
Dataset and Preprocessing:
Input topologies are resized to 256 x 256 for the encoder.
The renderer is trained on 64 x 64 patches sampled from topologies, along with their coordinates and cell sizes. Training can also be performed on full images (full grid of coordinates and cell sizes for any given topology).
Batch Size and Epochs:
Batch size: 128.
Training epochs: 50.
Distributed Training and Precision:
Training utilizes Distributed Data Parallel (DDP) on 4 H100 GPUs.
Automatic Pytorch mixed precision training is employed.
Optimizer and Learning Rate Schedule:
AdamW optimizer is used for training.
Cosine schedule is used for learning rate with 200 steps of warmup. linearly increas- ing learning rate from 0 to 10-4 then gradually reducing it to 10 duringg training.
34

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F.3 Latent Diffusion Architecture
The latent diffusion model operates the latent encodings of the autoencoder, which have 64 64 resolution with channel. The architecture of the model follows UNet architecture, which can be seen in full in our publicly available code. For conditioning of the LDM model, we a problem encoder architecture similar what we described the main body of the paper:
Conditioning Mechanism
Time Embedding: Standard timestep embedding is used to encode the diffusion timestep t.
Problem Embedding (P): The TO problem definition P (boundary conditions, forces, volume fraction, aspect ratio, cell size) is encoded into a fixed-size problem embedding P using a dedicated FroblemEnnoode.
Boundary conditions (Shawndery) and forces (S represented as point clouds, are processed by separate BPOM modules (MLPs with mean/max//mnn pooling of point features concatenated).
Scalar and low-dimensional conditions (volume fraction VF, cell size aspect ratio a) are processed by individual MLPs.
The concatenated embeddings are passed through final MLP produce P.
The problem embedding is projected to the same dimension as the time embedding and added to it. This combined embedding conditions the ResNet and Attention blocks within the U-Net.
Combined Embedding:
For full details of the architecture, please refer to the code we provide.
F4
latent Diffusion Training
The LDM is trained on latent codes obtained from the pre-trained autoencoder, paired with their corresponding TO problem specifications P.
Dataset:
Consists latent tensors and their associated problem definitions (forces, bound-
ary conditions, volume fraction, etc.). The dataset loader handles stochastic dropping of conditions for classifier-free guidance.
Diffusion Process:
DDPM noise schedule is used with a "velocity" target, and a cosine noise schedule for training.
-
For inference, a DDIM noise scheduler is employed for faster sampling.
Training Objective:
The model is trained to predict the velocity of the diffusion process, conditioned on the problem embedding P. The objective error between the true and predicted velocity, as described Equation 6 of the main paper:
minimize the mean squared
LLDM E vo(2t,t|P)]{?
where VaLe -
Classifier-Free Guidance: Enabled by randomly setting the problem conditioning P to a null embedding during training with a specified probability, specifically, we hide all of 50% of the time and hide boundary conditions and forces each with probability of 25% separate from the full 59% hiding.
Optimizer and Learning Rate:
AdamW optimizer.
Initial learning rate: 10-4, with 200 warmup steps for the autoencoder.
cosine learning rate scheduler anneals the learning rate to a final value of 10-5,
Batch Size and Epochs:
Batch size: 64.
Training epochs: 50.
Distributed Training and Precision:
Training utilizes Distributed Data Parallel (DDP) on 2 H100 GPUs.
Mixed precision training is employed.
35

---

E55
Computational Resources
All experiments and training were conducted using H100 GPUs. Autoencoder training takes 2 days to complete implementation, and the diffusion model takes 4 days to train. We had multiple training runs, but we do not have an exact estimate of the time needed. Inference time indicated for the RTX 4090 GPU in our model, which was used for inference experiments.
F.6 Dataset Visualization
Here we provide few examples of the Open'TO dataset generate. We visualize forces and boundary conditions, and can be seen that OpenTO contains samples that include non-repeating complex problem definitions with internal, edge, distributed, and point boundary conditions and forces.
Constrained in Y
Constrained in X
Applied Load
Figure 16: Samples from OpenTO Dataset.
36

---

Constrained In
Constrained in X
Applied Load
Figure 17: Samples from OpenTO Datasct.
37

---

Constrained In
Constrained in X
Applied Load
Figure 18: Samples from OpenTO Datasct.
38

---

Constrained
Constrained in X
Applied Load
Figure 19: Samples from OpenTO Datasct.
39

---

Constrained In
Constrained in X
Applied Load
Figure 20: Samples from OpenTO Datasct.
40

---

Constrained In
Constrained in X
Applied Load
Figure 21: Samples from OpenTO Datasct.
41

---

Constrained In
Constrained in X
Applied Load
Figure 22: Samples from OpenTO Datasct.
42

---

Constrained Y
Constrained in X
Applied Load
Figure 23: Samples from OpenTO Datasct.
43

---

Constrained In
Constrained in X
Applied Load
Figure 24: Samples from OpenTO Datasct.
44

---

Constrained in
Constrained in X
Applied Load
Figure 25: Samples from OpenTO Datasct.
45

---

E.7 Combining Prior Data With OpenTO
We generate 2M samples described above and provide labels (configuration information) for 700,000 samples, and provide 10,000 test samples based the same procedural generation process. Finally, it is important note that we also merge the 194.000 data points used prior work [34] into our dataset, expanding it to 2.2M total samples.
G Break-Even Analysis
(where C' is computational cost), which for OAT-like models is dominated by data generation. Our data generation cost (168 H100-days) vastly exceeded our final training run (16 H100-days). Ignoring minor preliminary development runs, the calculation yields a break-eves even hard to justify; however, we argue that this metric does not capture the full picture of deep learning's computational efficacy. This is because it ignores the hardware efficiency of parallel data generation and, more importantly, fails to capture the value of democratizing fast TO for users without large compute, enabling greater design exploration and more experiments. This metric points to over million uses to justify total computational run time of less than month. Although highlighting high development costs, we believe this argument justifies the large break-even point.
Critics cite the break-even point [62], Camp Cnferes
Cre
data-generane tett-gnnnntnnnommnaaeee
2.32 million uses. This cost seems
H
Code & Data
The code and data for OAT be found at: http://gghhhacccocmmmmmm////oppt
46


Reviews:
[{"review_id": "uBS8iLh9ha", "reviewer": "Reviewer_ukV1", "summary": "This paper presents a foundation framework of structural topology optimization by handling arbitrary boundary conditions, domain shapes, and resolutions. It is a pre-trained latent diffusion model designed for minimum-compliance structural design, marking a foundational advance. In this paper OpenTO Dataset is established including 2.2M samples with arbitrary boundary conditions, shapes, resolutions and forces.  It achieves a 10x reduction in compliance error compared to prior methods. Generates high-resolution topologies maintaining efficiency on the inference.", "questions": "1. E_min in Sections A.1.3 and A.1.4 should provide a more detailed discussion of their interrelationship.\n2. The mathematical symbols used in the vθ formula (line 223) describing the inverse denoising process require clarification. Would you please provide a detailed explanation of this formulation?\n3. In the experimental results, the mean volume fraction error (VFE) was reported. However, another metric VFE (Med) (median volume fraction error), was also used in Reference [31] but not mentioned in this study. Could the authors explain why this metric was not included?", "limitations": "yes.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 4, "originality": 4, "strengths_and_weaknesses": "Strengths:\n1. This topology optimization model can handle arbitrary boundary conditions, resolutions, and aspect ratios, overcoming limitations of prior fixed-grid approaches.\n2. This model achieves very low compliance error without the need for any direct optimization. It is efficient to generates high-resolution topologies with scalable inference.\n3. This paper introduces an OpenTO dataset which contains 2.2 millions structures. The procedural data generation is very clear in Appendix B.\n4. This method could handle mixed shape/resolution data easily.\n\nWeaknesses:\n The computational complexity of the method remains uncharacterized from a mathematical perspective.", "was_revised": false, "final_justification": "", "rebuttal": "We thank Reviewer ukV1 for their positive assessment and rating. We are grateful for the acknowledgment of our model's ability to generalize to arbitrary conditions with high accuracy and the value of the OpenTO dataset. We will address the specific questions regarding the mathematical details and the VFE to help address the reviewer's concerns and improve the paper. We hope the reviewer will be satisfied with our explanation and proposed modifications.\n## Calrifying $E_{\\text{min}}$\nIn TO formulation $E_{\\text{min}}$ is added to the overall Young's modulus so that in parts of the mesh or design space where no density is assigned, i.e. $\\rho_e=0$ the local stiffness does not become zero, resulting in a singular matrix. To prevent this, a very small $E_{\\text{min}}<<E_{\\text{solid}}$ is added to prevent absolute zero stiffness in any part of the mesh. In our data generation process and evaluations of all methods, we set $E_{\\text{min}}=10^{-9}$, while $E_{\\text{solid}} = 1$.\nSince there may be a lack of clarity on this matter, we propose to add a sentence to the final camera-ready version to address this. Specifically, we propose adding the following changes:\n\nIn Appendix A.1.3, after line 658, we will add this clarification:\n\n> Note that the choice of $E_{\\text{min}}$ is purely to avoid singular values and is selected to be $10^{-9}$ in our data generation compared to the much larger value of $E_{\\text{solid}}=1$. This choice is informed by prior literature recommending such a value as a good balance between stable optimization and accuracy of simulations [47].\n\nFurther, we will add in Appendix A.1.4, after line 671, we will add:\n> Since in out implemtation we set $E_{\\text{min}}=10^{-9}$, the choice of $\\rho_{\\text{min}}=0$ is used, although numerically any design element in a mesh which recieves a 0 density will effectively have a Young's modulus of $10^{-9}$ which given our choice $E_{\\text{solid}}=1$ would be equivalent to setting $E_{\\text{min}}=0$ and $\\rho_{\\text{min}}=10^{-9}$.\n\nWe hope that these additions clarify the exchangeability of setting a lower bound above zero on the densities or setting a zero lower bound and the same value as the minimum Young's modulus when $E_{\\text{solid}}=1$. Again, these choices are the common values used in most SIMP implementations, and we hope that clarifying this will help with replicating any data generation efforts.\n## Detailing The Classifier Free Guidance Formulation\nWe thank the reviewer for their feedback. We are happy to add the full detailed explanation of the classifier-free guidance scaling. \nWe propose changing the sentence (line 222-224): \"We can apply this during inference, with guidance scale $w$, by performing the inverse denoising process with $\\hat v_\\theta(z_t,t, P)\\;=\\;v_\\theta(z_t,t \\mid \\varnothing)\\;+\\;w\\;\\Bigl(v_\\theta(z_t,t \\mid P)- v_\\theta(z_t,t, \\varnothing)\\Bigr)$, rather than $v_\\theta$, which allows us to adjust the guidance strength.\" to the better clarify the inference with classifier-free guidance.\n\nThe proposed replacement is as follows:\n> Classifier-free guidance can be applied in our approach by adjusting the denoising velocity. Normally, in conditional denoising, one would use the predicted velocity at time step $t$, $v_\\theta(z_t,t \\mid P)$, which predicts the velocity for the current noisy sample $z_t$ given problem embedding $P$. When applying classifier-free guidance we can also compute the un-conditional denoising, namely $v_\\theta(z_t,t \\mid \\varnothing)$, and rather than just using $v_\\theta(z_t,t \\mid P)$, we measure the shift in velocity direction as a result of conditioning, $v_\\theta(z_t,t \\mid P)-v_\\theta(z_t,t \\mid \\varnothing)$ and amplify this shift by an over-relaxation factor, $\\omega$, leading to a final denoising velocity $\\hat v_\\theta(z_t,t, P)\\;=\\;v_\\theta(z_t,t \\mid \\varnothing)\\;+\\;w\\;\\Bigl(v_\\theta(z_t,t \\mid P)- v_\\theta(z_t,t, \\varnothing)\\Bigr)$, which can be used in place of $v_\\theta(z_t,t \\mid P)$, mimicking what Ho and Salimans [15] propose for classifier-free guidance.\n\nWe thank the reviewer for helping improve the clarity and readability of the paper and hope that the above explanation is satisfactory.\n## Inference Time Provides A Better Measure Of Efficiency\nWe thank the reviewer for pointing out formal computational analysis in our work. We believe the reported inference times are a much more practical measure of computational cost rather than a formal big $O$ notation. Thus, we refrained from a more tedious analysis of the computational cost in a formal manner. However, we are happy to add a brief discussion on computational complexity to the paper. However, we would like to clarify why we take this stance.\n\nFormally, the convolutional neural networks will have a constant resolution, thus a specified cost associated with the $64\\times 64$ latent space diffusion and $256\\times 256$ input and output of the auto-encoder. In big $O$ notation, this would be seen as a constant since we run the denoising with a fixed number of steps for all cases, thus $O(1)$. The neural field after the convolutional decoder simply scales by the number of pixels. In TO/FEA terms, this would be element count, thus it becomes overall $O(N)$, with $N$ being the number of elements or pixels. This formal analysis, however, does not to cover the reality that all of the deep learning aspects, which boil down to $O(N)$, can be run in a highly parallelized pipeline on the GPU. For the optimization problem, one has to solve a linear system of equations for each iteration of optimization, which naively is $O(N^3)$ for a genearal $N \\times N$ matrix (Note in FEA the size of the system is actually $\\text{Nodes} \\times \\text{dof}$ which becomes $O(N)$ when constants are taken care of). But at this point many experts in the numerical linear algebra field would point out that the problems we solve in FEA are structured and highly sparse, and thus can be solved with a multi-grid solver, which in theory reduces the linear system solve cost to a best-case scenario of $O(N)$[1-3]. But this again does not cover the fact that multi-grid solvers are preconditioned conjugate gradient solvers which operate by solving the system iteratively, thus making them not parallelizable, and therefore much slower. Which brings us back to the wall-clock being a much more reasonable measure of computational efficiency in this specific case. Regardless, we acknowledge that this nuance is not immediately clear. Given this, we will add the following brief discussion to the the paper, mentioning this high-level analysis and the reasoning for inference time being a better overall metric. Specifically, we propose the following change to line 378:\n\n> ...over iterative optimization. It's important to mention that a high-level analysis of computational cost in this case would result in an $O(N)$ complexity for OAT, where $N$ is the number of pixels/elements for a given sample. This is because the non-constant cost of OAT is in the neural field renderer, which would have a complexity of $O(N)$. This is compared to the expensive FEA simulations at each step of the optimization, yielding an $O(N^3)$ complexity (Solving a linear system of equations). However, given that FEA matrices in this setting are highly sparse and the meshes in our work are structured, specialized multi-grid solvers can, in theory, reach an $O(N)$ efficiency [1-3]. This high-level analysis is however, fails to realize the inference efficiency of deep learning models compared to conventional optimization, given that it will not take into account the highly parallelized inference of deep learning models on the GPU compared to the iterative nature of linear system solvers. This makes inference time an overall better measure of performance in practical consumer hardware.\n\n[1] Erik A. Träff, et. al. Simple and efficient GPU-accelerated topology optimisation: Codes and applications. Computer Methods in Applied Mechanics and Engineering\n[2]Haixiang Liu, et. al. Narrow-band topology optimization on a sparsely populated grid. ACM Trans. Graph.\n[3]Niels Aage, et. al. Giga-voxel computational morphogenesis for structural design. Nature.\n\n## On Median Reporting Of Volume Fraction Error\nThe main reason we omit this is that we did not explicitly discuss this aspect of the results, given that no further insight was gained from this, and the overall lower median was covered in the CE discussion. Most notably, removing these values helped make the tables more readable and helped report results in a more concise way. The code we provide includes the full benchmarking, which reports the medians of VFE. If you are curious about the values themselves, below are the full results of OAT in each table:\n\n**Table 1**\n| Model | CE % | CE % Med | VFE % | VFE % Med |\n| :--- | :--- | :--- | :--- | :--- |\n| OAT (Ours) | 0.169 | -0.016 | 2.21 | 2.03 |\n| OAT (Ours) + 5 | 0.0438 | 0.001 | 0.91 | 0.78 |\n| OAT (Ours) + 10 | 0.0329 | 0.0142 | 0.29 | 0.21 |\n\n**Table 2**\n| Model | CE % | CE % Med | VFE % | VFE % Med |\n| :--- | :--- | :--- | :--- | :--- |\n| OAT (Ours) | 0.89 | 0.23 | 1.42 | 1.22 |\n| OAT (Ours) + 5 | 0.34 | 0.11 | 0.53 | 0.38 |\n| OAT (Ours) + 10 | 0.042 | 0.016 | 0.089 | 0.076 |\n\n**Table 3**\n| Model | CE % | CE % Med | VFE % | VFE % Med |\n| :--- | :--- | :--- | :--- | :--- |\n| OAT (Ours) | 0.42 | 0.046 | 1.90 | 1.66 |\n| OAT (Ours) + 5 | 0.19 | 0.053 | 0.73 | 0.51 |\n| OAT (Ours) + 10 | 0.079 | 0.056 | 0.21 | 0.098 |\n\n**Table 4*\n| Model | CE* % | CE* % Med | VFE* % | VFE* % Med | Failure Rate % |\n| :--- | :--- | :--- | :--- | :--- | :--- |\n| OAT (Ours) | 10.06 | 4.49 | 1.03 | 0.97 | 48.55 |\n| OAT (Ours) + 5 | 7.37 | 3.37 | 0.70 | 0.65 | 21.46 |\n| OAT (Ours) + 10 | 5.69 | 2.32 | 0.67 | 0.59 | 15.51 |\n\nWe hope that this clarifies the reasoning for omitting these values. If the reviewer believes this adds additional insight, we are happy to add this to the camera-ready paper."}, {"review_id": "TFH4e29pnU", "reviewer": "Reviewer_77iF", "summary": "The paper proposes OAT (Optimize Any Topology), a resolution-invariant framework that fuses a shared autoencoder, a convolutional local implicit-field renderer, and a conditional latent diffusion model to synthesize near-optimal topology designs for arbitrary aspect ratios, resolutions, volume fractions, loads, and constraints. Leveraging the 2.19 million-example OpenTO dataset covering 2 million distinct boundary conditions, OAT attains sub-1 % compliance error on 64x64 and 256x256 datasets and cuts inference time to faster than SIMP and roughly more accurate than prior neural baselines such as NITO and TopoDiff.", "questions": "see weaknesses.", "limitations": "see weaknesses.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**Strengths**\n\n1. **Large, publicly released dataset.** The paper introduces the OpenTO corpus, containing 2.19 M samples and 2 M distinct boundary conditions, which offers the community an unprecedentedly broad and reproducible benchmark for training and evaluating topology-optimization models.\n    \n2. **State-of-the-art accuracy.** On $64\\times 64$–$256\\times 256$ datasets, OAT achieves compliance errors below $1\\%$ without post-optimization, outperforming prior generative baselines such as TopoDiff and NITO by roughly one order of magnitude.\n    \n3. **High versatility.** Thanks to its resolution-invariant latent representation, OAT handles arbitrary aspect ratios, mesh sizes, volume fractions, loads, and boundary constraints without retraining, markedly simplifying deployment across heterogeneous design scenarios.\n    \n\n**Weaknesses**\n\n1. **Attribution of gains to the new dataset is unclear.** OAT is evaluated on OpenTO, whereas competing methods use their original training data; the reported superiority may therefore stem from higher-quality data rather than algorithmic advances. A fair comparison would retrain existing generative approaches on OpenTO (minor architectural adjustments should suffice) to isolate the methodological benefit of OAT itself.\n    \n2. **High failure rate under fully random conditions.** A single forward pass yields $\\approx48$% invalid topologies on the unrestricted OpenTO benchmark, and even after ten SIMP refinement steps the failure rate remains $\\approx16$%, which may limit reliability in safety-critical workflows.\n    \n3. **Dependence on post-optimization.** The recommended 5–10 SIMP iterations needed to “heal” minor deviations contradict the ambition of one-shot optimal generation and diminish real-time performance in high-throughput settings.\n    \n4. **Complex architecture and integration cost.** Combining an autoencoder, an implicit-field renderer, and a conditional diffusion model increases engineering overhead and GPU-memory management complexity compared with single-stage UNet- or NeRF-based generators, potentially impeding adoption in industrial pipelines.", "was_revised": true, "final_justification": "I believe this paper presents a valuable new dataset and proposes a correspondingly more general method, so I have a positive view of the work. The author's rebuttal has also largely addressed the concerns I raised. However, since I am not very familiar with this field, it is difficult for me to justify increasing my score further.", "rebuttal": "We thank Reviewer 77iF for their encouraging review and for highlighting how OAT introduces a versatile platform for deep learning in TO problems and achieves SOTA performance by a large margin. We also appreciate the recognition of OpenTO as a valuable dataset. We appreciate the insightful points regarding the performance gains and dataset, failure rates, and model complexity. These are important considerations that we hope to address here to provide a better understanding of our method's capabilities and limitations.\n## SOTA Of-The-Shelf Models Can't Adapt To OpenTO\nThe question of using existing methods with the OpenTO to extract dataset gains compared to architectural gains is a noteworthy one. To respond to the reviewer briefly, adapting existing methods other than NITO is not a simple matter of training them on the new data. This is why we believe OAT is truly the first model to attempt to solve the most generalized variant of the TO problem. And in the case of NITO, we indeed provide experiments of a NITO variant as large as OAT trained on OpenTO, which, as clearly evident in Figure 3 and numerous examples in Appendix D, leads to very low-quality samples, further highlighting the strength of OAT as an architecture. There are primarily two reasons why an adaptation of SOTA is not very simple, and why the need for OAT arises in the first place. Firstly, existing approaches work on fixed resolution and square domains. Secondly, existing methods simply do not permit easy adaptation of the widely varying boundary conditions and forces that OpenTO includes.\n\n### SOTA Struggles With General Aspect Ratio and Resolution\nOne of the major hurdles in most prior works is their inability to generalize to random resolutions. Note that the OpenTO dataset is not just random in its aspect ratios, but it also has varying resolutions, which can reach up to 1024 pixels in a row or column. This means that one must train existing methods such as TopoDiff or DOM at their native $64\\times64$ resolution by resizing and centering everything, which, given the very low resolution of $64\\times 64$ (minimum pixel count in OpenTO), would most likely yield very poor quality samples and thus not provide a fair comparison. Alternatively, the models would have to be scaled to a larger resolution, such as $256\\times 256$, for which training a diffusion model is highly costly and in most cases not even effective at generating high-quality samples, see [31] and their experiments on full diffusion training at a resolution $256\\times 256$. This resolution is going beyond most SOTA diffusion models for image generation, e.g., stable diffusion, which tackles this matter by reducing the size of the diffusion space using an AutoEncoder, such as in our approach. Therefore, we truly must emphasize that existing frameworks are using off-the-shelf diffusion models, which, simply put, are not easily adapted to the OpenTO dataset, which involves fully random resolution and aspect ratios. Despite this, we do train a larger version of NITO, which can be adapted to this data and show that the vast diversity of OpenTO hinders its performance in generating high-quality samples, while our proposed framework is capable of producing high-quality, realistic topologies.\n\n### Boundary Conditions and Problem Representations Are Non-Trivial\nThe second major hurdle in adapting existing methods lies in how they represent TO problems. We briefly allude to this issue in the paper (lines 163-169). All prior works, except NITO, represent TO problems in a peculiar manner. Rather than the raw representations of the boundary conditions and loads, and volume fraction, prior works use image-based representations. Specifically to represent boundary conditions, these prior methods run a single FEA simulation on the domain with full density (all pixels black with density 1) and use the stress and strain fields as a representation of boundary conditions in image-like form. This means they compute said fields and pass them as a channel in their diffusion or GAN models. Given that OpenTO has a fully variable shape and resolutions, obtaining consistently sized fields for these different problems will not be trivial, as one has to interpolate and pad such fields to a fixed resolution, which may lose significant fidelity. Furthermore, prior works are designed for simple problems with a single load and finite repeated boundary settings (42 boundary configurations), which makes it easier to decipher the boundary conditions from physics fields. In OpenTO, just looking at stress fields may not provide such granular information for highly complex and non-regular boundary configurations and shapes. Furthermore, one of the reasons prior works have slower inference is their need for one FEA run to represent boundary conditions, which OAT overcomes. [31] discusses this matter in more detail if the reviewer is interested in a more in-depth analysis of this issue.\n## A Different Viewpoint On High Failure Rates\nWe acknowledge that the high failure rate can be a major limitation and agree that it should be addressed. However, we must highlight that we have conducted an inference-time scaling experiment on best of $N$ sampling, which highlights how this matter can be handled. To this end, we have explicitly discussed this as a major pain point for future work (see lines 410-415) and made the case that RL frameworks, which are becoming a common theme in many foundation models, can hopefully aid in improving this aspect of our model. The reason we believe RL frameworks will be effective in this scenario is that we see in examples such as those shown in Figure 3 that the model is capable of producing high-quality samples; however, small inaccuracies are the major reason for the high failure rate. However, our model can be sampled many times for each problem at a very low cost compared to SIMP (Table 5). Thus, we conduct this new best of $N$  experiment, which we plan to add to the manuscript. In this experiment, we sample OAT $N$ times and report metrics on the best of the $N$ for each individual sample in the test set (Not the whole run). Below are the results of this experiment:\n\n**Best of N Sampling Experiment (Note: one single run - so numbers are slightly different from Table 4)**\n| N | 1 | 2 | 4 | 8 | 16 | 32 | 64 | 128 | 256 |\n| :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- |\n| Failure Rate |  49.03 | 45.83 | 41.66 | 34.37 | 28.12 | 20.83 | 19.79 | 15.34 | 9.86 |\n| CE* % Mean |  10.85 | 9.51 | 8.71 | 7.33 | 7.59 | 6.34 | 4.39 | 3.19 | 3.08 |\n| VFE* % Mean |  0.92 | 0.99 | 1.04 | 1.11 | 1.18 | 1.25 | 1.33 | 1.37 | 1.35 |\n\nWe see that as we sample OAT more, we reduce the failure rate significantly, verifying the above claim of small inaccuracies causing failures (even a single missed pixel can miss a boundary condition), and showing how RL and rejection sampling refinements of OAT can be effective at reducing the failure rate and improving performance.\n\nWe hope the addition of this new experiment addresses the primary concern raised by the reviewer and provides a rich perspective on this limitation of OAT.\n## Post Optimization Cost Is Much Less Than Full Optimization\nSimilar to our discussion on the high failure rate, we do explicitly acknowledge that this additional optimization does come at a cost of slower inference. However, we would like to highlight that in optimization, typically 200-300 steps of optimization are needed for convergence, given the non-linear and non-convex nature of the problem with sensitivity to initialization and the underlying non-linear optimizer. Specifically, looking at Table 5 of the paper, we do note that although post optimization does increase the cost of inference, it still provides an order of magnitude acceleration compared to the conventional SIMP optimizer. However, in our discussion of the efficiency, we do note that this use of post optimization does come at a notable cost, especially at higher resolutions, and agree with the reviewer on this limitation. We do believe that, like the proposed future work on how the failure rate can be addressed, the use of rejection sampling and RL-inspired policy optimization frameworks can be effective at raising OAT's performance on this benchmark. As shown above, although best of $N$ does not significantly reduce compliance error like it does failure, there is a trend towards reduction of error, which gives hope for further improvements in RL, like policy optimization. Noting this is not immediately clear and discussed in our conclusion, we propose a minor change to the language of the conclusion around this future work (aside from adding the new experiment) to put into perspective the potential downstream improvement that OAT provides. Specifically, we propose changing the sentence starting on line 411:\n\n>Firstly, we see that currently OAT faces a notable failure rate and relies on post optimization on fully random benchmarks, which future work will focus on addressing through promising directions such as reinforcement learning[33,43] based on physics models and optimizer guidance in diffusion models[9]\n\nWe hope the discussion above satisfies the reviewers' concerns on this matter.\n\n## On Complex Architecture and Integration Cost\nWe agree with the reviewer that OAT introduces a deep learning framework into the workflow. However, we argue that this integration cost is a minor engineering hurdle compared to the significant existing complexity of setting up the FEA, physics, and optimization for any TO problem. Since our provided code and checkpoints can readily use this existing setup, the effort should be minimal. Given the significant acceleration OAT provides (Table 5) and the increasing trend of integrating deep learning into commercial engineering software, we believe the benefits justify this cost. We hope this clarifies the balance of integration cost and inference acceleration."}, {"review_id": "JSnUPedCRQ", "reviewer": "Reviewer_9u8P", "summary": "Topology optimization is the task of optimizing an engineering design with an evaluation metric that depends on the solution of a PDE, such as the optimization of an airfoil to minimize the drag resulting from the flow field obtained by solving the Navier-Stokes Equations. Such optimization is typically costly and is performed with expensive adjoint-based optimization. In place of this, the paper trains a foundation model for topology optimization for a variety of domain sizes and shapes. In the process, they also develop an extensive dataset of topology optimization tasks and corresponding solutions.", "questions": "- It seems like data were generated at random in the proposed OpenTO dataset through procedural generation; however, it was not described how this generation is performed. In particular, I would imagine many domain with purely random generation would be implausible, so what are the mechanisms to cull or avoid these domains?", "limitations": "Yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "The primary contribution from this paper is a generally usable dataset of topology optimization tasks. While the component parts of the proposed model are standard, the assembly into a foundation model likely required substantial effort to get functioning, as we elaborate on below.\n\n**Quality**\n\nThe bulk of the work was in considering extensive empirical assessments of the proposed model against other topology optimization approaches. The authors do highlight well the deficiencies in their proposed model, which is more broadly an issue of using black-box machine learning predictors for directly predicting the solutions to such optimization tasks. The assessments are thoroughly conducted against number of different generated benchmark tasks, assessing for the compliance, failure rate, and sampling speed. These three metrics seem quite comprehensive as metrics for assessment and were assessed against some competitor methods.\n\n**Clarity**\n\nThe presentation is well-delivered and writing clear. Figures are also well-illustrated, such as the workflow diagram in Figure 2 and failure diagram in Figure 3. Exposition was easy to follow along throughout the paper.\n\n**Significance**\n\nThe work does contribute significantly in developing a comprehensive suite of topology optimization problems. This alone seems to be a worthwhile contribution, although the additional contribution of the foundational model serves as a good starting point for further exploration in this direction. The pointing out of deficiencies in this direct prediction approach around boundary effects is also a worthwhile contribution. \n\n**Originality**\n\nWhile no individual piece of the pipeline is inherently novel or distinct from their incarnations in previous works, the overall construction of a topology optimization foundation model from these pieces was a novel undertaking as was the generation of plausible topology optimization problems for training the model. More novelty in the foundation model would be interesting to pursue, but this feels like work that would be better suited to further work that builds upon this first step.", "was_revised": true, "final_justification": "My initial read of the paper suggested that both the proposed foundation model and dataset were worthwhile contributions to the broader community. Given that there are not many *conceptually* novel aspects of the work (in the sense that each part of the OAT foundation model has been leveraged in other models in the past and similarly for the data generation process), I was initially uncertain of how worthwhile the contribution to the broader community would be. However, reading the reviews from reviewers more intimiately familiar with the empirical side of this community, it seems like these contributions are wholly worthwhile despite the conceptually straightforward nature of it.\n\nI am, therefore, increasing my score to reflect this.", "rebuttal": "We thank Reviewer 9u8P for their balanced review and for recognizing the significant effort involved in creating both the dataset and the model. We appreciate the positive comments on the paper's clarity and the thoroughness of our empirical assessments. We will provide a more detailed explanation of the procedural generation and curation process for the OpenTO dataset to address the excellent question raised, and we hope that the rebuttal and the new experiment provided in response to other reviewers will convince the reviewer to raise their score.\n\n## On Procedural Data Generation\nWe thank the reviewer for bringing up this important point of discussion. We agree that it is important to be very clear on how the data was generated and what pipeline was used for OpenTO. We omitted the full discussion on the generation process in the main body of the paper because this involves many different nuances and details, which would not benefit the readers immediately, and we believe this would have lengthened the paper and reduced its readability. As such, in section 4.5 on the dataset, we briefly describe the features of the dataset and refer the readers to Appendix B, where the full details of the procedural generation are given. In Appendix B, we describe in full detail how OpenTO was created. On your concern about many domains being physically invalid, this is very much true in many random configurations, which is why in section B.3 we describe the validity check for a randomly generated problem. Mainly, we look at whether the generated configuration is physically fully constrained and the resulting FEA matrix is non-singular, thus making the problem solvable.  Beyond this physical feasibility check, we do not perform any other curations and make this deliberate choice informed by other works in deep learning for engineering design tasks, which show that fully random data, so long as physically valid, provides a great pathway for developing models that capture the fundamental physics and engineering requirements. A few examples of such datasets that have been procedurally generated are datasets for meta-materials[1], linkage mechanisms[3], or, more recently, the Alpha geometry dataset, procedurally generated by Google DeepMind [3] for geometry problems. The success of these validated yet not deeply curated datasets informed our decision to limit curation to only physically valid configurations, rather than curating for realism or relatedness to real-world problems. This is simply because we hope OAT can serve as a pre-trained foundational model that can be adapted to many downstream tasks, and thus, capturing the underlying physics is of great importance, and we believe that less curation could be helpful in this kind of generalization. We hope that this and the existence of the detailed procedural generation process in the appendix have cleared the concerns of the reviewer, and the reviewer would consider raising their score if they are satisfied with our response to this concern.\n\n[1] Lee, D., Chan, Y., Chen, W. (., Wang, L., van Beek, A., and Chen, W. (November 3, 2022). \"t-METASET: Task-Aware Acquisition of Metamaterial Datasets Through Diversity-Based Active Learning.\" ASME. J. Mech. Des. March 2023; 145(3): 031704. https://doi.org/10.1115/1.4055925\n[2] Heyrani Nobari, A, Srivastava, A, Gutfreund, D, & Ahmed, F. \"LINKS: A Dataset of a Hundred Million Planar Linkage Mechanisms for Data-Driven Kinematic Design.\" Proceedings of the ASME 2022 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. Volume 3A: 48th Design Automation Conference (DAC). St. Louis, Missouri, USA. August 14–17, 2022. V03AT03A013. ASME. https://doi.org/10.1115/DETC2022-89798\n[3] Chervonyi, Y., Trinh, T. H., Olšák, M., Yang, X., Nguyen, H., Menegali, M., … Luong, T. (2025). Gold-medalist Performance in Solving Olympiad Geometry with AlphaGeometry2. arXiv [Cs.AI]. Retrieved from http://arxiv.org/abs/2502.03544"}, {"review_id": "OZAqTCf70k", "reviewer": "Reviewer_zRq4", "summary": "This paper introduced Optimiza Any Topology (OAT), a foundation model for structural topology optimization (TO) with linear elasticity, together with OpenTO, a new, large-scale dataset with $2.2 \\times 10^6$ optimized solutions\nacross $2 \\times 10^6$ unique boundary conditions (BCs). OAT is shown to outperform previous deep learning methods for TO on multiple datasets and is the first generative model to be able to generate solutions for a wide range \nof resolutions, aspect ratios, loads, and fixture configurations.\n\nOAT is a latent diffusion model with a neural field decoder that can handle arbitrary resolution outputs, a significant limitation of previous methods like Topodiff. A conditional diffusion model generates variable geometries in \nlatent space, the problem specifications are encoded using MLPs and Boundary Point Order-invariant MLPs (BPOMs) for point-cloud representations of forces and fixtures.\n\nOn multiple benchmarks, OAT demonstrates state of the art (SoTA) performance, reducing the compliance error compared to the numerical baseline method (SIMP) by up to $90\\%$ compared with previous models and maintains fast inference times on a single GPU.", "questions": "- You don't just consider Dirichlet BC but also Neumann, right? At least in appendix A this should be mentioned at some point for completness.\n- Table 5 reports SIMP runtimes: Are these run on the CPU (single core or multiple cores?) or on the GPU?\n- Can you provide the exact computational cost and break-even points? \n- Failure cases: You report failure cases of OAT (see W2). Do these only appear in OOD test cases? Does OAT consistently fail on these cases or does it produce valid shapes when queried multiple times? Are there (OOD) where OAT fails completely even after multiple queries? \n\nI'm open to discussion and changing my rating if:\n\n- the computational costs/break-evens are included correctly \n- the authors address my points from the Weaknesses section.\n\nOpen question: OAT is a neural surrogate that approximates the entire TO process in a single step from BC to optimized structure. The expensive part in solving many large TO problems are typically not the SIMP updates but the FEM solves. Where do you see the advantages/tradeoffs between the OAT approach and instead developing a FEM neural surrogate to speed up SIMP?\nFor me this seems similar to GANs vs Diffusion/Flow matching, where it has turned out to be much better to transform the noise-to-data distribution in multiple steps instead of in a single stepas in GANs.", "limitations": "yes", "rating": 2, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "## Strengths\n- OpenTO is a new TO training dataset with a diverse set of loading conditions, applied forces, and prescribed volumes.\n- OAT is, because it is trained on OpenTO, able to handle a diverse range of BCs/forces.\n- A comparison against both SoTA data-driven models (Topodiff, ...) and the standard numerical method (SIMP).\n- Tests of OOD problems and documentation of failure cases. \n- Paper is well written: Clear structure, good overview, Figure, good Tables\n\n## Weaknesess\n- My fundamental criticism: While the results of OAT compared with earlier models like TopoDiff are definitely impressive and a clear step forward, I believe the term \"foundation model\" is overclaiming. OAT only applies to linear elasticity problems and only to 2D problems; this would all be considered fairly toy-problems by modern SOTA TO research. Compared to foundation models in other science applications (e.g. weather modelling), OAT can't be considered to be a foundation model. And even in this constrained setting it struggles to generalize to OOD cases, as shown in Table 4 (48.55% FR without PP). \n - The failure case (Figure 3, right) looks like OAT has simply memorized it's training dataset and is generating a training datapoint which had loading points close to the test case. Seems likely given the size of OpenTO.\n- Lack of proper reporting of computational costs. [1] suggests that DL methods for TO should provide a clear overview of the entire computational costs of generating the training data, training the model and inference: Only inference costs are measured exactly and a rough time given for training (2 days for the autoencoder, 4 days for the diffusion model without accounting for multiple training runs used for hyperparameter tuning etc.). \n- [1] further suggests DL TO models should provide breakeven estimates, so how many model queries are needed to amortize the dataset generation and training costs. These breakeven calculations are not provided.\n\n[1] WOLDSETH, Rebekka V., et al. On the use of artificial neural networks in topology optimisation. Structural and Multidisciplinary Optimization, 2022, 65. Jg., Nr. 10, S. 294.", "was_revised": true, "final_justification": "In my opinion, despite the massive scaling of model and especially dataset size, OAT is still plagued by the same issues as other data-driven TO: a 52% success rate on novel problems (aka 48% failure rate). I feel this core problem is not adequately addressed. I stand by  my concern of overclamining.\n\nI guess I'm just opposed to the approach, unlike the other reviewers.", "rebuttal": "We thank Reviewer zRq4 for their thorough review. We value the detailed feedback about defining the foundation model, analysis of failure cases, and the need for computational cost analysis. In the limited rebuttal length, we hope to address each point. (Note, we will refer to the reviewer's reference [1] as [58])\n## Clarifying The Foundation Model Claims\nWe believe we can justify the foundation claims and adjust the paper to better scope and reflect this. According to the widely accepted Stanford definition by Bommasani et al.[1], A foundation model is trained on broad data and can be adapted to diverse downstream tasks [1]. Specifically, the authors mention, \"A foundation model is itself incomplete but serves as the common basis from which many task-specific models are built via adaptation\". OAT meets the above criteria within the domain of structural topology optimization (STO) and provides diverse paths for downstream adaptations. Below, we explicitly describe how OAT can be deemed a foundation model in this context:\n\n1. Broad Data: OAT is trained on OpenTO with 2.2M samples covering unique boundary conditions, diverse aspect ratios (1:1 to 10:1), and resolutions (64 to 1024 px). Previous models were trained on an order of magnitude less data.\n2. Generalization: OAT generates optimized topologies for arbitrary boundary conditions and resolutions without retraining. In contrast, no previous data-driven method has achieved this generalizability (33-34 generalizability score based on [58]).\n3. Scale and Efficiency: It has 730M parameters and achieves fast inference (<1 second on a single GPU). It also provides >10x reduction in compliance error compared to prior neural methods without post-optimization.\n4. Adaptability: Like many other foundation models in computer science, OAT provides many paths for downstream adaptation. Like CLIP, a widely regarded foundation model which was trained on large image-text pairs and adapted to various downstream vision-language tasks, OAT’s learned problem embeddings could be used for other applications involving linear elasticity, such as rapid compliance prediction without full TO or other physics-based downstream tasks. This is enabled by the generality of OAT in handling boundary conditions and its large-scale training. Moreover, OAT’s latent diffusion architecture can be adapted. While trained for minimum compliance, it could be fine-tuned or guided for other objectives like stress minimization, making the pre-trained model a powerful starting point. Finally, OAT's autoencoder, trained on 2.2 million topologies, provides a rich latent space for complex topologies. This expensive-to-train component can be reused as a neural operator, like a representation of topologies in classical optimization, or to train smaller, specialized diffusion models for new TO problems at a fraction of the cost.\n\nOn the \"2D linear elasticity\" point, we acknowledge and plan to explicitly clarify that our current foundational claim is scoped specifically to this. Importantly, the Stanford definition does not mandate immediate multi-physics or 3D coverage to be deemed foundational; rather, it emphasizes breadth and adaptability within the targeted domain. The above arguments validate this. Moreover, foundational engineering models often begin with targeted domains (e.g., AlphaFold for protein folding or Finite-Flow FM for CFD) before extending to broader contexts. Indeed, future extensions to 3D or multi-physics represent straightforward adaptations leveraging the existing OAT backbone. We plan to explicitly clarify our claim to \" 2D linear elasticity\" and will define \"foundation model\" according to [1] in Section 1 to avoid confusion and scope our claim in Section 6. We hope this addresses the reviewer's main concern, and the reviewer will consider raising their score in light of this proposed clarification.\n## On Computational Cost and Breakeven\nWe appreciate the reviewer for pointing out this criticism of DL methods by the TO community. We do have some concerns about the metric of breakeven, but we see value in adding this analysis to the paper. We propose adding a subsection 5.4 after the inference discussions on breakeven analysis. The proposed contents of this section will be as follows:\n\n>Critics of data-driven TO often bring up the breakeven point [58], $\\tau=\\frac{C_{\\text{train}}}{C_{\\text{SIMP}}-C_{\\text{inference}}}$ where $C$ is **computational cost**, which for models like OAT is dominated by data generation. Our data generation cost (3 weeks on 8 H100s) vastly exceeds our final training run of 16 H100-days (Appendix E). We ignore the cost of our preliminary development runs—two runs totaling roughly 1/10th of the final training cost (run for 10 epochs at 1/2 resolution)—as the calculation is dominated by data generation, leading to a conventional break-even of $\\tau\\approx 2.32$ million uses. Note that this metric does not truly capture the full picture of the break-even computational efficacy of deep learning methods. This is because it ignores the hardware efficiency of parallel data generation versus single runs and, more importantly, fails to capture the value of democratizing fast TO for users without large compute, enabling far greater design exploration and more experiments. Given a total computational run time of less than one month, OAT enables fast TO for anyone using our trained model checkpoint, unlocking foundation model capabilities and fine-tuning opportunities. We emphasize that, as with LLMs such as ChatGPT, the one-time expense of producing 2.2 M optimized exemplars is amortized over all future queries. No single user is expected to ‘pay back’ the training cost; instead, the community collectively benefits from near-zero marginal cost per design query once the model is trained.\n\nWe hope that this analysis clarifies the scope and contributions of the work. We hope that by clearly showing the break-even point, our justification for it, and clarifying experimental runs and their cost, the reviewer will consider raising their score.\n## On OOD, Overfitting Concerns, and Sampling\nWe thank the reviewer for their valuable feedback. We address the concerns about overfitting and out-of-distribution (OOD) performance by arguing that failures stem from problem sensitivity rather than memorization. We support this with a new experiment.\n\nFirst, memorization is statistically infeasible.\n1. **No Data Overlap:** We have verified that no test samples exist in the training set.\n2. **Combinatorial Space:** The problem space is immense. Even for the smaller problems (4225 nodes in 64x64 configuration), there are $\\approx 2^{4225}$ possible boundary configurations, making it highly improbable for our model to find near-matches within only 2M training examples (this does not even consider the fully random domain shape and resolutions, and loads). The visual diversity in Appendix E.6 further supports this.\n\nWe hypothesize that OAT generates diverse, nearly-valid solutions, but the high sensitivity of the problem can cause some to fail (Figure 3, which is a test OOD example). To demonstrate this, we performed a new \"best of $N$\" sampling experiment, generating $N$ solutions per problem and selecting the best.\n\n**Best of N Sampling Experiment (Note: one single run - so numbers are slightly different from Table 4)**\n| N | 1 | 2 | 4 | 8 | 16 | 32 | 64 | 128 | 256 |\n| :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- |\n| Failure Rate | 49.03% | 45.83% | 41.66% | 34.37% | 28.12% | 20.83% | 19.79% | 15.34% | 9.86% |\n| CE* % Mean | 10.85 | 9.51 | 8.71 | 7.33 | 7.59 | 6.34 | 4.39 | 3.19 | 3.08 |\n| VFE* % Mean | 0.92 | 0.99 | 1.04 | 1.11 | 1.18 | 1.25 | 1.33 | 1.37 | 1.35 |\n\nThe results show that as we sample more, the failure rate drops from 49% to under 10%, and CE improves. This demonstrates two key points:\n1.  OAT learns a distribution of plausible solutions, proving it is not memorizing training data.\n2.  This sampling method is a powerful way to use OAT to find valid, high-quality designs in parallel (a potential alternative to post-optimization).\n\nWe will add this experiment to the final manuscript and include a discussion. We believe it resolves the concerns regarding OOD generalization and overfitting. \n## Response to Minor Points\nWe thank the reviewer for their suggestions, which will improve the paper's clarity. To address the missing explicit reference to Neumann boundary conditions and body forces, we will make two updates:\n1.  In Appendix B, Section B.2 (after line 699), we will add the following sentence:\n    > OpenTO also includes loads which we refer to as distributed loads, which are typically characterized formally as Neumann boundary conditions when loads are a result of stress/pressure applied at the boundaries, and body forces when an internal load is applied to part of the domain interior (often seen in electromagnetic forces or gravitational body forces).\n2.  In Appendix A (line 636), we will change the wording to \"...surface tractions (i.e., Neumann Boundary Conditions)...\".\n\nTo clarify the SIMP inference times are CPU run times using MKL compiled CHOLMOD as the solver, we will add the GPU times (our own fast GPU multi-grid solver) and update the table caption (this was originally omitted as GPU acceleration in 2D is negligible).\n* The added SIMP-GPU speeds are **9.13s for 64x64** and **68.30s for 256x256**.\n* The following text will be added to the caption:\n    > \"SIMP optimizer times are reported using an Intel 14-900K CPU (RTX4090 is used for SIMP-GPU), and all SIMP inference speeds are measured for 150 steps of optimization (which we found was the overall average iteration count to convergence)\"\n\nWe hope these changes address the raised concerns.\n\n## Open Question\nGiven space, we will discuss this during the discussion period.\n\n[1] Bommasani, R., et al (2022). On the Opportunities and Risks of Foundation Models."}]

Author rebuttal:

Meta-review: The paper introduces a foundation modeling approach to topological optimization, plus a large new data set of optimized solutions across different boundary conditions.

Both the problem, and the question of foundational approaches in this area, are of clear and timely interest to the machine learning community. All reviewers agree that there are some interesting contributions here, although opinions are somewhat divided regarding significance. Personally, I do share reviewer zRq4's concern regarding overclaiming: On the one hand, there are contributions and results here, on the other hand, the claims made about these results seem rather strong. All in all, however, the majority of reviewers believe those contributions warrant publication.
Decision: Accept (poster)