Aligning Optimization Trajectories with Diffusion Models for Constrained Design Generation
Authors: Giorgio Giannone
Akash Srivastava
Ole Winther
Faez Ahmed
Venue: NeurIPS 2023 poster (2023)
Keywords: diffusion models
engineering design
generative optimization
trajectory matching
TL;DR: conditional diffusion model and trajectory matching for topology optimization

Abstract: Generative models have significantly influenced both vision and language domains, ushering in innovative multimodal applications. Although these achievements have motivated exploration in scientific and engineering fields, challenges emerge, particularly in constrained settings with limited data where precision is crucial. Traditional engineering optimization methods rooted in physics often surpass generative models in these contexts. To address these challenges, we introduce Diffusion Optimization Models (DOM) and Trajectory Alignment (TA), a learning framework that demonstrates the efficacy of aligning the sampling trajectory of diffusion models with the trajectory derived from physics-based iterative optimization methods. This alignment ensures that the sampling process remains grounded in the underlying physical principles. This alignment eliminates the need for costly preprocessing, external surrogate models, or extra labeled data, generating feasible and high-performance designs efficiently. We apply our framework to structural topology optimization, a fundamental problem in mechanical design, evaluating its performance on in- and out-of-distribution configurations. Our results demonstrate that TA outperforms state-of-the-art deep generative models on in-distribution configurations and halves the inference computational cost. When coupled with a few steps of optimization, it also improves manufacturability for out-of-distribution conditions. 
DOM's efficiency and performance improvements significantly expedite design processes and steer them toward optimal and manufacturable outcomes, highlighting the potential of generative models in data-driven design.

Aligning Optimization Trajectories with Diffusion Models for Constrained Design Generation
Giorgio Giannone
Akash Srivastava MIT-IBM Watson Al Lab
Massachusetts Institute of Technology Technical University of Denmark
exashsri@mit.. edu
ggiorgio@mit... edu
Ole Winther
Faez Ahmed Massachusetts Institute of Technology faezêmit edu
Technical University of Denmark University of Copenhagen olwi@dtu.c dk
Abstract
Generative models have significantly influenced both vision and language domains, ushering in innovative multimodal applications. Although these achievements have motivated exploration scientific and engineering fields, challenges emerge, particularly in constrained settings with limited data where precision is crucial. Traditional engineering optimization methods rooted physies often surpass generative models in these contexts. To address these challenges, introduce Diffusion Optimization Models (DOM) and Trajectory Alignment (TA), a learning framework that demonstrates the efficacy of aligning the sampling trajectory of dif- fusion models with the trajectory derived from physics-basee based iterative optimization methods. This alignment ensures that the sampling process remains grounded in the underlying physical principles. This alignment eliminates the need for costly preprocessing, external surrogate models, extra labeled data, generating feasible and high-performance designs efficiently. We apply our framework to structural topology optimization, fundamental problem in mechanical design, evaluating its performance on in- and out-of-distribution configurations. Our results demonstrate that TA outperforms satte-of-the-att deep generative models on in-distribution configurations and halves the inference computational cost. When coupled with a few steps of optimization, it also improves manufacturability for out-of-distribution conditions. DOM's 'ss efficiency and performance improvements significantly expe- dite design processes and steer them toward optimal and manufacturable outcomes, highlighting the potential of generative models in data-driven design.
Introduction
The remarkable progress large vision [27, 20, 43,44,,88 85] and language models [30, 21, 78] has ushered in unparalleled capabilities for processing unstructured data, leading to innovations in multimodal and semantic generation [79, 67, 18, 80]. This momentum in model development has influenced the rise of Deep Generative Models (DGMs) in the realms of science [51, 61] and engineering [84, 101], especially constraint-bout
problems like structural Topology Optimization (TO [95]), offering potential to make the design process faster.
Many engineering design applications predominantly rely on iterative optimization algorithms. These algorithms break down physical and chemical phenomena into discrete components and incrementally enhance design performance while ensuring all constraint requirements are met. As an example, topology optimization aims to determine the optimal material distribution within a given design space,
37th Conference on Neural Information Processing Systems (NeurIPS 2023).

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Optimization Trajectory
Algnmen
Trajactory Alignment
K6-a
x
Rlxu 4))
Matching
Sampling Trajectory
(a) Sampling and Optimization Trajectory
(b) Trajectory Alignment
Figure Trajectory Alignment. Intermediate sampling steps in Diffusion Optimization Model are matched with intermediate optimization steps. doing so, sampling path biased toward optimization path, guiding the data-driven path toward physical trajectories. This leads to significantly more precise samples.
under specified loads and boundary conditions, achieve the best performance according to set of defined criteria, such as minimum weight or maximum stiffness. Traditional iterative methods like SIMP [15] are invaluable but grapple with practical challenges, especially for large-scale problems, owing to their computational complexity.
Recent advancements have sought to address these challenges by venturing into learning- -baeed methods for topology optimization, specifically DGMs. These models, trained on datasets compris- ing optimal solutions across varying constraints, can either speed or even substitute traditional optimization processes. They also introduce diverse set of structural topologies by tapping into large datasets of prior designs. The dual advantage of generating variety of solutions and accommo- dating numerous design variables, constraints, and goals renders these learning- based methodologies especially enticing for engineering design scenarios.
However, purely data-driven approaches in generative design often fall short when benchmarked against optimization-ban based methods. While data-driven techniques might prioritize metrics such as re- construction quality, these metrics might not adequately gauge adherence engineering performance and constraint requirements. The core of this challenge lies in the data-driven methods' omission of vital physical information and their inability to incorporate iterative optimization details during inference. This oversight can compromise the quality of the solutions, particularly when navigating complex constraints and stringent performance requirements. Consequently, there's emerging need for methodologies that amalgamate the strengths of both data-centric and physics-informed techniques more adeptly navigate engineering applications. Paving the way forward, structured generative models have made notable strides in this domain, exemplified by innovative techniques like TopologyGAN [69] and TifoDiff [62].
Limitations. Structured generative models, despite their recent advancements in engineering designs, grapple with some pressing challenges. For one, these models often require additional supervised training data to learn guidance mechanisms that can improve performance and manufacturability [84]. In the context of Diffusion Models [43], executing forward simulations is not one-off task; it requires repetition, often in the order of tens to hundreds times, to derive an appropriate topology [62]. Moreover, integrating physical data into these models isn't straightforward. It demands time- consuming FEA (Finite Element Analysis) preprocessing step during both the training and inference phases, which is pivotal for computing critical parameters like stress and energy fields[62, 69]. As result, the sampling process is slow, and the inference process is computationally expensive. Such overheads challenge the scalability and adaptability of these models. Consequently, the touted benefits of data-driven techniques, notably fast sampling and quick design candidate generation, are somewhat diminished.
Proposed Solution. introduce a conditional diffusion model that synergistically combines data- driven and optimization-basee techniques. This model is tailored to learn constrained problems and generate candidates in the engineering design domains (refer to Fig. 3). Instead of relying on computationally heavy physics-based exact solutions using FEM, our method employs cost-effective physics-informer informed approximations to manage sparsity conditioning constraints. We introduce a

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Trajectory Alignment (TA) mechanism in the training phase that allows the model to leverage the information the trajectory that was used by an iterative optimization-basee based method in the training data, drastically cutting down the sampling steps required for solution generation using diffusion models. Moreover, our framework can amplify performance and manufacturability in complex problems by integrating few steps of direct optimization. This method strikes a balance between computational efficiency and precision, offering adaptability to novel design challenges. By bridging the gap between generative modeling and engineering design optimization. powerful tool for tackling complex engineering problems. At its core, our findings highlight the value of diffusion models benefitting from optimization methods' intermediate solutions, emphasizing the journey and not just the final outcome.
solution emerges as
Contribution. Our contributions are the following:
(i) We introduce the Diffusion Optimization Models (DOM), a versatile and efficient approach to incorporate performance awareness generative models of engineering design problems while respecting constraints. The primary objective of DOM is to generate high-quality candidates rapidly and inexpensively, with focus on topology optimization (TO) problems. DOM consists of
Trajectory Alignment (TA) leverages iterative optimization and hierarchical sampling to match diffusion and optimization paths, distilling the optimizer knowledge in the sampling process. As result, DOM achieves high performance without depending on FEM solvers or guidance and can sample high-quality configurations in as few as two steps.
Dense Kernel Relaxation, an efficient mechanism to relieve inference from expensive FEM pre-processing and
Few-Steps Direct Optimization that improves manufacturability using few optimiza-
tion steps.
(ii) We perform extensive quantitative and qualitative evaluation in- and out-of of--off-iiiuution. showing how kernel relaxation and trajectory alignment are both necessary for good perfor- mance and fast, cheap sampling. We also release large, multi-fidelity datasel of sub-optimal and optimal topologies obtained by solving minimum compliance optimization problems. This dataset contains low-resolution (64x64), high-resolution (256x256), optimal ( (120k), and suboptimal (600K) topologies. To our knowledge, this is the first large-scale dataset of optimized designs that also provides intermediate suboptimal iterations.
2 Background
Here we briefly introduce the Topology Optimization problem [14], diffusion models [105, 44, 98], class of deep generative models, conditioning and guidance mechanisms for diffusion models, and deep generative models for topology optimization [69, 62]. For more related work see Appendix A.
The Topology Optimization Problem. Topol- ogy optimization is computational design ap- proach that aims to determine the optimal ar- rangement of structure, taking into account set of constraints. Its objective is to identify the most efficient utilization of material while en-
Constrained Optimization
Lcads
Volsze Praction
Performance
Boundryy Conditions
suring the structure meets specific performance requirements. One widely used method in topol- ogy optimization is the Solid Isotropic Mate- rial with Penalization (SIMP) method [13]. The SIMP method employs density field to model the material properties, where the density in- dicates the proportion of material present in particular region. The optimization process involves iteratively adjusting the density field, considering constraints such as stress or deformation. In the context of a mechanical system, common objective to solve a generic minimum compliance problem. This problem aims to find the distribution of material density, represented as E R". that minimizes the deformation of the structure under prescribed boundary conditions and loads [59]. Given set of design variables x {xx
requiremi
Figure 2: In topology optimization, the objective is to find the design with minimum compliance under given loads, boundary conditions, and volume fractions.
1 where
3

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n is the domain dimensionality, the minimum compliance problems can be written as:
min c(x) = F
vxx) = 1²xx <
(I)
x
The goal is to find the design variables that minimize compliance c(x) given the constraints. F is the tensor of applied loads and (xx) is the node displacement, solution of the equilibrium equation (x))U(x) = F where K(x) is the stiffness matrix and is a function of the considered material. v(x) is the required volume fraction. The problem is a relaxation of the topology optimization task, where the design variables are continuous between 0 and 1. One significant advantage of topology optimization is its ability create optimized structures that meet specific performance requirements. However, a major drawback of topology optimization is that it can be computationally intensive and may require significant computational resources. Additionally, some approaches topology optimization may be limited in their ability to generate highly complex geometries and get stuck in local minima.
Diffusion Models. Let Xo denote the observed data Xn E R.D. Let X1...., XT denote T latent variables in RR. We now introduce, the forward or disfusion process q, the reverse generative process po, and the objective L. The forward or diffusion process is defined as [44]: ((X1r(X0)) = q(1, (xo) IT= (= q(x, (1- The beta schedule 31, 32,... Br is chosen such that the final latent image Xy is nearly Gaussian noise. The generative or inverse process pe defined as: pe (Xo, X1:7) po (o(xo x1 )p(x) I (o(xxx1|Xx)) The neural network [ t) is shared among all time steps and is conditioned on The model is trained with weighted version of the ELBO that relates to denoising score matching [105]. The negative ELBO can be written as:
T
Po (Xo, XI:T)
Eg
log
= Lo + Σ LL-1 + LT. t=2
(2)
( xxo
where Lo = Eg(xs (xo) |- log p(xo [x]] is the likelihood term (parameterized by a discretized Gaus- sian distribution) and, if .1....3r are fixed, Lr = KLI[g (x)), p(xx)] is constant. The terms I,-1 for t = 2, .... T can be written as: - Ex [xx 1 xn) p(xx--|xx;)] The terms LI-T-1 1 can be rewritten prediction of the noise added to in ((x, xo). Parameterizing Hi using the noise prediction Eg, we can write
(x) = Bg(r) [wellee(xx e)) <1
where We 20fax(1-az which corresponds to the ELBO objective [50, 56].
Conditioning and Guidance. Conditional diffusion models have been adapted for constrained engineering problems with performance requirements. TopoDiff [62] proposes to condition on loads, volume fraction, and physical fields similarly to [69] learn a constrained generative model. In particular, the generative model can be written as:
(x Xx, c, g) = -oxx-11/o0Xx c) + Σ gp. 7),
(4)
where c is conditioning term and is function of the loads I, volume fraction v, and fields f. ic h(1, v, f). The fields considered are the Von Mises stress d um Vof 011 + 0 + 30 and the strain energy density field W = (011 + 022f22 + 2012(12)/2 Here Tig and 6ij are the stress and energy components over the domain. g is a guidance term, containing information to guide the sampling process toward regions with low floating material (using classifier and gym) and regions with low compliance error, where the generated topologies are close optimized one (using a regression model and g.). Where conditioning e is always present and applied during training, the guidance mechanism g is optional and applied only at inference time.
TofoDiff Limitations. TopoDiff is effective at generating topologies that fulfill the constraints and have low compliance errors. However, the generative model is expensive in terms of sampling time, because we need to sample tens or hundreds of layers for each sample. Additionally, given the model conditions on the Von Mises stress and the strain energy density, for each configuration of loads and

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boundary conditions, we have to preprocess the given configurations running FEM solver. This, other than being computationally expensive and time-consuming, relies on ine-grained knowledge of the problem at hand in terms of material property, domain, and input to the solver and performance metrics, limiting the applicability such modeling techniques for different constrained problems in engineering or even more challenging topology problems. The guidance requires the training of two additional models (a classification and regression model) and particularly useful with out-of-distribution configurations. However such guidance requires additional topologies, optimal and suboptimal, to train the regression model, assuming that we have access to the desired performance metric on the train set. Similarly for the classifier, where additional labeled data has to be gathered.
Diffusion Optimization Model
Direct Optimization
Conditionnng
Trajectory Alignment
Korsed
+
Derimint on (Feu Stepe!
pv(t)11/49) Af(ann pelaa))) n)
Figure 3: The DOM pipeline with conditioning and kernel relaxation (top left) and trajectory alignment (top right). The Diffusion Optimization Model generates design candidates, which further refined using optimization tools. After generation step (left side). we can improve generated topology using few steps of SIMP (5/10) to remove floating material and improve performance (right side). See Appendix 10 for comparison of the inference process for DOM and TopoDiff [62].
Method
To tackle such limitations, we propose Diffusion Op- timization Models (DOM), a conditional diffusion model to improve constrained design generation. One of our main goals is improve inference time with-
wit wit krenet bomm
oet xstepell
0.40
konc!
out loss in performance and constraint satisfaction. DOM is based on three main components: (i) Tra-
||x0 (xp, Eg)
jectory Alignment (Fig. and Fig. 4) to ground sam- pling trajectory in the underlying physical process; (ii) Dense Kernel Relaxation (Fig. 5) to make pre- processing efficient; and (iii) Few-Steps Direct Opti- mization to improve nut-of-distribution performance (Fig. for an overview). See appendix for algo- rithms with and without trajectory alignment. Empirical Methodology. We would like highlight that our primary contribution, trajectory alignment, predominantly empirical. While do make assump- tions about the optimization and sampling trajectory and utilize TA. have not established compre- hensive theoretical framework ensuring convergence of the regularized loss the score optimized by a diffusion model. Nonetheless, our empirical findings sampling process to the desired solutions.
3.34
0.32
10
100
100
310
400
500
000
Iterations
Figure Distance between intermediate sampling steps in DOM optimization steps with and without Trajectory Alignment. Given a random sampling step and the corresponding optimization step stt) mod (t, n) where clean space, using the
[2, 10). We
compute the matching approximate posterior to obtain an estimate for x" given x, and the noise prediction eg. Then we compute the distance ||x (xe, <u) ope ll2- provide evidence for the convergence of the
Trajectory Alignment (TA). Our goal is to align the sampling trajectory with the optimization trajectory, incorporating optimization in data-driven generative models by leveraging the hicrarchical
5

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sampling structure of diffusion models. This aligns trajectories with physics- physics-based information, as illustrated in Fig. la. Unlike previous approaches, which use optimization as pre-pree pre-procsssing or post- processing steps, trajectory alignment is performed during training and relies upon the marginalization property of diffusion models, i.e., ((x/(xx) = q(x1:t Xo)dxx===1, where x, VazXa + (1 -4) c, with ~ (0, T). The trajectory alignment process can match in clean space (matching step 0), noisy space (matching step t). performance space, and leverage multi-fidelity mechanisms. At a high level, TA is regularization mechanism that injects sampling step, forcing to be close to the corresponding optimization step in terms distance. This process provides consistency mechanism [108, 104, 97] over trajectories and significantly reduces the computational cost of generating candidates without sacrificing accuracy.
aptimization-informess prior at each
Aligument Challenges. The alignment of sampling and optimization trajectories is challenging due to their differing lengths and structures. For example, the optimization trajectory starts with an image of all zeros, while the sampling path starts with random noise. Furthermore, Diffusion Models define a Stochastic Differential Equation (SDE. [106]) in the continuous limit, which represents a collection of trajectories, and the optimization trajectory cannot be directly represented within this set. To address these issues, trajectory alignment comprises two phases (see Figure lb): a search phase and matching phase. In the search phase, we aim to find the closest trajectory, among those that can be represented by the reverse process, to the optimization trajectory. This involves identifying a suitable representation over trajectory that aligns with the optimization process. In the matching phase, we minimize the distance between points on the sampling and optimiza- tion trajectories to ensure proximity between points and enable alignment between trajecto- ries.
Trajectory Search. We leverage the approxi- mate posterior and marginalization properties of diffusion models to perform a trajectory search, using the generative model as a parametric guide to search for a suitable representation for align- ment. Given an initial point Xy, we obtain an approximate point by sampling from the posterior distribution g(x,) xo). We then pre- dict (x(x)) with the model and use it to obtain x" (xe, eg (x:)). In a DDPM, x is an approximation of xo and is used as an intermediate step to sam- ple x using the posterior functional form q(x x*). In DOM, we additionally leverage x to transport the sampling step towards suitable representation for matching an intermediate optimiza- tion step xtte((t) corresponding to using some mapping. Trajectory alignment involves matching the optimization trajectory, which is an iterative exact solution for physics-based problems, with the sampling trajectory. which is the hierarchical sampling mechanism leveraged in Diffusion Mod- els [43] and Hierarchical VAEs [100]. In practice, in DOM we sample x, = v vā,xo + (1 - āz)e from (xx, (X0) and run a forward step with the inverse process (x c) conditioning on the constraints to obtain the matching representation x for step t:
Figure 5: Comparison of iterative (left), sparse (center), and dense single-step (right) conditioning fields for Constrained Diffusson Model. Unlike the expensive iterative FEA method, the physics-inspired fields offer a cost-effective, single-step approximation that's 's domain- agnostic and scalable.
X ~ g(8 17 (x:;
(5)
7 (Xx, = (x: v1 âi (xt, c))/ Vor.
Trajectory Matching. Then we match the distribution of matching representation (xx |x,, co) for sampling step L with the distribution of optimized representations g(x" S(tt-))opptt at iteration s (corresponding to step t - 1) conditioning on the optimizer S. In general, given that the sampling steps will be different than the optimization steps, we use s(t) 1 n, x (1 11-12 t/T), where n, is the number of optimized iterations stored, and then select the integer part." We then can train the model as a weighted sum of the conditional DDPM objective and the trajectory alignment regularization:
Lzox Egtxx xx)
KlL[g(x-11 xo) (x x,, c)] + [L (x,, ((xx((-)))lopt)) (6)
This mechanism effectively pushes the sampling trajectory at each step to match the optimization trajectory, distilling the optimizer during the reverse process training. In practice, following practice
for example, if n, 5 and T' = 1000, 1 For (T" T 200). and for steps (200 1).

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in DDPM literature, the distribution variances are not learned from data. For the trajectory alignment distributions, we set the dispersion the same values used in the model. By doing so we can rewrite the per-step negated lower-bound as weighted sum of squared errors:
Egte)
[wg |so (Xo, e), c) - <|12] + ac |x (x,, Eo)
opt
(x.e)
clean
where LTA is the trajectory alignment loss for step t, and L c) is conditional DDPM loss for step t. This is the formulation employed for our model, where we optimize this loss for the mean values, freeze the mean representations, and optimize the variances in a separate step [68].
Alignment can also be performed in alternative ways. We can perform matching in noisy spaces, using the marginal posterior to obtain a noisy optimized representation for step t - 1, g(x opt (xx()) )
and then optimize L'ossyy = Os || ||x__ x;-1 113 Finally, we can match in performance space: this approach leverages an auxiliary model fo similar to (consistency models) and performs trajectory alignment in functional space, / = apl If (x(-1) - P=(t-1)lle l2 where we match the performance for the generated intermediate design with the ground truth intermediate performance (s(t-1) for the
optimized x opt x((--)) We compare these and other variants in Table 6.
Dense Conditioning over Sparse Constraints. All models are subject to conditioning based on loads, boundary conditions, and volume fractions. In addition, TopoDiff and CopoDiff-GUIDED undergo conditioning based on force field and energy strain, while TopoDift-FF and DOM are conditioned based on dense kernel relaxation, inspired by Green's 's method [36, 34], which defines integral functions that are solutions to the time-invarian Poisson's 's Equation [33, 41]. More details are in Appendix D. The idea to use the kernels as approximations to represent the effects of
the boundary conditions and loads as smooth functions across the domain (Fig. 5). This approach
avoids the need for computationally expensive and time-consuming Finite Element Analysis (FEA) to provide conditioning information. For load or source I, a sink or boundary and r = ||x -x|||2
(x; - x2)² (xj x2)², we have:
Ki(X, xG)) Σt. (11---0/1xxxx0 B 16(xxx;;0) Σb afllx x x / /maxx E /x/
/101 p(xi)
(8)
where p is the module of generic force in 2D. Notice how, for r - 0, Ki x:) + p. and r - oc. (1xx,x) x1) + We notice how closer to the boundary the kernel is null, and farther from the boundary the kernel tends to 1. Note that the choice of a parameters in the kernels affects the smoothness and range of the kernel functions. Furthermore, these kernels are isotropic, meaning that they do not depend on the direction in which they are applied. Overall, the kernel relaxation method offers a computationally inexpensive way to condition generative models on boundary conditions and loads, making them more applicable in practical engineering and design contexts.
Few-Steps Direct Optimization. Finally, we leverage direct optimization to improve the data- driven candidate generated by DOM. In particular, by running few steps of optimization (5/10) we can inject physics information into the generated design directly, greatly increasing not only performance but greatly increasing manufacturability. Given sample from the model Xg ~ po (xo Xx )Po (x1:r,, we can post-process it and obtain Xg = oxt(Xf, n) an improved design leveraging steps of optimization, where E [5, 10). In Fig. 3 we show a full pipeline for DOM.
4 Experiments
Our three main objectives are: (1) Improving inference efficiency, and reducing the sampling time for diffusion-based topology generation while still satisfying the design requirements with a minimum decrease in performance. (2) Minimizing reliance force and strain fields conditioning information, reducing the computation burden at inference time and the need for ad-hoc conditioning mechanisms for each problem. (3) Merging together lcarning- baaed and optimization-base methods, refining the topology generated using a conditional diffusion model, and improving the final solution in terms of manufacturability and performance.
Setup. We train all the models for 200k steps on 30k optimized topologies on a 64x64 domain. For each optimized topology, we have access to a small subset (5 steps) of intermediate optimization
7

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- 
D
SH neooe
Few Sew-
Figure 6: Few-Step sampling for Topology generation. Top row: Diffusion Optimization Model (DOM) (DOM) with Trajectory Alignment. Middle row: CopoDiff-GUIDEE Bottom now: The Orimizaai result. DOM produces high-quality designs as few as two steps, greadly enhancing inference efficiiney compante requiring 10-100 steps. Trajectory Alignment helps DOM generate dear-ooratnn scometricn swiftly, improving few-step sampling in conditional diffusion models for topology optimization. See appendix Fig. 13 for more examples.
Mamde
to previous models
Trajec
sampligg
steps. We set the hyperparameters, conditioning structure, and training routine as proposed in [62]. Appendee
G for more details. For all the models (Table 1) we condition on volume fraction and loads. For TopoDiff, we condition additional stress and energy fields. For TopoDiff-FF [37], a variant of TopoDiff' conditioning To TopoDiff- TopoDiff-GUIDED GUIDED leverages a compliance regressor and floating material classifier gut guidance. We
a kernel relaxation, we condition on boundary conditions and kernels.
use reduced number of sampling steps for all the experiments,
Dataset. We use dataset of optimized topologies gathered using SIMP as proposed in [66, 62]. Together with the topologies, the dataset contains information about optimal performance. For each topology, we have information about the loading condition. boundary condition, volume fraction, and optimal compliance. Additionally, for each constraint configuration, a pre- pe-ppoocesing step computes compliance
compun the force and strain energy fields (see Fig. 5) when needed. Appendix F for more details on the dataset and visualizationn visualizations.
Evaluation.
We evaluate the model using engineering and generative metrics. In particular, we consider metries that evaluate how well our model fulfills: physical constraints using wrt prescribed Volume Fraction (VFE); engineering constraints, as manufacturabilityy as measured by Floating Material (FM); performance constraints, as measured by compliance error (CE) wrt the optimized SIMP solution; sampling time constraints (inference constraints) as mea- sure by sampling time (inference and pre- pr-pesssssn)) We consider two scenarios of increasing complexity: (i) In- distribution Constraints. The constraints in this test set are the same as those of the training set. When measuring performance on this set, we filter generated configurations with high compliance. (ii) diitrii-ittiitiionn Constraints. The constraints in this test set are different from those of the training set. When measuring performance on this set, we filter generated configurations with high compliance. The purpose of these tasks is to evaluate the generalization capability of the machine cooommnn
Table 1:
Comparative study of generative models in topology optimization consider- ing factors like conditional input (COND), finite element method (FEM), and guidance (GUID). Unlike others, the DOM model op- crates without FEM preprocessing or GUID- ANCE. More visualizations and optimization trajectorics are in the Appendix.
COND wlo FEM w/o GUID
TopologyGAN [69]
x
x
[62]
/
X
TopoDiff [62]
x
X
DOM (ours)
/
capabils
learning models in- and out-of-distribution. By testing the models on different test sets with varying levels of difficulty, i importandy. we want to understand how important the role of the force field and energy unknown constraints.
assess how well the models can perform new, unseen data. More
meeryystiat strain is with
In- In-Distribution Constraints. Table 3 reports the evaluation results in terms of constraints satisfac- tion and performance for the task of topology generation. In Table 2 we report metrics commonly models in terms of fidelity (IS, sFID, P) and diversity (R). We see how such metries are all close and is challenging to gain any understanding just by rely- aaneereeeeeee
employed to evaluate the quality of tenerative
ing
g on classic generative metrics when evaluating constrained design generation. These results justify the need for a evaluation that considers the performance and feasibility of the senerated design.
8

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Generative metrics in- distribution metries. Precision denotes the fraction of generated topologies that are real- istic, and recall measures the fraction of the training data manifold covered by the model.
Table 2:
In Table DOM achieves high performance and is at least 50 % less computationally expensive at inference time, not requiring FEM preprocessing or additional guidance through surrogate models like TopologyGAN and TopoD>- iff. We also compare with Consistency Models [104], Diffusion Model that tries to predict its input at each step. DOM can be seen generalization of such method when a trajectory is available as a ground truth. Overall, DOM with Trajectory Alignment is competitive better than the previous proposal in terms performance on in-distribution constraints, providing strong evidence that TA is an effective mechanism to guide the sampling path toward regions of high performance.
sFID 4
Pf
R f
cDDPM Topo
3.41
40.30
0.73
0.85
3.57
36,20
0.80
0.86
TopoDiff-G
3.63
35.96
0.79
0.86
DOM w/o TA
3.48
37.54
0.76
0.86
DOM w/ TA
3.68
36.73
0.77
0.85
Table 3: Evaluation of different model variants on in distribution constraints. CE: Compliance Error. VFE: Volume Fraction Error. FM: Floating Material. We 100 sampling steps for all diffusion models. We see that DOM w/ TA competitive with SOTA on topology generation. being computationally % less expensive inference time compared to Tifo Trajectory Alignment greatly improves performance without any additional inference cost. See appendix Fig. 11 for confidence intervals.
STEPS
CONSTRAINTS AVG CE
MDN L
VFE 1
FM L
INFERENCE (6)
TopologyGAN [69]
FIELD
48.51
2.06
11.87
46.78
3.37
Conditional DDPM [68]
100
RAW
60.79
3.15
1.72
8.72
2.23
Consistency Model [104]
100/1
KERNEL
10.30
2.20
1.64
8.72
2.35
FF [37]
100
KERNEL
1,92
2.05
8.15
TopoDiff' [62]
100
FIELD
5.46
0.80
1.47
5.79
5.54
TopoDiff-GUIDED [62]
100
FIELD
5.93
0.83
1.49
5.82
5.77
DOM w/o TA (ours)
100
KERNEL
13.61
1.79
1.86
7.44
2.35
DOM w/ TA (ours)
100
KERNEL
4.44
0.74
1.52
6.72
2.35
Table 4: Evaluating sampling topologies with few steps (2-10) for TopoDiff and DOM. G: Guided using regression and classifier guidance. AVG CE: average compliance error. MDN CF: median compliance error. VFE: volume fraction eтror. FM: floating material. INF: inference time. UNS: unsolvable confi configurations. I.D: load disrespect. DOM largely outperforms TopoDiff the few sampling step regimes, showing Trajectory Alignment's effectiveness as a grounding mechanism. DOM can generate reasonable topologies in just two sampling steps, where TopoDiff and DOM w/ TA fail completely, even presenting cases of load disrespect.
STEPS SIZE
AVG % CE MDN % CE % VFE
% FM
INF (6)
% UNS 4 % LD
in-distro
TopoDiff-G
239M
681.53
436.83
80,98
98.72
3.36
2.00
15.92
DOM (ours)
121M
22.66
1.46
3.34
33.25
0.17
(494.94
2.11
0.00
TopoDiff-G DOM (ours)
239M
43.27
15.48
2.76
77.65
3.43
1.44
121M
11.99
0.72
2.27
20.08
0.24
(-93.00
2.77
0.00
TopoDiff-G DOM (ours)
239M
6.43
1.61
1.95
20.55
3.56
0.00
0.00
10
121M
4.44
0.57
1.67
11.94
0.35 90.17 1771159 %))
0.00
0.00
out-distro
TopoDiff-G DOM (ours)
239M
751.17
$48.26
81.46
100.00
3.36
1.90
16.48
121M
79.66
10.37
3.69
44.20
0.17 (- 94, %))
2.80
0.00
TopaDiff-Ci DOM (ours)
239M
43.50
19.24
2.58
79.57
3.43
2.20
0.00
121M
38.97
5.49
2.56
0.24
%6)
1.40
TopoDiff-G DOM (ours)
239M
10.78
2.55
1.87
21.36
3.56
2.10
0.00
121M
32.19
3.69
1.78
14.20
0.35
90.17
17
0.40
0.00
Generation with Few-Steps of Sampling. Table 4 compares two different algorithms, TopoDiff- GUIDED and DOM, in terms of their performance when using only few steps for sampling. The table shows the results of the in and both having STEPS values of 2, 5, and 10, and SIZE of 239M and 1 We can see that DOM outperforms by large margin TopoDiff-G when tasked with generating a new topology given few steps, corrohorating our hypothesis that aligning the sampling and optimization trajectory an effective mechanism to obtain efficient generative models that satisfy constraints. DOM outperforms
comparison, with TopoDiff - and DOM

---

CopoDiff-UUIDEE even being 50 % smaller, without leveraging an expensive FEM solver for conditioning but but relying on cheap dense relaxations, making it 20/10 times faster at sampling, and greatly enhancing the quality of the generated designs, providing evidence that Trajectory Alignment an effective mechanism to distill information from the optimization path. In Fig. 6, we provide qualitative results to show how DOM (top row) is able to generate reasonable topologies, resembling the fully optimized structure running SIMP for 100 steps (bottom row), with just two steps at inference time, where the same model without TA or Topo are not able to perform such task. Overall these results corroborate our thesis regarding the usefulness of trajectory alignment for high-quality constrained generation. For more experiments on inference Appendix B.
Merging Generative Models and Optimiza- tion for Out-of-Distribution Constraints. Table 5 shows the results of experiments on out-of-distribution constraints. In this scenario, employing FEM and guidance significantly en- hances the performance of TopoDiff. Condi- tioning on the FEM output during inference can be seen as a form of test-time conditioning that can be adapted to the sample at hand. However, merging DOM and few iterations of optimiza- tion is extremely effective in solving this prob- lem, in particular in terms of improving volume fraction and floating material. Using the combi- nation of DOM and SIMP is a way to impose the performance constraints in the model without the need for surrogate models or guidance.
Table 5: Nitoo--Distribution Scenario Comparison: TopoDiff-G outperforms DOM due to its adaptive con- ditioning mechanism, which leverages expensive FEM- computed fields. However, DOM coupled with few steps of direct optimization (5/10) greatly surpasses Topo in performance and manufacturabilityy This un- derscores the effectiveness of integrating data-driven and optimization methods in constrained design creation.
STEPS MDN % CE J.
VFE.
FM
TopoDiff-FF TopoDiff-G
100
16.06
1.97
8.38
100
1.82
1.80
6.21
DOM
100
3.47
1.59
8.02
DOM SIMP DOM SIMP
100+5
1.89
1.77
10.19
100+10
1.15
1.10
2.61
Trajectory Alignment Ablation. The core contribution of DOM is trajectory alignment, method to match sampling and optimization trajectories of arbitrary length and structure map- ping intermediate steps to appropriate CLEAN (noise free or with reduced noise using the model and the marginalization properties of DDPM) representations. However, alignment can be performed in multiple ways, leveraging NOISY representation, matching performance (PERF), and using data at a higher resolution to impose consistency (MULTT). In Table 6 we per- form an ablation study, considering DOM with and without kernel relaxation, and leveraging different kinds of trajectory matching. From the
Table 6: Ablation study with and without kernel and trajectory alignment. We explore different ways match the sampling and optimization trajectory and we measure the Median Compliance Error. TA: trajectory alignment. CM: Consistency Models [104].
KERNEL TA MODE IN-DISTRO OUT-DISTRO
DOM
X
3.29
8.05
DOM
X
V
CLEAN
1.11
9.01
CM
V
2.20
5.25
DOM
V
X
1.80
5.62
DOM
V
MULTI
34.95
54,73
DOM
NOISY
2.08
6.23
DOM
PERF
2.41
6.82
DOM
V
CLEAN
0.74
3.47
table, we see that using dense conditioning is extremely important for out-of-disttibution performance, and that matching using CLEAN is the most effective method in and fitt-fffdistibutionnn In Fig. 4 we report visualization of the distance between sampling and optimization trajectory during training. From this plot, we can see how the kernel together with TA helps the model to find trajectories that closer to the optimal one, again corroborating the need for dense conditioning and consistency regularization.
Conclusion
We presented Diffusion Optimization Models, a generative framework to align the sampling trajectory with the underlying physical process and leam efficient and expressive generative model for constrained engineering design. Our work opens new avenues for improving generative design in engineering and related fields. However, our method is limited by the capacity to store and retrieve intermediate optimization steps, and, without a few steps of optimization, it underperforms out-of-distribution compared to FEM-conditional and guided methods.
10

---

Acknowledgment
We would like to thank Noah Joseph Bagazinski, Kristen Marie Edwards, Amin Heyrani Nobari, Cyril Picard, Lyle Regenwetter and Binyang Song for insightful comments and useful discussions. We extend our gratitude to the anonymous reviewers for their constructive feedback and discussions that greatly enhanced the clarity and quality of the puper. OW's work was funded in part by the Novo Nordisk Foundation through the Center for Basic Machine Learning Research in Life Science (NNF200C0062606) and by the Pioneer Centre for AI, DNRF grant number PI.
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[117] S. Xu, Y. Cai, and G. Cheng. Volume preserving nonlinear density filter based on heaviside functions. Structural and Multidisciplinary Optimization, 41:495-505: 2010.
[118] L. Xue, J. Liu, G. Wen, and H. Wang. Efficient, high-resolution topology optimization method based on convolutional neural networks. Frontiers of Mechanical Engineering, 16(1) (60099996,, 3 2021.
[119] A. Yildiz, N. Öztürk, N. Kaya, and E. Öztürk. Integrated optimal topology design and shape optimization using neural networks. Structural and Multidisciplinary Optimization, 25(4):251 260, 10 2003.
[120] S. Yoo, S. Lee. S. Kim, K. H. Hwang. J. H. Park, and N. Kang. Integrating deep learning into cad/cae system: generative design and evaluation of 3d conceptual wheel. Structural and Multidisciplinary Optimization, 64(4):2725- 2747, 10 2021.
[121] Yu, Hur, J. Jung, and Jang. Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinay Oprimization, 59(3):787-799, 2019.
[122] J.-Y. Zhu, Park, P. Isola, and A. Etros. Unpaired image-to-image translation using cycle-consistent adversarial networks. In Proceedings of the IEEE international conference on computer vision, pages 2223-225 2032-223,,20 2017.
16

---

A Related Work
Engineering design is the process of creating solutions to technical problems under engineering requirements [22, 91, 23]. Often, the goal is to create highly performative designs given the required constraints. Topology Optimization (TO [14]) branch of engineering design and is critical component of the design process in many industries, including aerospace, automotive, manufacturing, and software development. From the inception of the homogenization method for TO, a number of different approaches have been proposed, including density-based [13, 86, 64], level-set [3, 111], derivative-based [99]. evolutionary [115], and others [19]. The density- methods are widely used and use nodal-based representation where the lovel-set leverages shape derivative to obtain the optimal topology. To improve the final design, filtering mechanisms have been proposed [117, 39, 94], Hybrid methods are also widely used. Topology Optimization has evolved as a more and more intensive computational discipline, with the availability of efficient open-source implementations [92, 93, 46, 59, 4]. See [59] for more on this topic. See [95, 96] for comprehensive review of the Topology Optimization field.
Topology Optimization.
Generative Models for Topology Optimization.
Following the success Deep Learning (DL) in vision, surging interest arose recently for transferring these methods to the engineering field. In particular, DL methods have been employed for direct-design [1, 5, 11, 60, 110], accelating the optimization process [7, 19, 52, 107, 118], improving the shape optimization psttprocesssig [42, 119], super-resolusion [32, 65, 120], sensitivity analysis [6. 75, 10, 89J. 3d topologies [54, 87, 11 and more [26, 24, 25, 29]. Among these methods, Generative Models are especially appealing to improve design diversity in engineering design [2, 72, 70, 71J. In TO, the work of [47, 82, 83, 55, 109] focus on increasing diversity leveraging data-driven approaches. Additionally, Generative Models have been used for Topology Optimization problems conditioning on constraints (loads, boundary conditions, volume fraction for the structural case), directly generating topologies [81, 90, 40] training
dataset of optimized topologics, leveraging superresolution methods to improve fidelity [121, 58), using filtering and iterative design approaches [74, 73, 17. 35J to improve quality and diversity. Methods for 3D topologies
have also been proposed [12, 55J. Recently, GAN [69] and DDPM-based [62] approaches, conditioning on constraints and physical information, have had success in modeling the TO problem. For comprehensive review and critique of the field, see [113].
Conditional Diffusion Models. Methods to condition DDPM have been proposed, conditioning at sam- pling time [28]. learning class-conditions score [106]. explicilly conditioning on class information [68]. features [38, 91, and physical properties [116, 45]. Recently, Topo [62] has shown that conditional diffusion models with guidance [31] are effective for generating topologies that fulfill the constraints and have high manufacturability and high performance. TopoDiff relics on physics information and surrogate models to guide the sampling of novel topologies with good performance. Alternatives to speed up sampling in TopoDiff' have been recently proposed [37], trading performance for fast candidate generation. Improving efficiency and sampling speed for diffusion models is an active research topic, both reducing the number of sampling steps 168, 103, 102, 8J, improving the ODE solver [48, 112, 57J, leveraging distillation [88, 63]. and exploiting autoencoders for dimensionality reduction [85, 77. 76J. Reducing the number of sampling steps can also be achieved by improving the property of the hierarchical latent space, exploiting a form of consistency regular- ization [ [108, 97, 122] during training. Consistency Models [104] proposes reconstructing its input from any step in the diffusion chain, effectively forcing the model to reduce the sampling steps needed for high-quality sampling. We similarly want to improve the latent space properties but leverage trajectory alignment with a physical process. Recently, energy consstrcodt diffusion models [1 14] have been proposed to regularize graph learning learn expressive representations for structured data.
17

---

http://///////.oooo////////////a -iffunnio
Additional Experiments
B.1 Sampling Topologies in Few-Steps
Table Evaluating sampling topologies with few steps (2-10) for TopoDiff and DOM. G: Guided using regression classifier guidance. AVG CE: average compliance error. MDN median compliance error.
VFE: volume fraction error. FM: floating material, INF: inference time. UNS: unsolvable configurations. LD:
load disrespect.
STEPS SIZE
AVG % CE
MDN & CE % VFE % FM
INF (6)
% UNS + % LD
in-distro
TopaDiff-G DOM (ours)
239M 121M
681.53
436.83
80.98
98.72
3.36
2.00
15.92
22.66
1.46
3.34
33.25
0.17
(-94.90
2.11
0.00
TopoDiff-G DOM (ours)
239M
43.27
15.48
77.65
3.43
1.44
121M
11.99
0.72
2.27
20.08
0.24
93.00
2.77
0.00
TopoDiff-G DOM (ours)
239M
6.43
1.61
1.95
20.55
3.56
0.00
0.00
121M
4.44
0.57
1.67
11.94
0.35 90.17 %6)
0.00
0.00
out-distro
TopoDiff-G DOM (ours)
239M
751.17
$48.26
81.46
100.00
3.36
1.90
16.48
121M
79.66
10.37
3.69
44.20
0.17
94. %6)
2.80
0.00
TopoDiff-G DOM (ours)
239M
43.50
19.24
2.58
79.57
3.43
2.20
0.00
121M
38.97
5.49
2.56
26.70
0.24
93.00 %))
1.40
TopoDiff-G DOM (ours)
239M
10.78
2.55
1.87
21.36
3.56
2.10
121M
32.19
3.69
1.78
14.20
0.35
90.17
0.40
0.00
B.2 Inference Time
With the previous experiments, proved that data-driven approach, biased towards the physical process, distill optimization process sample novel topology a few steps. We then provide experiments for and out-of-distribution and ablate our choice of kernel relaxation and trajectory alignment mechanism. However, the final goal of data-dria ditt-driven design is to Iearn a general tool for fast candidate generation. In Table 8 we compare inference time for different models and, more importantly, for low- (64) and high-resolution (256). Given the computational burden of training such models at high resolution, (around 5% the training), and we use them for sampling. With this experiment, we want to emphasize how fast DOM can perform inference compared to SOTA model. We choose to run all the generative models and optimized for 100 steps: this setting is not suited for DOM. because as we have seen the model excels in the few-step sampling task. Table presents comparison of various models based different factors such resolution (RES), size, preprocess time, postprocess time, sampling, and inference time.
train all the models for only 10k steps
Table 8: Inference time for different models at low and high resolution. For all the diffusion models we sample 100 steps and for SIMP we iterate for steps. *We report optimization time for full comparison, but is important to emphasize that SIMP runs on CPU and the DDPM-based models on GPU. A AT; ATinference
(T (Toodel T topodi 1) /Ttr)//TTooot (opodiff)
RES
SIZE
PREPROCESS POSTPROCESS SAMPLING INFERENCE AT ferencece
TifoDiff'
121M
3.31
0.00
2.23
5.54
0.00
TopoDiff-GUIDER DOM (ours)
239M
3.31
0.00
5.77
4.15
121M
0.12
0.00
2.35
57.58 %
SIMP
18.12
0.00
18.12
227.07 %6
TopoDiff
256
553M
31.84
0.00
7.78
38 68.62
+ %
MoDDif--GUIDED
256
1092M
31.84
0.00
8.46
40.30
+ +4.35 %
DOM (ours)
256
553M
0.31
0.00
7.78
8.09
79.05 %
SIMP
256
0.00
316.02
0.00
316.02
718.28 %"
18

---

C
Algorithms
Algorithm DOM with Trajectory Alignment
Require: Optimized Topologies Xn
Require: Constraints C (BC, L, VF)
Require: Intermediate Optimization Steps X
while Training do
Sample batch (Xo, C, x
Compute Dense Relaxation K(be,1) 1)
Compute Conditioning c = ())
Sample t, x
Compute x, ~ (ixx(x(Xxx) (Xo)
Forward Model 60 (x,, c)
Compute Loss L c) = || c) E|1
Trajectory Search 8²(x; (g) (xi v1 āt co (x,,
Trajectory Matching LTA = || (xx, En) x xt)) l1
Compute Loss Cpon(0) = 1x, c) + LTA
Backpropagate (oL)an(
end while
Algorithm 2 DOM without Trajectory Alignment
Require: Optimized Topologies X
Require: Constraints C = (VC,
while Training do
Sample batch (xo, c)
Compute Dense Relaxation k: K(be,)) 1)
Compute Conditioning c = (k: c)
Sample c
Compute x, ~ q(,, (xo)
Forward Model eo (Xe; c)
Compute Loss L. (x, c) = c)
Compute Loss Loon (0) L4-1 (),
Backpropagate f & (eLDou(f)
end while
19

---

import numpy up
import torch
def Cogptt..kemeelolaalccccc......dd
axis):
size latch (-))
if aris "x":
ix xx th
(batch_ [0] != 0)
10
coord - xx
elif axis "y":
12
ix 1
13
yy th. arguhere
[1] 0)
14
coord =
if (coor))
17
return batch_load sampee [ix],
18
19
x_grid th tensor ([i for i in range (size)])
y-grid th tenser ([j for j in ranze
22
pernel_load
23
for 1 in range (len (coord)):
x-grid = th. tensor ([i for i in raize(sizz)]
y-grid tensor ([j for in range [size)]
26
# distance
27
x_grid = x_grid [oord[]]]0] y-grid = y-grid coord[1l[i]
28
30
grid th. meshgrid(xxgrrd y-grid)
31
32
r_load = th. sqrt [grid + grid[1]+*2)
if axis "x"; p batch_load_samme [0] [coord [1) [0] coord [1] [1]]
35
36
elif axis =- "y";
37
= batch load, sample [1] [coord [o] coord [1] [1]]
kernel kernel_load kernel P
exp(- 1/r_ l/r_load=*2)
40
41
return kernel_load. coord
Listing Dense Kemel Relaxation for Sparse Loads.
20

---

D Physics-based Conditioning on Constraints
Seeking more efficient Luffusion-basee topology generation while reducing dependency on force and strain fields, our approach is to approximate boundary conditions and loads using kernels. These kernels approximate the impact of constraints on the domain. The kemel architecture opt for draws inspiration from Green't method [36, 34, 16, 41J. This method establishes integral functions solutions to the time-invarinn aaae-ivvarinnPoisson Equation [33, 41J. an extended form Laplace's 's Equation addressing point source excitations. Expressed mathematically, Poisson's Equation is given as V / (a) = h, with / as the force term and f representing a general function over domain X. This equation underpins numerous natural phenomena, with notable case being force part O. This particular brings forth Laplace's Equation, which is frequently used in heat transfer scenarios.
Green's 's method, part of larger family of methods, offers structured approach to address partial differential equations, even when domain specifics unknown. The solutions derived using this method termed Green't functions [53]. Even though such solutions can generally complex, for broad range physical issucs where constraints and forces are approxxxxxi,,, straightforward functional framework can be established, hinging the source and sink concept.
Consider, for instance, laminar domain like beam plate, restrained in a feasible manner. If point force applied (such downward pressure beam's 's edge plate's center) xy, this can be captured using the Dirac delta function, 6(x xy). The delta function is highly discontinuous but has powerful integration properties, specifically f f zy dz = f(zy) within domain X. The solution for the time-invariant Poisson's s Equation, when focusing on pinpoint forces, can be represented as a Green's 's function solution. This solution is primarily influenced by the distance from where the force is applied. Specifically:
1
(xx, x²)
(9)
4T
2
x²
24 21|2 |zy 22 To approximate the forces and loads impacting our topologies, we utilize kernel approximation grounded in Green's functions. While this model may preciscly map all loads and boundary conditions, does furnish us with computationally efficient conditioning data that aligns with core physical and engineering constraints. Harnessing these concepts, we aspire to enrich the conditioning data for the model, aiming to elevate generative models constrained by such conditions.
where = |x-5/1
21

---

Table 9:
Design and Modelling requirements for constrained generative model for topology optimization. Our goal is to improve requirements that are challenging fulfill. work, we focus on improving Floating Material, reducing Compliance Error, and reducing Sampling Time.
CLASS
METRICS
GOAL
Hard-constraint
Loads Disrespect
Feasibility
Hard-constraint
Floating Material
Manufacturaiiiity
Soft-constraint
Volume Fraction
Min Cost
Functional Performance
Compliance Error
Max Performance
Modeling Requirements
Sampling Time
Fast Inference
E Tables and Visualizations
Mean Complance Generated 1opo ogies
130
Mean Compl ance Optimized lopo lgita
Median Comslince Genertted Tapo ligles
Median Complance Optimised Topetig
% Sample cc
Compliance
00M
D0M we
2.01
0.02
0.03
0.M4
0.05
Threshold (t)
% train set oc
(a) Ratio of generated samples with CR < f.
(h) Ratio of the training set with compliance t.
% Saauffcturrbiltyy
0.99
0.98
kznd
pot
0.96
[|x* Eg)
0.95
0.36
0.34
0.83
% Performance
Iterations
(c) Manufacturability vs Performance.
(d) Distance sampling and optimization path.
Table 10: Conditioning or guiding variables for different optimization methods and model configurations.
Load
BC
Kernel Load
Kernel BC
Force Field
Energy Field
VF
Performance
[14]
TopologyGAN [69]
x
TifoDiff [62]
/
TopoDiff-FF (37)
/
/
x
/
DOM (ours)
/
X
/
X
DOM Trajectory Aligament (ours)
/
/
DOM Optimizer (ours)
/
X
/
22

---

generated topologies
optimized topologies
0.10
0.08
0.06
Density
0.04
0.02
0.00
5
10
15
20
25
30
Compliance
Figure 8: Histogram empirical distribution compliance for generated and optimized topologies.
23

---

28
R
Figure 9: DOM Sampling, Conditional Samples from a Diffuusion Optimization Model trained with Trajectory
Alignment. For each row, we select constraint configuration (boundary condition, loads, volume fraction) and sample the model 10 times. We use 100 sampling steps. We repeat this process 12 times. During sampling, DOM is standard conditional DDPM and does not have access the optimization trajectory.
Sampling
TopoDII--UIDED Sampling Inference
Cardssaing
Aa4
Cotple Regesaor
pvfsui #, m Misur po(swo) + de o)
wisscin
decaval
Figure 10: DOM Sampling Process at Inference Time. DOM (left) and TopoDiff-Guided Guided (right) sampling process during inference. DOM is conditional diffusion model. DOM is conditioned on the boundary conditions, loads, volume fraction, and kernels. After conditioning, we sample DOM with standard ancestral sampling like any DDPM. Contrary to TopoDiff (right), DOM does not rely on expensive preprocessing to compute energy and strain fields and auxiliary models for guidance. DOM can generate samples in a few steps (2/5) contrary to TopoDiff-G which requires hundreds of steps generate reasonable topologies.

---

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diowcc
Figure 11: Confidence interval for design requirements in-distribution constraint configurations.
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Figure 12: Examples of generated topologies with good performance.
25

---

Soo W
Tigure 13: Compartsonn DOM w/ TA and DOM w/o TA generation with 2 steps.
Figure 14; Comparison DOM w/ TA and TopoDiil Sepiif-Guuii generation with 2 steps.
OP
Figure 15: Comparison DOM and YopoDiff-GUIDE generation with 100 stops.
26

---

Figure Left: x from intermediate generation steps for DOM w/ TA (top block), DOM TA (middle block), SIMP iterations (bottom block). Right: Prediction of x from 10 intermediate generation steps for DOM w/ TA (top block), DOM w/o TA (middle block), SIMP iterations (bottom block).
27

---

Dataset
We build dataset optimized topologies and intermediate optimization steps at low-resolution (64x64) and high-resolution (256x256). In particular:
50K low-resolution optimized topologies w/ constraints.
60K high-resolution optimizer topologies w/ constraints.
250K low-resolution intermediate steps [10, 20, 30, 50, 70] w/ constraints.
300K high-resolution intermediate steps [10, 20, 30, 50, 70] w/ constraints.
In Figure 17, 18, 19, 20 show examples of intermediate steps 10, 20, 30, and optimized topologies high-resolution.
Training Topologies
Figure 17: Intermediate optimization output after 10 iterations. Resolution: 256x256.
28

---

Training Topologies
Figure 18: Intermediate optimization output after iterations. Resolution: 256x256.
29

---

Training Topologies
Figure 19: Intermediate optimization output after 30 iterations. Resolution: 256x256.
30

---

Training Topologies
Figure 20: Optimized output. Resolution; 256x256.
31

---

G Experimental Details
Table 11: Relevant Hyperparameters for baselines and DOM on 64x64 datasets.
Topo
TopoDift-G
DOM w/o TA
DOM w/ TA
Dimension
1x64x64
Ix64x64
1x64x64 30K
1x64x64
Model Set
30k
30K
30K
Guidance Set
150K
Intermediate Set
150K
Test Configurations
1800
1800
1800
1800
Batch size Architecture
64
64
64
64
Unet
Unet
Unet
Unet
Iterations
200K
200K
200K 2e-1
200K
Learning rate
2e-1
2e-
2e-1
Loss
Le
Le
Le
L, TTA
Optimizer
Adam
Adam
Adam
Adam
32


Reviews:
[{"review_id": "2W9Ok9Arzy", "reviewer": "Reviewer_Db6Y", "summary": "The paper introduces Diffusion Optimization Models (DOM) to generate feasible and high-performance designs in structural topology optimization. DOM is based on diffusion model, and a key design component is trajectory alignment method, in which trajectory in diffusion-based optimization is biased towards given ground-truth optimization path, to inject knowledge on physical constraints into the model in a data-driven manner.  The paper also introduces techniques for training, such as trajectory search to find a suitable representation for matching an intermediate optimization step, new conditional objective with trajectory alignment regularization, and dense kernel relaxation to relieve inference from expensive pre-processing. Experiments in in- and out-of-distribution domains demonstrate DOM outperforms and compares competitively against state-of-the-art methods. Experiments on few-steps of sampling show DOM takes a fraction of the other method’s inference time.", "questions": "* Are small iteration steps of optimization (5/10) enough to produce optimized topologies? Can you give an intuition behind choosing the iteration steps?\n* What is the reason of having high-resolution topologies w/o constraints as dataset? (In Appendix)\n* What does SIZE in Table 3 mean? \n", "limitations": "Yes, the limitations are discussed in the main text. ", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The paper is placed in the current growing literature on deep generative models for topology optimization to reduce the iteration times in optimizing topology while fulfilling constraints associated with engineering. The exposition for the motivation is written crisply and is generally easy to follow. Additional details on backgrounds for DOM are also explained in Appendix, which helps readers get a clear picture of motivation and architecture of the proposed method. The experiments in in-distribution domains show that DOM can reduce inference time while achieving the lowest error (for some metrics). For out-of-distribution problems, DOM itself is comparable to baselines and outperforms them with additional iterative optimization method. ", "weaknesses": "For in-distribution problem, although DOM outperforms other baselines in most of the metrics, it is still unclear how much the post-processing optimization have an impact on reducing those metrics. As also reported in Table 4, although DOM+SIMP(100+10) achieves more than 3-fold improvement in CE and FM against the baseline DOM, the post-processing optimizer still has a possibility to have an impact on reducing errors. It is also concerning that it would have relatively high impact in the case of few-steps of sampling, so it is difficult to access how much DOM itself contribute to decreasing the score in the metrics. I also couldn’t find detail of the optimizer.\n\n**Minor comments**\n\n- Figure 3 is unclear. For example, how the figure named “conditioning” on the upper left is related to Figure 5? What are the dot and lines in blue?\n- Some table numbers in the main text of Chapter 4 seems to be inconsistent.\n   - Missing Table 8. Should be Table 4 (Appendix) perhaps? \n   - Missing Table 10.\n   - Missing Table 5. Should be Table 4 perhaps?\n- A sentence starting from Setup. (Line 291) should be in a distinct paragraph.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "## **Authors**: *We emphasize that we do not use post-processing for any results or visualizations in the paper except to reduce floating material in Table 4*.\n\n>Strengths:\nThe exposition for the motivation is written crisply and is generally easy to follow. Additional details on backgrounds for DOM are also explained in Appendix, which helps readers get a clear picture of motivation and architecture of the proposed method.\n\n**Authors:** Thank you for the appreciation of our work and exposition.\n\n>Weaknesses:\nFor in-distribution problem, although DOM outperforms other baselines in most of the metrics, it is still unclear how much the post-processing optimization have an impact on reducing those metrics. \n\n**Authors:** **We want to clarify that we do not employ any post-processing for the in-distribution results, showing the efficacy of just DOM without post-processing**. For all the tables and figures in the paper (specifically, Table 2 and Table 3), we utilize a standard DOM and perform sampling while conditioning on the given constraints. \nIn general, the numerical optimizer (SIMP) will improve the results when used as a post-processing step for our model and all the baselines. **For this reason, in Table 4, our goal is not to substitute the optimizer but to augment it by improving scalability and speed using diffusion models**. We only use direct optimization to demonstrate the power of combining a learning method with optimization when dealing with out-of-distribution scenarios.\n\n>It is also concerning that it would have relatively high impact in the case of few-steps of sampling, so it is difficult to access how much DOM itself contribute to decreasing the score in the metrics. I also couldn’t find detail of the optimizer.\n\n**Authors:** **We emphasize that we do not use post-processing for any results or visualization in the paper except to reduce floating material in Table 4. Hence, the large improvements you observe in the experiments (Table 2, 3, Appendix Table 3) are attributed to Trajectory Alignment**. In Appendix F (Figs 10, 11, 12, 13), we present the intermediate optimized topology using only SIMP (details for the optimization problem in Section 2). It is important to note that with a limited number of SIMP steps (without initializing with a generated topology from DOM), the optimized topologies appear coarse and lack fine details and structure.\nRegarding Table 4, our aim is not to eliminate the numerical optimizer but rather to accelerate the generation process by integrating generative models and optimizers. The numerical optimizer provides physical guarantees but can be slow and scale poorly with problem resolution. In contrast, DOM is fast in terms of preprocessing and sampling, but it lacks guarantees, especially in out-of-distribution scenarios. By combining these approaches, we gain generality and benefit from the strengths of both methods.\n\n>Figure 3 is unclear. For example, how the figure named “conditioning” on the upper left is related to Figure 5? What are the dot and lines in blue?\n\n**Authors:** The dots and lines represent the \"raw\" sparse boundary conditions, while the green arrow signifies a load. **These constraints serve as conditioning for the model to generate a feasible topology that satisfies the specified constraints**.\n\n> Questions:\nAre small iteration steps of optimization (5/10) enough to produce optimized topologies? Can you give an intuition behind choosing the iteration steps?\n\n**Authors:** **Few iterations of SIMP are not enough, especially as the problem resolution increases**. Appendix F, figures 10, 11, 12, and 13 depict the evolution of optimized topologies with an increasing number of iterations. Even after 30 iterations, many details are missing, and only the general structure is formed. After 10 iterations, we can only achieve a generic structure without any fine details. Additionally, in Appendix B.3, Table 3 demonstrates how SIMP's computational cost scales with resolution (64 -> 256). This issue is typical with numerical solvers, where each iteration step is expensive and relatively slow.\n\n\n> What is the reason of having high-resolution topologies w/o constraints as dataset? (In Appendix)\n\n**Authors:** The potential uses are many, including training an unconditional model, conducting FID computations on a held-out test set, and utilizing them for data augmentation. **It's important to mention that constraints are available for a significant portion of the dataset, comprising 660K samples**.\n\n>What does SIZE in Table 3 mean?\n\n**Authors:** The model's parameter count differs between TopoDiff-GUIDED and DOM. **Specifically, TopoDiff-GUIDED has twice the number of parameters as DOM**. This increase is due to the utilization of two additional surrogate models (regression and classification models), which must be separately trained on a labeled dataset to support the guidance mechanism.\n"}, {"review_id": "13bo7lYL1V", "reviewer": "Reviewer_q5Ex", "summary": "The paper proposes a model that combines engineering optimization with conditional diffusion generative models for constrained design generation. The main idea is to match the sampling trajectory of the diffusion model with the trajectory derived from the traditional physics-based model.", "questions": "1) In line 63, the author mentions that \"in DDPM-based models, one has to run the forward models tens/hundreds of times to obtain a suitable topology.\" Can you elaborate on what the term \"topology\" refers to in this context?\n\n2) When introducing the diffusion model, in line 145, the term $\\mu_\\theta(x_t, t)$ is referred to as a neural network without mentioning whether it is the mean of distribution p or q. It is only clear in equation 4.\n\n3) In line 213, the author mentions the \"continuous limit.\" What do you mean by the \"continuous limit\"?\n\n4) In line 231, is there a typo where the posterior is written as q(x_{t-1}|x_t, x^{\\theta})?\n\n5) It is a bit confusing in terms of the notation. In line 228, it is mentioned that $x^{\\tilde{\\theta}}(x_t, \\epsilon_t(x_t))$ is an approximation of $x_0$, but in equations 5 and 6, it seems that $x^{\\tilde{\\theta}}$ represents the inferred $x_{t-1}$.\n\n6) It is still not very clear to me how the trajectory search occurs. After obtaining $\\epsilon_{\\theta}$, how is $x^{\\tilde{\\theta}}$ found? And why can it be matched with a step that is mod((t-1), n_s)?\n\n7) If I understand correctly, one needs to store the intermediate steps of trajectory optimization. To obtain these intermediate steps, does one still need to run the full optimization? If so, this removes the justification that the model helps to eliminate the computational complexity of traditional topology optimization algorithms. If only part of the optimization is performed, does the trajectory matching only apply to the early stage of sampling?", "limitations": "They have discussed the limitation of the work and there is no potential negative societal impact.", "rating": 6, "confidence": 2, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "Strength:\nThe idea of using a physics-based trajectory to guide the sampling process of the diffusion model is very interesting.\nThe introduced kernel relaxation to replace the expensive finite element preprocessing is also interesting.", "weaknesses": "Weakness:\nThe model still needs extra steps of direct optimization. \nThe presentation of the paper could be improved, some parts are rather confusing.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "## **Authors**: *We emphasize that trajectory alignment is performed only during training and that DOM does not require extra steps of optimization*.\n\n> Strength: The idea of using a physics-based trajectory to guide the sampling process of the diffusion model is very interesting.\n\n**Authors**: thanks for acknowledging the novelty of our approach and our contribution. We are the first to leverage intermediate cheap optimization steps to improve generative design leveraging a conditional diffusion model.\n\n>Weakness: The model still needs extra steps of direct optimization. The presentation of the paper could be improved, some parts are rather confusing.\n\n**Authors**: **The model does not need extra steps of optimization, with the exception of Table 4 where our goal is to merge generative models and optimization**. We want to emphasize that we employ additional optimization steps exclusively for the out-of-distribution experiments in Table 4. For all other results in the paper (including Tables 2 and 3, figures, and the appendix), DOM does not rely on direct optimization. Instead, when showcasing samples from a few steps, DOM utilizes standard ancestral sampling. The extra steps of optimization are an optional tool for challenging situations (like out-of-distribution scenarios).\n\n>Questions:\nIn line 63, the author mentions that \"in DDPM-based models, one has to run the forward models tens/hundreds of times to obtain a suitable topology.\" Can you elaborate on what the term \"topology\" refers to in this context?\n\n**Authors**: When we mention \"topology,\" we are referring to **samples generated using a DDPM-based model designed for generative topology optimization**, such as TopoDiff. These topologies represent shapes that adhere to specific constraints provided as conditioning. \n\n>In line 213, the author mentions the \"continuous limit.\" What do you mean by the \"continuous limit\"?\n\n**Authors**: Diffusion Models (DDPM) can be trained using either a discrete number of steps T or continuous time with T approaching infinity. **In this setting (T -> inf), a DDPM can be seen as an SDE (Stochastic Differential Equation), allowing us to utilize tools from ODE (Ordinary Differential Equation) theory for sampling**. This process is analogous to NeuralODE with stochasticity, enabling efficient and effective generation of samples from the model.\n\n\n>In line 231, is there a typo where the posterior is written as q(x_{t-1}|x_t, x^{\\theta})?\n\n**Authors**: Thank you for bringing this up. **The formula is indeed correct**. In DDPM, it is possible to construct a posterior $q(x_{t-1} | x_t, x_0)$. Both the posterior and the model share the same functional form, represented by a Gaussian distribution. This allows us to utilize the posterior as an intermediate step for sampling. Given $x_t$ and the model $\\epsilon_{\\theta}$, we can create a learned approximation for $x_0$ and sample $x_{t-1}$ from the model. This approach facilitates the sampling process within DDPM and contributes to its efficiency and effectiveness.\n\n\n>It is still not very clear to me how the trajectory search occurs.\n\n**Authors**: In a standard DDPM, $\\tilde{x}^{\\theta}$ serves as an approximation for $x_0$ and is used for sampling the model. In DOM, we leverage this representation for trajectory alignment. **During trajectory search, we create a distribution around $x^{\\theta}$ and then match this distribution with a specific intermediate optimization step**. This process facilitates trajectory alignment, which is a crucial aspect of the DOM approach. **Please see also the algorithms in Appendix D**.\n\n>If I understand correctly, one needs to store the intermediate steps of trajectory optimization. To obtain these intermediate steps, does one still need to run the full optimization? \n\n**Authors**: The trajectory optimization regularization occurs exclusively during training. **At inference time, the model functions as a standard conditional diffusion model, eliminating the need for any intermediate steps or additional storage**.\nRemember that during the model's training, final optimized topologies are used. Obtaining these topologies necessitates running the complete optimization process, and the intermediate steps are gathered as a natural byproduct. **As a result, there is no extra effort or computational cost associated with collecting these intermediate steps during the training process**. See also the general response on top.\n\n>If so, this removes the justification that the model helps to eliminate the computational complexity of traditional topology optimization algorithms. If only part of the optimization is performed, does the trajectory matching only apply to the early stage of sampling?\n\n**Authors**: **Trajectory alignment regularization is exclusively applied during training**. During inference, we sample from a standard conditional diffusion model, conditioning on a given set of constraints. The strength of our approach lies in the ability to guide the sampling trajectory toward the optimization trajectory, effectively grounding the sampling process in the optimization path. This is achieved without the need for additional Finite Element Method (FEM) pre-processing, guidance models, or data augmentation, making the technique efficient and self-contained."}, {"review_id": "OYp1M2t4uw", "reviewer": "Reviewer_hnNz", "summary": "The paper presents a diffusion-based method for topology optimization. The method seems to base on TopoDiff, but uses trajectory alignment instead of guidance to produce plausible results even under few-step sampling.", "questions": "- Is the kernel relaxation a novel contribution, or is this adapted from another paper?\n- The frame alignment between t and s is an odd choice-- why use a mod? Wouldn't that mean there could be a sudden flip from optimizer end state to starting state in the middle of the diffusion process?\n", "limitations": "The paper does mention the poor OOD generalization as a limitation which is appreciated. But there's probably a bit of a larger point there, it seems both DOM and even the TopoDiff variants studied here do require a training set very close to the test scenarios, which is a fundamental problem if the goal is to replace classical solvers with learned ones.", "rating": 5, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "Diffusion models seem like a natural fit for solving inverse problems, but it's still open what is the most efficient and practical way to do so. This paper explores trajectory alignment as an alternative to guidance.", "weaknesses": "The paper doesn't read very well, especially section 3. It's clearly meant as an extension to TopoDiff, and it's often hard to figure out which aspect of it is novel, and what is previous work. Figure 1, 2 don't really help for understanding the method. The description of kernel relaxation (both in the main text and appendix) is very hard to understand without any background in topology optimization. I assume the 'simulation trajectory' comes from running the simulator S (which I assume is SIMP?), but this doesn't seem to be described anywhere. In general, the paper assumes a lot of domain knowledge, and doesn't do enough to explain its relevance to the ML community. While there's plenty of room to study the intersection of engineering problems and ML at NeurIPS, the key interest of the community is ML methods which can be applied widely-- this is a bit of a missed opportunity. Finally, related work absolutely needs to be part of the main text and can't be moved into the appendix.\n\nI'm not a domain expert in topology optimization, and It's a bit hard for me to judge the relevance of the results. The main point of comparison seems to be TopoDiff; visually, the results in the appendix seem very similar, and scores in Tab. 2 are very similar as well. OOD generalization without SIMP seems even worse than TopoDiff. The post-processing with a few steps of SIMP seems a bit sketchy-- you could presumably do this with the output of other models as well, and the whole point of the paper is to avoid running SIMP. Relatedly, it would be great to actually compare performance against a GT solver like SIMP, as the method the paper wants to replace.\nThe main benefit of DOM seems to be inference speed as it can produce reasonable designs after very few diffusion steps. If this is the case though, the paper needs to make a more concerted argument and comparisons about inference time, and why that matters. It would e.g. be great to see a line plot with x axis \"wall clock inference time\" and y axis \"score\" for both DOM and Topodiff. Fig 6 is interesting, but 2 steps is of course pretty extreme, in this regime a diffusion model isn't that different from a deterministic feed-forward UNet predictor. Which is also something that would have worth comparing to-- if TA with a few steps works, can you just directly predict the optimized solution from the conditioning?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "## **Authors**: *Topology Optimization is a hard problem without analogy in the machine learning domain. Our goal is not to replace classical solvers but to augment them.*\n\n>Strengths:\nDiffusion models seem like a natural fit for solving inverse problems, but it's still open to what is the most efficient and practical way to do so. This paper explores trajectory alignment as an alternative to guidance.\n\n**Authors**: Thanks for correctly assessing the core goal of our work.\n\n>Weaknesses:\nIt's clearly meant as an extension to TopoDiff, and it's often hard to figure out which aspect of it is novel, and what is previous work. \n\n**Authors**: **DOM is not an extension but a new paradigm**. TopoDiff+G is SOTA for Topology Optimization. However, TopoDiff uses expensive FEM preprocessing that scales poorly with resolution and relies on surrogate models, labeled data, and many sampling steps. Instead, DOM leverages Trajectory Alignment during training, demonstrating that a conditional diffusion model can generalize without relying on FEM preprocessing, guidance models, large labeled datasets, and extensive sampling steps.\n\n>The description of kernel relaxation (both in the main text and appendix) is very hard to understand without any background in topology optimization. I assume the 'simulation trajectory' comes from running the simulator S (which I assume is SIMP?), but this doesn't seem to be described anywhere. \n\n**Authors**: We provide a detailed description of the TO problem, including the optimizer (S is SIMP) and the formulation in Section 2 (Background - The Topology Optimization Problem). We will enhance clarity regarding kernel relaxation.\n\n>I'm not a domain expert in topology optimization, and It's a bit hard for me to judge the relevance of the results. \n\n**Authors**: **Our results are novel and relevant to the engineering design field. We provided the first example of a diffusion model leveraging optimization trajectories to improve generative design**. We are the first to show that such an approach is effective and performs well.\nThe significance of our findings can be summarized in three key aspects:\n* DOM enables sampling high-quality topologies in a few steps, making it widely applicable.\n* TA serves as a powerful regularization tool, utilizing intermediate optimization steps for better results —a novel insight in this field.\n* Our method avoids the need for auxiliary models, labeled data, or costly pre-processing, making it suitable for large-scale generation and relevant for various vision-based problems in generative design.\n\n>OOD generalization without SIMP seems even worse than TopoDiff. The post-processing with a few steps of SIMP seems a bit sketchy-- you could presumably do this with the output of other models as well, and the whole point of the paper is to avoid running SIMP. \n\n**Authors**: **The goal of this paper is to provide a scalable, fast, and effective diffusion model for TO. Our goal is not to substitute but augment numerical optimization**. Achieving OOD generalization with a pure learning-based method is challenging. TopoDiff has an advantage in this regard since it utilizes FEM conditioning during pre-processing and auxiliary models for guidance. This allows the FEM solver to dynamically adapt the conditioning information for each new constraint configuration.\n\n>Relatedly, it would be great to actually compare performance against a GT solver like SIMP, as the method the paper wants to replace. \n\n**Authors**: We do. The Compliance Error (CE) is computed with the SIMP performance as ground truth for all the models. $CE(x) = Compliance_{model}(x) - Compliance_{SIMP}(x) / Compliance_{SIMP}(x)$. Compliance is a standard performance metric for TO.\n\n> The main benefit of DOM seems to be inference speed as it can produce reasonable designs after very few diffusion steps. If this is the case though, the paper needs to make a more concerted argument and comparisons about inference time, and why that matters.\n\n**Authors**: See App B (Table 1,3) for details. **We specifically emphasize the inference and sampling time vs steps (2, 5, 10) using the same hardware**. This analysis clearly shows the efficiency of our approach.\n\n> if TA with a few steps works, can you just directly predict the optimized solution from the conditioning?\n\n**Authors**: **No, it is not possible to directly predict the topology**. DOM is a generative model, trained using 1000 steps. During inference, we skip steps. There have been many attempts to reconstruct/regress the topology, but results have been poor in terms of engineering performance. The challenge in generative design is balancing generative and engineering metrics. DOM excels by performing on par with the SOTA in generative metrics while rapidly generating high-performance topology.\n\n>Questions:\nIs the kernel relaxation a novel contribution, or is this adapted from another paper?\n\n**Authors**: **Yes, the kernel relaxation that we propose is a novel contribution**. We improve and simplify the conditioning proposed in TopoDiff-FF. While TopoDiff-FF presents a learnable kernel relaxation using an unnormalized kernel, we further develop this concept by proposing a normalized kernel without any learnable components. See Listings 1.\n\n>But there's probably a bit of a larger point there, it seems both DOM and even the TopoDiff variants studied here do require a training set very close to the test scenarios, which is a fundamental problem if the goal is to replace classical solvers with learned ones.\n\n**Authors**: **Topology Optimization is a hard problem. Our goal is not to replace but augment solvers**. The in-distro test set utilizes the same boundary conditions but different loading and volume fractions. This is a challenging testing scenario in engineering design. **For instance, in this scenario, GANs generate ~50% unfeasible topologies due to load disrespect and floating material issues, making the designs impossible to manufacture**."}, {"review_id": "HzZ16hVmQM", "reviewer": "Reviewer_dj1b", "summary": "This work studies the use of diffusion models for constrained topology design, a task where physics-based optimization methods still outperform deep learning approaches. The authors introduce diffusion optimization models (DOMs), a diffusion-based approach to generate topology candidates rapidly and efficiently. The main contribution is \"trajectory alignment\" (TA), which attempts to \"align\" the intermediate expected predictions (through Tweedie's formula) of the noise/score estimator $\\epsilon_\\theta(x_t, c)$ with the intermediate topologies of the physics-aware optimization process. The authors evaluate DOM on a dataset of optimized topologies with the SIMP solver, and demonstrate improvements over previous data-driven methods, both in terms of quality and efficiency.\n\n**Edit** Following the rebuttal/discussion period and in light of the changes proposed by the authors, I upgrade my score from 3 to 6. ", "questions": "* At line 315, \"(i) In-distribution constraints. The constrainsts in this test set are the same as those of the training set.\" Does this means that the training and test sets are the same?\n* Why are 1-step generation models such as GANs or consistency models slower than 100-steps models?\n* When varying the number of sampling steps (e.g. Table 3), are the models trained for that specific number of steps or is there a gap between training and evaluation? If the latter how is that gap handeled?\n* What is the proportion of the inference time allocated to pre-processing?\n* At line 243, is the $s(t-1)$ formula correct? $s(t) = n_s - \\lfloor n_s \\frac{t}{T} \\rfloor$ seems to match the footnote better.", "limitations": "See weaknesses.", "rating": 6, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 3, "strengths": "* The subject of generative modeling under constraints is interesting.\n* The idea of \"aligning\" the sampling process of a diffusion model with another process is interesting and worth studying.\n* The proposed method demonstrates improvements over previous methods for a dataset of optimized topologies.", "weaknesses": "* The proposed method is not mathematically sound. The modified training objective ($\\mathcal{L}_{DOM}$) does not lead to a proper diffusion model in the general case, meaning that simulating the reverse process has no guarantees to generate samples from the target distribution. Additionally, aligning trajectories with fundamentally different time axes (optimization steps against perturbation steps) seems dubious and is not formally motivated by the authors.\n\n* I suspect that the proposed method works for the application only because the mapping \"constraints -> optimal topology\" is (almost) deterministic, leading to a regressive task instead of a generative task. Therefore, the diffusion model learns to transform all noise to the same topology, given the constraints, which explains the very low number of sampling steps. The effectiveness of the method cannot be correctly assessed without a proper generative task. An example of such task would be to generate topologies without conditioning on the constraints, that is sampling from $p(x) = \\int p(x, c) ~ dc$ instead of $p(x | c)$. Because the \"optimization trajectory\" is still available, the proposed method should be applicable. Furthermore, if the application is a generative task, the authors should generate several topologies for each set of constraints (both with DOM and SIMP).\n\n* The link drawn with consistency models (lines 205 and 335) is hazardous. First, consistency models do not \"predict their input at each step\" as the authors write (line 334) but are forced to be consistent along trajectories, such that they eventually define a mapping \"noise -> data\". Second, the proposed trajectory alignment pushes all trajectories to be the same, given the constraints, which is vastly different and in no way a \"generalization\".\n\n* At line 89, the authors claim that DOM \"respect constraints\". I beleive that neither trajectory alignment nor dense kernel relaxation guarantee the satisfication of constraints. Only the \"few-steps direct optimization\", which is not part of the diffusion procedure, could guarantee to respect constraints.\n\n* The manuscript has many vague or confusing statements and concepts, which are introduced without definition or context. Follows a non-exhaustive list of such statements: design process, line 4; iterative optimization, line 5; optimization trajectory, line 10; firmly grounded in the underlying physical principles, line 12; two steps, line 13; sample tens or hundreds of layers for each sample, 168; physics-based information, line 192; multi-fidelity mechanisms, line 201; close to the corresponding optimization step in terms of distance, line 204; among those that can be represented by the reverse process, line 217; lines 219-220; transport, line 232; suitable representation, line 232; some mapping, line 234; optimize this loss for the mean values, freeze the mean representations and optimize the variances in a separate step, line 253; volume fraction, line 309; compliance error, line 312. Additionaly, some part of the manuscript are verbose and redundant, notably lines 25-28, the trajectory alignement section, and the lines 234-237. \n\n* There is an overwhelming amount of details related to the application (topology optimization) that distracts the reader from the methodological contribution.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "## **Authors**: *We would like to clarify that DOM is a well-defined conditional diffusion model. TA is used only during training. At inference time, we sample DOM using standard ancestral sampling and the model does not have access to any intermediate optimization step. See the attached PDF for examples*.\n\n> Strengths:\nThe subject of generative modeling under constraints is interesting.\nThe idea of \"aligning\" the sampling process of a diffusion model with another process is interesting and worth studying.\n\n**Authors**: Thanks for mentioning such aspects of our work. The connection between generative models and engineering design is deeply underexplored and we think that trajectory alignment is important for both fields.\n\n>Weaknesses:\nThe modified training objective does not lead to a proper diffusion model in the general case, meaning that simulating the reverse process has no guarantees to generate samples from the target distribution. \n\n**Authors**: **DOM is a well-defined conditional diffusion model, trained using a reweighted lower bound to the likelihood**. \n\n* The epsilon parameterization (for each step $t$) is: $w_t || \\epsilon_{\\theta}(x_t) - \\epsilon  ||^2_2$. If $w_t=1$, we reweight the bound giving more importance to large $t$, focusing on global structure (see Alg. 1 in arxiv/2006.11239).\n\n* We train the model variance following IDDPM (Eq.16 arxiv/2102.09672), where at each iteration the model mean is trained using the epsilon-parameterization, then the mean is frozen and the variance is optimized using the ELBO. (lines 247-253) \n\n* The regularization term introduced by TA in Eq.6 is always non-negative and is added to the loss (Eq.7) or equivalently subtracted from the lower bound, providing still a proper objective for DDPM. Notice that we choose the same variance for the distribution over $\\tilde x^{\\theta}$ and $x^{opt}_{s}$, so TA does not act on the model variance but only on the model mean. Based on these points and the samples, it should be clear that DOM is generative and TA is a regularization mechanism that does not collapse the model to a deterministic mapping.\n\n* **Trajectory alignment is used only during training**. At inference time, we sample DOM using ancestral sampling without any intermediate optimization step. We have undertaken rigorous empirical validation of our model (in-distribution, out-of-distribution, inference efficiency, scalability, generative and engineering metrics).\n\n>The effectiveness of the method cannot be correctly assessed without a proper generative task.\n\n**Authors**: **The provided samples (see PDF and general response) were generated by DOM trained w/ TA and sampled with standard ancestral sampling**. We have also evaluated various generative metrics in Appx B.4 to support our claims. With these samples, we demonstrate that our model generates a diverse range of samples while adhering to the given constraints (10 samples in each row).\n\n>Because the \"optimization trajectory\" is still available, the proposed method should be applicable.\n\n**Authors**: **This assumption is wrong. During inference, we do not have intermediate optimization steps for the test set**. Our sampling process (at inference time) relies only on conditioning the constraints and the kernels.\n*We understand that the nomenclature (\"sampling trajectory\") can be confusing and we will be glad to clarify this point in the final version of the paper*.\n\n>The link drawn with consistency models (lines 205 and 335) is hazardous.\n\n**Authors**: Thank you for bringing up this point. **We agree that DOM and CM are different approaches. Our intention is to provide a parallel to build intuition**. Our method, trajectory alignment, does not aim to force all trajectories to be identical. Instead, we strive to bring them as close as possible to each other while learning a generative model for data distribution. \n\n>At line 89, the authors claim that DOM \"respect constraints\".\n\n**Authors**: **DOM w/ TA is a robust and effective model for topology optimization**. While numerical solvers provide guarantees, through empirical evaluation our model consistently generates designs that largely adhere to the imposed constraints while achieving high levels of performance.\n\n> Questions: Does this means that the training and test sets are the same?\n\n**Authors**: **Training and test conditioning configurations are different**. The in-distro scenario poses a challenging generalization task because the TO problem is highly non-linear. SOTA GANs generate >50% of samples that do not fulfill the constraints in the in-distro scenario. In the in-distro test set, boundary conditions match those from the training set, but loadings and volume fractions differ. This is standard evaluation practice for generative models (as proposed in TopoDiff).\n\n\n> Why are 1-step generation models such as GANs or consistency models slower than 100-steps models?\n\n**Authors**: **GANs are slower because of the field computations during pre-processing. Without pre-processing, GANs perform extremely poorly for topology optimization**.\nIn Table 2, we sample consistency models with 100 steps to have a fair evaluation. We also show that CM underperforms compared to DOM in appx Table 2.\n\n> When varying the number of sampling steps (e.g. Table 3), are the models trained for that specific number of steps or is there a gap between training and evaluation? If the latter how is that gap handeled?\n\n**Authors**: **No, we train one model with T=1000 and then vary the sampling steps at sampling time**. We use the IDDPM skipping approach (w/o using DDIM).\n\n> What is the proportion of the inference time allocated to pre-processing?\n\n**Authors**: **For TopoDiff, >50% for 64x64 res, and >70% for 256x256 res. For DOM is <1%**. In appx Table 3 we show this proportion, showing the advantage of our method. Pre-processing is time-consuming and problem-specific, whereas our approach is cheap and applicable to a general set of vision-based problems."}]

Author rebuttal:
We would like to thank the reviewers for the feedback. After reading the reviews, we feel to clarify three raised issues:

- Q1) Is DOM a generative model? 

#### **Yes, DOM is a conditional diffusion model trained with the standard epsilon parameterization and a regularization term (TA). The attached PDF showcases its generative capabilities using conditional samples**. We show 12 different constraint configurations (rows) and 10 samples (columns) for each constraint.


- Q2) Is TA used during inference? 

#### **No, TA is a training regularization technique. Inference sampling uses the standard ancestral sampling with DOM as its foundation. All results and visualizations use this standard approach**.


- Q3) Are our findings significant for engineering design and machine learning? 

#### **Our results are extremely significant for design. We pioneered a scalable method integrating optimization steps in a diffusion model for deep generative design**. Specifically, DOM's trajectory alignment helps **improve performance by > 25%** on in-distribution scenarios compared to SOTA (Table 2) and at the same time **reduces inference time by 50 %** for small resolution, **80% for large resolution** (Appx - Table 3), and more than **90% in the few sampling-step regimes** (Table 3). Our approach speeds up sampling and shows the efficacy of broadening the impact of diffusion models for topology optimization and other vision-centric engineering design problems. This directly impacts many industries doing structural optimization. It removes reliance on guidance, large labeled datasets for constraints, and costly FEM pre-processing. 

--------
We also want to clarify our contributions: 

* **Novelty and relevance to the community**. DOM w/ TA is the first work to showcase that intermediate optimization steps can be used to regularize the sampling trajectories during training without relying on auxiliary models, or expensive pre-processing. This is an important contribution to the community.
* **Merging Optimization with Diffusion Models**. Contrary to other works, that try to substitute the optimization solvers and leverage only the generative model (like TopoDiff-GUIDED or TopologyGAN), we show how merging the two is a powerful approach for OOD scenarios, **reducing un-manufacturable topology by 3 folds**. Numerical solvers leverage decades of research in physics in engineering but are limited by scalability and get stuck in local minima. Diffusion Models are scalable and can help to warm-start the solver. We hope these points clarify the significance of the contributions.
* **Dataset**. We release the **largest publicly available dataset** on topology optimization and intermediate steps (see appendix F) with the paper, which can help many new researchers in generative models and generative design.

----------------

Here we extend our response, emphasizing a critical aspect of the inference process for DOM. We apologize for the confusing terminology (optimization trajectories and sampling trajectories) and we will clarify this point in the final paper.

* **Sampling at inference time with DOM**. **During sampling at inference time, DOM operates as a standard conditional diffusion model**. This means that when generating new samples, DOM adheres to the typical conditional sampling procedure (ancestral sampling in DDPM), where it takes the given constraints into account and generates samples accordingly. You can see our implementation in `scripts/sample_kerner_relaxation.py`.

* **Trajectory Alignment (TA) is solely confined to the training phase of DOM**. TA is a regularization technique employed during training to ensure that the sampling path aligns well with the observed data distribution. This alignment improves the model's ability to generate realistic and meaningful samples while adhering to the provided constraints.

* **DOM with Trajectory Alignment (TA) refers to DOM that has been "trained" with TA (see appendix algorithm 1)**. During inference, we employ standard ancestral sampling for DOM, using constraints and kernels (refer to algorithm 2 in arxiv/2006.11239). The alignment of "sampling" and "optimization" trajectories acts as training regularization, detailed in section 3 with a regularized loss for TA training. This sync harnesses DDPM's hierarchical structure and stepwise marginalization through $q(x_t | x_0)$, steering training samples towards the optimization trajectory. Whenever we mention DOM w/ or w/o TA, we are referring to its training process. This distinction will be highlighted in our final paper.

* **We do not utilize the optimization trajectories at any point during evaluation**. All results, including experiments, figures, tables, and models, are validated using the same standard sampling procedure. The test sets are the same as those provided by TopoDiff (arxiv/2208.09591) for fair assessment. When we refer to DOM with TA, we specifically mean DOM trained with TA, which is a regularization mechanism applied solely during training. 

* To reduce the number of sampling steps, we utilize the approach and code from improved DDPM (section 4 - arxiv/2102.09672), without leveraging DDIM. This method enables us to effectively skip steps in diffusion models. We maintain consistency by using the same model for sampling with different numbers of steps (e.g., 2, 5, 10, 100, etc.), ensuring a fair and unbiased comparison across various sampling scenarios.

Regarding the selection of the step in DOM with TA, we apologize for any confusion caused. You are correct in pointing out the mismatch with the footnote: this is a residual of an idea (cyclic trajectory matching) that we explored but ultimately did not utilize. Instead, we employ the function $s(t) = \text{mod}(T, t) \times n_s / T$ and then select the closest integer, which aligns with your suggestion. We will rectify this mismatch in the final version of our work. Thank you for bringing this to our attention.

Meta-review: ### Summary 

The paper introduces Diffusion Optimization Models (DOM) to address challenges in constrained design generation, combining the power of generative models with engineering optimization principles. DOM is built upon diffusion models and employs an essential technique called Trajectory Alignment (TA) to align the sampling trajectory of diffusion models with the trajectory derived from traditional physics-based methods. This alignment ensures that the generative process remains grounded in the underlying physical principles, making it suitable for environments with limited data and a need for high precision. The paper introduces several techniques to train DOM effectively, including trajectory search to identify a suitable representation for matching an intermediate optimization step, a conditional objective with trajectory alignment regularization, and dense kernel relaxation to expedite inference without expensive preprocessing. Experimental results are on both in-distribution and out-of-distribution configurations demonstrate that DOM outperforms state-of-the-art deep generative models by 25% in terms of performance and reduces inference computational cost by half. The DOM can efficiently combine learning and optimization trajectories, generating high-quality designs in just a few steps. This approach significantly improves engineering performance and inference efficiency, paving the way for the widespread application of generative models in data-driven design processes.

### Decision 
At the end of the rebuttal the reviewers were mostly positive about this paper. The experiments in this paper are encouraging and the problem that is being studied is interesting. Majority of the reviewers suggested this paper either for accept or borderline accept. The authors have done a really good job addressing the concerns raised by the reviewers. The reviewers have raised some important concerns, in particular on clarity of the paper, I would recommend reviewers to address those concerns in the camera-ready version of the paper.
Decision: Accept (poster)