Feature-Mapping Topology Optimization with Neural Heaviside Signed Distance Functions
Authors: Aleksandr Kolomeitsev
ANH-HUY PHAN
Venue: ICML 2025 poster (2025)
Keywords: Neural Heaviside Signed Distance Functions
Autoencoder Architectures
Topology Optimization
Primary area: applications->everything_else
TL;DR: Feature-Mapping Topology Optimization with Neural Heaviside Signed Distance Functions

Abstract: Topology optimization plays a crucial role in designing efficient and manufacturable structures. Traditional methods often yield free-form voids that, although providing design flexibility, introduce significant manufacturing challenges and require extensive post-processing. Conversely, feature-mapping topology optimization reduces post-processing efforts by constructing topologies using predefined geometric features. Nevertheless, existing approaches are significantly constrained by the limited set of geometric features available, the variety of parameters that each type of geometric feature can possess, and the necessity of employing differentiable signed distance functions. In this paper, we present a novel method that combines Neural Heaviside Signed Distance Functions (Heaviside SDFs) with structured latent shape representations to generate manufacturable voids directly within the optimization framework. Our architecture incorporates encoder and decoder networks to effectively approximate the Heaviside function and facilitate optimization within a unified latent space, thus addressing the feature diversity limitations of current feature-mapping techniques. Experimental results validate the effectiveness of our approach in balancing structural compliance, offering a new pathway to CAD-integrated design with minimal human intervention.

Accelerating the development of manufacturable structures with minimal human input is a significant concern in the Feature-Mapping Topology Optimization with Neural Heaviside Signed Distance Functions
Aleksandr Kolomeitsev
Anh-Huy Phan
Abstract
design industry. While tools like Computer-Aided Design (CAD) have reduced early design errors and expedited the creation of documentation, there remains need for en- hanced automation, particularly in design decision-making.
Topology optimization plays a crucial role in de- signing efficient and manufacturable structures. Traditional methods often yield free-form voids that, although providing design flexibility, intro- duce significant manufacturing challenges and require extensive post-processing. Conversely, feature-mapping topology optimization reduces post-processing efforts by constructing topolo- gies using predefined geometric features. Never- theless, existing approaches are significantly con- strained by the limited set of geometric features available, the variety of parameters that each type of geometric feature possess, and the neces- sity of employing differentiable signed distance functions. In this paper, we present a novel method that combines Neural Heaviside Signed Distance Functions (Heaviside SDFs) with struc- tured latent shape representations to generate manufacturable voids directly within the opti- mization framework. Our architecture incorpo- rates encoder and decoder networks to effectively approximate the Heaviside function and facili- tate optimization within a unified latent space, thus addressing the feature diversity limitations of current feature-mapping techniques. Experi- mental results validate the effectiveness of our approach in balancing structural compliance, of- fering new pathway to CAD--integrated design with minimal human intervention.
Component design relies on meeting strength characteristic requirements and manufacturabilitt criteria, meaning that components must be tailored specific, predefined manu- facturing methods.
Topological optimization allocates material within de- sign space to meet strength or thermal requirements. How- ever, it encounters challenges, including complex geome- tries that necessitate advanced manufacturing techniques and extensive post-processing, which can diminish the time advantages gained from minimal engineer involvement in the topology optimization process.
Recent advancements have addressed manufacturabiilty concerns in topology optimization. In additive manufactur- ing, manufactuuabilittyfouuuu approaches, such as stress- mininization topology optimization, tackle practical issues by incorporating connectivity constraints into the optimiza- tion process (Chao et al., 2021). Additionally, projection- based algorithms enforce nozzle size restrictions in mate- rial extrusion basod additive manufacturing, ensuring com- pliance with production constraints (Carstensen, 2020).
Innovative computational techniques, including performance-awaa diffusion models, surpass GAN-based methods in manufacturabiittyawaaa topology optimization by integrating constraint guidance into the optimization process (Mazé & Ahmed, 2023). Furthermore, machine learning approaches automate the adjustment of geometric constraints and optimize building orientations for addi- tive manufacturing, thereby enhancing the feasibility of topology-optimized structures (Mohseni & Khodaygan, 2024). Recent studies, such as (Zehnder al., 2021), indicate that neural topology optimization with implicit representations and self-supervised learning match the performance of state-of-the-ar mesh-based solvers.
1. Introduction
Artificial Intelligence Center, Laboratory of Intelligent Signal and Image Processing, Skolkovo Institute of Science and Technol- ogy, Moscow, Russia. Correspondence to: Aleksandr Kolomeit- sev balkkanndrkllomiitss. @kootecc.rus.
Multi-scale design methods further enhance manufactura- bility by bridging the gap between macro- and micro-level design features, facilitating manufacturable topology opti- mization while reducing computational costs (Groen, 2018;
Proceedings of the 42nd International Conference Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Feature-Mappinn Topology Optimization with Neural Heaviside Signed Distance Functions
Feature Contour
SDF Surface
Heaviside Surface
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Figure 1. Example of SDF and Heaviside function for quadrangle geometric feature
Xia & Breitkopf, 2015; Zhu et al., 2017). Additionally, filtering ensure
In response to these challenges, we propose leveraging an variational autoencoder-based model to approximate the Heaviside function, thereby providing a unified represen- tation for diverse geometric features within a single la- tent space. We illustrate this approach using examples of ellipses, triangles, and quadrilaterals. Furthermore, we demonstrate how the decoder of the trained model can be utilized for topology optimization by treating the la- tent representation of geometric feature as its parameters. The code and models publicly available https: g i thub. com A eexanderriii 0970119972222 200770222/SSSD--O-
techniques like cdlvetion-diiftsion--eee machinable designs by applying efficient computational fil- ters to unstructured meshes (Höghøj & Träff, 2022).
Feature-based modeling incorporates geometric primitives to produce CAD-compatible designs during topology opti- mization, thereby facilitating direct manufacturability (Liu & Ma, 2015). Moreover, constructive solid geometry (CSG) approaches optimize structural boundaries using implicit representations, enhancing scalability and control over complex geometries (Mei et al., 2008).
Thus, topology optimization is increasingly transitioning from a conceptual tool to a practical enabler of manufac- turable designs that align with real-world constraints.
2. Related Work
Topology optimization involves the optimization of mate- rial distribution within design domain to achieve maxi- structural performance under specified constraints. has been applied across various fields, including structural engineering (Izumi et al., 2024), fluid dynamics (Li et al., 2023), and thermal systems (Yu et al., 2020).
Building this potential, our research aims integrate advanced optimization techniques with machine learning paradigms to overcome the inherent limitations of current meapure-mapping methods in topological optimization. By addressing these challenges, we strive to enhance the scala- bility, llcxibility, and overall effectiveness of topology opti- mization processes, ultimately leading the development of more efficient and manufacturable structural designs.
SIMP. One the most widely utilized methods in topol- ogy optimization the Solid Isotropic Material with Penal- ization (SIMP) method (Bendsøe Kikuchi, 1988), which employs continuous density variable (or pseudo-density) to interpolate material properties between solid and void states, where values close to zero correspond to the void state and values near one represent the solid state. Typi- cally, each pseudo-density variable is associated with a spe- cific element within the finite element method (FEM) dis- cretized domain (see Appendix B). The objective function is commonly formulated to minimize compliance while ad- hering volume fraction constraint. Two of the most prevalent approaches for solving the optimization problem formulated in the SIMP framework are the Optimality Cri- teria (OC) method and the Method of Moving Asymptotes (MMA), both of which utilize adjoint differentiation for
To generate manufacturable solutions through topology op- timization that replicate those created by engineers using CAD, geometric features must consist of closed chains of linear and are segments connected tangentially, without re- stricting the number elements in the chain. However, ex- isting feature-mappi mapping methods cannot accommodate such geometric representations. Although it is possible to plicitly express the Signed Distance Function (SDF) (see Fig. 1) within these methods, calculating gradients at seg- ment contacts presents significant challenges. Moreover, these methods lack the ability to seamlessly transform one type of geometric feature into another.
2

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Feature-Mapping Topolegy Optimization with Neural Hcaviside Signed Distance Functions
sensitivity analysis (Bendsøe & Sigmund, 2004).
2020) introduces Neural Distance Fields to represent dis- tance fields for non-closed surfaces, while (Jiang al., 2020) employs neural networks to represent SDFs with dif- ferentiability for solving inverse problems such as 3D re- construction.
Feature Mapping Methods. Feature mapping methods op- timize the sizes and positions of geometric features within the design domain to ensure that FEM elements inside the feature are designated as either material (Norato et al., 2015) or voids (Saxena, 2010; Zhang et al., 2017a; Wein al., 2020). Various studies have represented geometric features using hyperellipses (Guo et al., 2014; Zhang al., 2017b; Sharma et al., 2017; Norato, 2018), 8), B B-splines (Lee & Kwak, 2008; Lee et al., 2007: Kim et al., 2008), and Bézier curves (Wang & Yang, 2009). In a recent work (Padhy et al., 2025), the authors introduce polygon-based geometric features constructed by combining half-spaces.
3. Backkrrund
The table of all notations used in this paper is shown in Appendix A.
3.1. FEM Discretization for 2D Elasticity Problem
The finite element method (FEM) is employed to discretize the design domain for the 2D elasticity problem by dividing it into mesh of finite elements, typically bilinear quadri- laterals or triangular elements. Each element is associated with a set of nodal degrees of freedom corresponding to displacements the x and y directions. The equilibrium equations for the elasticity problem are derived using the principle of virtual work, leading to the following system of linear equations
To achieve a smooth transition of pscudo-density the feature boundaries, the Heaviside Signed Distance Func- tion (SDF) is utilized. The SDF value for point is de- fined the distance to the feature boundary, with a pos- itive sign if the point resides inside the feature and a negative sign otherwise. This SDF value is transformed into a pscudo-density value through a Heaviside function, which can be implemented using hyperbolic tangent func- tion (Wein & Stingl, 2018), polynomials (Zhang et al., 2016; Dunning, 2018), trigonometric functions (Norato et al., 2015). Many feature mapping methods incorporate various smooth combination functions to ensure continu- ous derivatives with respect to high-level parameters, such as the Tenissllmeier-Steihhauuee function (Shapiro, 2002) or smooth R-functions (Chen et al., 2007).
Ku=f
where K is the global stiffness matrix, u is the displace- ment vector, and is the external force vector. The stiffness matrix K is assembled from the elemental stiffness matri- ces, which are functions of the material properties and the geometry of each element. As example, the assembling of the stiffness matrix is shown in Appendix B.
By maintaining smoothness throughout the computational process SDF-based methods can leverage range of gradient-based optimization techniques (Zhang & Norato, 2018). In these approaches, derivatives with respect to parameters are calculated using the chain rule, analogous to the techniques used in SIMP with the adjoint method (Tröltzsch, 2010). It is noteworthy that some proposed methods employ a non-differenniaabl maximum function, necessitating the use of stochastic optimization methods such as genetic algorithms (Wang & Yang, 2009), evolu- tionary strategies (Bujny et al., 2017), Bayesian optimiza- tion (Sharpe et al., 2018), and support vector regression (Lei et al., 2018) as alternative solutions.
3.2. SIMP Method Problem Formulation
The Solid Isotropic Material with Penalization (SIMP) method is widely used approach in topology optimization for determining the optimal material distribution within a design domain. The primary objective is to minimize the compliance C' of the structure, defined as the work done by the external forces
min C(p)= = uf
(2)
subject to the equilibrium equation
K(p)u=
(3)
Neural Network Approximation of the Heaviside SDF. Neural networks have been employed approximate the Heaviside function for boundary transitions in paramet- ric level-set methods for topology optimization (Deng & To, 2021), Beyond topology optimization, neural net- works are utilized to approximate SDFs in various appli- cations. A prominent example is DeepSDF (Park et al., 2019), which encodes 3D shapes and represents contin- uous signed distance fields. MetaSDF (Sitzmann et al., 2020) enhances SDF representation by leveraging previ- ously learned knowledge. Additionally, (Chibane et al.,
and a volume constraint
eePe S V
Vimax
(4)
where Pr E [Pmin; 1] represents the pseudo-density variable for element e, and Ue is the volume of element e. N is the total number of elements, and Vmax is the maximum allow- able volume, Pmin 2 0 is the minimum allowable pseudo- density. The stiffness matrix K K(p) is interpolated using the
3

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Feature-Mapping Topology Optimization with Neural Heaviside Signed Distance Functions
SDF and Heaviside Function. For a specified geometric feature m define subset of parameters 8m, the signed distance function (SDF) is defined as
X
Shape Encoder
X- --Dooder
a(g. (5m)) sdf (Sm 6) - d(E, (5((s))
e (smm)
NNene
NN₂
(7)
otherwise
Heaviside
where d(c, ())(sm) represents the distance from point & to the boundary of feature m. (2mm) the domain occupied by the geometric feature, parameterized by 8m-
Decoder
NN NNttowviidee
The motivation for using the Heaviside function is to scale the distance function to the range [0, 1], which is conve- nient for the optimization process and for training the neu- ral network. We approximate the Heaviside function using a sigmoid function
Figure 2. Neural Heaviside SDF Model
SIMP penalization scheme
N K(p) Σ Ke
(5)
H H8m, 5)
(8)
1 P (#m
where Kg is the element stiffness matrix of the solid ma- terial of the element e and > is the penalization fac- tor that encourages the solution towards a 0-1 (black-and- white) design.
where 3 controls the steepness of the sigmoid function (see Fig. 1), typical value is 3 20.
By employing neural approximation, we obtain differen- tiable approximation of the Heaviside function in a new la- tent space, such that H (sm: E) z NN (Zm E), where 2m is a latent representation of the feature parameters 8m mapped using shape encoder.
3.3. Feature-Mapping Topology Optimization (FMTO) Problem Formulation
Feature- Feature-Mapping Topology Optimization (FMTO) extends traditional topology optimization by incorporating geomet- ric features into the design process. In FMTO, pseudo- densities are parameterized using the defining attributes of these geometric features, such scale, offset, and rotation angle. The optimization problem can thus be formulated as
Neural Heaviside SDE The proposed model is based variational autoencoder-based architecture (Kingma & Welling, 2022) consisting of a shape encoder and several decoders (see Fig. 2). During training, we establish a com- mon latent space for different geometric features by mini- mizing multiple loss functions.
min, C(s) = = ((((s) K.,u
Shape Code. The shape code Xm is vector of 15 ele- ments organized in one-hot manner: for each feature type, specific segment of the code is activated while the remain- ing elements are set to zero, except for the first element Xm [0], which serves as a normalized label. For instance, in our experiments, an ellipse labeled as 0, triangle as 0.5, and quadrangle 1. For an ellipse, Xm [1] parame- terizes the semi-axis ratio, with the major axis fixed 0.25. In the case of triangle, Xm [2, 3] specify the coordinates of its movable vertex, while the other vertices are fixed at (-0.5, -0.5) and (0.5, -0.)). The elements Xm|[4 : 6) rep- resent the rounded radii of the triangle's vertices. For a quadrangle, [7,778) and Xm (9, 10] denote the coordinates of the two top vertices, respectively, while the bottom ver- tices remain fixed in the triangle. Finally, Xm 14] specify the rounded radii of the quadrangle's vertices. This scheme is illustrated in Figure 3.
s.t.
K(p(s))u= ff
Σ (o) Vmax
where s represents the set of feature parameters. The fea- ture mapping involves defining correspondence between the design variables 8 and the element pseudo-densities p. typically through a Signed Distance Function (SDF) rep- resentation. This mapping ensures that the geometric fea- tures are smoothly integrated into the design domain, facil- itating a more controlled and manufacturable optimization process.
4. Method
4.1. Neural Signed Distance Function
In the proposed method, we use a neural approximation of the Heaviside function of the Signed Distance Function (SDF) of geometric features to represent voids in Feature Mapping Topology Optimization.
Shape Encoder. The encoder receives a shape code Xm for a geometric feature m, and outputs a latent vector, 2m = NN (xm).

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Feapure-Mapping Topolegy Optimization with Neural Heaviside Signed Distance Functions
0.5
V3, R3
V7, R7
V6. R6
b
+
V1, R
V2, R2
V4, R4
V5, R5
U3
U6
U7
x())
6/0.25
T3
V/
R1
R2
R
T6
y/
1/7
R4
Rs
R6
Rx
Figure 3. Assembly of Shape Code for Various Geometric Features. Here, represents a normalized label, b denotes the minor axis an ellipse, and R, indicate the vertices and rounded radii of the i-th polygon corner, and æ; and y: are the coordinates of the i-th vertex of the polygon.
Heaviside Decoder. The decoder takes the latent vec- tor 2m concatenated with the point coordinates & as an input, and outputs the value of H that point, H NNHcavisid (2m.
struction decoder
B D
Lyecon
(x
NN eNN (x:)))
(10)
B
4.2. Feature Mapping Topology Optimization with Neural Heaviside SDE. Inference
Reconstruction Decoder. This decoder is responsible for reconstructing the shape code from the latent space, Xm
NNreconZ2m)
During inference, we employ the pretrained Heaviside De- coder with fixed weights to approximate the Heaviside function for the geometric features represented by the latent vectors. This approximation also enables us to compute the gradients required to adjust these latent vectors with respect to the compliance objective function.
Loss Functions. During training, minimize the mean squared error (MSE) between the predicted and true values of H and Xm- The Kullback-Leibler divergence (Dxi.) is employed as regularization term to ensure that the latent distribution is close to the Gaussian distribution.
Problem Formulation. Unlike the classical approach to topology optimization, which specifies the final volume re- quirement as a constraint, our proposed method incorpo- rates this requirement into the objective function. This re- quirement becomes active only if the volume exceeds a specified threshold. Apart from this constraint, the prob- lem formulation remains similar classical FMTO, where the goal is to minimize compliance, and the optimization is performed with respect to high-level geometric feature variables that define the boundaries of voids
The total loss function is defined as
B H (x)))))
Σ
NN Heavisidd (NNeac (x)),
B
B
AKL
Σ
B
where the batch size, Axr. is the regularization param- eter for the KL divergence, qs (2.[x;) is the encoder distri- bution for the latent vector 2, given the shape code Xis and p(z;) is the prior (normal) distribution over the latent vector Zi-
N
Σ PL LuuuK,t KNu + Avna max of Vmnx) a=l
min (w.b.a.2)
s.t.
De = maxx 3
NN (Ee, Zm
is important to note that the reconstruction decoder is trained separately, while all other parts of the model remain frozen. The MSE loss is exclusively applied to the recon-
ε.. wm (Ec bm R(am) K(p)u
(11)
5

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Feature-Mapping Topolegy Optimization with Neural Heaviside Signed Distance Functions
where M is the number of geometric features, Wm and bm the scale and offset of feature m, R((m) is the rota- tion matrix, Om is the rotation angle, and 2m represents the shape variables for feature m in the latent space. Conse- quently, 8m is defined as 8m = (Wm, bm, Om: km J. &e is the centroid coordinates of the element e.
Refactoring Mechanism. To prevent the latent variable to be infeasible, realize refactoring mechanism, where every 5 iterations we update the value of 2m. using the re- construction Decoder and Encoder
mm = NNenc CNN (NNrecco ('m mld))
(15)
The motivation behind formulating the pseudo-density function to obtain combined Heaviside function that yields value close to I when point lies within one of the features and close to 0 otherwise, thereby creating void within the feature.
Additional implementation details include the implementa- tion of the KS function and the selection of its parameters. Further details are provided Appendix G.
6. Results
To achieve a smooth approximation of the maximum func- tion, we employ the Pebissllmeier-Stinhaane function, taking into account the p limits
6.1. Neural Heaviside SDF
To evaluate the performance of the Neural Heaviside SDF; we conducted several experiments using generated datasets and a comprehensive set of metrics.
max (NN Heavisiie (Se, 2m)) =0 M.L...M
Punin In
exp(7rs 2m))
The accuracy of the predictions for H and was assessed using Mean Squared Error (MSE). For the topology optimization process to converge effectively, is crucial to minimize noise in the predicted H. To evaluate this noise, we compute the norm of the gradient of the pre- dicted H using the finite difference method on grid points
Metrics.
Tks
(12)
where Yks is the smoothing parameter.
Gradient Chain Rule for High-Level Design Variables. To solve the optimization problem with the combined ob- jective function, we employ an iterative method using ad- joint differentiation. The sensitivity of the function J (sm) with respect to high-level design variable sm is computed as follows
N
Smth
IV Havisiide Es) - H (Xm, Ex) ll2
N,
1=1
(16)
where N, represents the number of sensor points. This metric was computed across each m-th feature in the test dataset and subsequently averaged.
0J 3sm
N
[J Ope
Ope
Σ
+ppe - - uKuuu)
(13)
dsm
5. Implementation Details
Datasets. Our datasets include training and testing datasets for the Heaviside Decoder and the Reconstruc- tion Decoder. The dataset for training the Heaviside De- coder contains samples of geometric features and randomly located points with their corresponding Hcaviside values. The dataset for training the Reconstruction Decoder con- tains only samples of shape codes. To evaluate the Smth metric, we generated dataset that includes feature code shapes and a grid of points with their Heaviside values. Fur- ther details can be found in Appendix E.
Variable Initialization. Variables are initialized such that, in the first iteration, geometric features form a regular grid of squares with maximum rounded radii. The initial value of 2m for the first iteration is computed via Shape Encoder.
Variable Limitations. To constrain the values of design variables, preventing them from becoming too small or too large and ensuring they remain within the design domain, we apply sigmoid reparameterization
Model Architecture. The architecture of the Encoder consists of a block fully-connected layers with batch nor- malization and LeakyReLU activation functions. The archi- tecture of the Reconstruction Decoder mirrors that of the Encoder. The Heaviside Decoder utilizes the architecture of the DeepSDF decoder (Park et al., 2019). For more de- tails on the model architecture, please refer to the Appendix C. To demonstrate the effectiveness of using the DeepSDF model as a decoder, we trained various model types, includ- ing Autoencoders (AE), Variational Autoencoders (VAE),
1
Sm.j (Smax.j Smin.j) I
+ Smin.i (14)
where 8max.j and Smin.j are the maximum and minimum allowable values of the design variable 8m.j. respectively; j == 1,
As a result, we minimize the objective function with respect to the logits of the design variables km.j-
6

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Feature-Mapping Topolegy Optimization with Neural Heaviside Signed Distance Functions
and Maximum Mean Diserepancy VAES (MMD-VAE), suring that each model had an equivalent number of pa- rameters in their respective blocks. A symmetric decoder model mirroring the encoder was also employed for com- parison with the DeepSDF Decoder.
Shape Claxs Ellose Tarngs
Stape Cass
Elipze Thange
Dimension'
LSSe D
The comparative results are presented in Table for model training runs. Additionally, we conducted t-test to confirm the superiority the best combination of training strategies and model architectures the other combina- tions. The results are presented in Appendix F.
5NE Dimenston
5NE Dimension
(a) First Strategy
(b) Second Strategy
Table 1. Comparison of different model architectures with and without DeepSDF as decoder and different training strategies.
Figure 4. Comparison latent representations of geometric fea- clusters for different training strategies. First strategy. Training latent representation using the Heaviside decoder. b) Second strategy. Training latent representation using the Recon- struction decnder.
MSE#v
MSEx
Smoothness
DeepSDF Decoder
0.000346
0.0002
0,00586
AE
Training latent via Heaviside Dec.
+ 1.81e-05
+ 3,4e-05
+ 9.95e-05
MMD
0,000368
0.000228
0.006
-VAR
+ 1.95e-05
3.52e-05
0.000102
0.000277
0.000202
0.00575
VAE
- 1.07e-05
2.82c-05
7.11e-05
Symmetric Decoder
4
0.0014
0.000229
0.0165
AE
(a) Heaviside contour
(b) Geometry Reconstruction
0.000268
+ 3.5e-05
0.000658
MMD
0.00134
0.000212
0.0161
Figure 5. Reconstruction of geometric features from simplex tran- sition between features in the latent space. a) Reconstruction of contour from Heaviside values. b) Reconstruction of geometry through the Reconstruction decoder.
-VAE
- 0.000233
- 3.08c-05
0.000667
0.00128
0.000117
0.0163
VAB
0.000201
+ 2e-05
+ 0.000574
DeepSDF Decoder
and then fine-tuning the Heaviside decoder on the trained latent representation.
0.000796
0.000226
0.0084
AE
Training latent via Reconstr. Dec.
0.000101
+ 5.04e-05
0.000286
MMD
0.00059
0.000195
0,00818
Notably, the first strategy achieves good metrics for both MSE and MSE In contrast, the second strategy yields better metrics for MSEx but does not achieve sufficiently good metrics for MSE We suggest that training the la- tent representation with the Heaviside decoder results in a latent space tailored specifically to the contours of geomet- ric features. Conversely, training with the Reconstruction decoder focuses the latent representation the shape code of the features without adequately considering their geome- try. This is indirectly supported by the distribution of latent vectors reduced to a 2D plane using t-SNE (I (Laurens van der Maaten Hinton, 2008), as shown in Figure
-VAE
+ 5.42e-05
+ 1.61e-05
+ 0.000403
0.000475
7.42e-05
0.00806
VAE
+ 3.51e-05
1.09e-05
0.000187
Symmetric Decoder
0.00155
0.000232
0.0175
AE
+ 0.000257
+ 5.27e-05
+ 0.00044
MMD
0.0015
0.000198
0.017
-VAE
+ 0.00023
2.21e-05
0.000704
0.00146
7.02e-05
0.017
VAE
0.000327
+ 4.82e-06
0.000888
In all cases, we observe improved metrics when adding DeepSDF compared to the symmetric decoder.
Based on these presumptions, we can consider a triangle as special case of quadrangle from the perspective of the Heaviside contour, where vertex connects collinear edges. This results in overlapping distribution clusters for these features (see Figure 6.1). Additionally, for ellipses, the transitional shapes to other features in our dataset are not well-represented,, leading to noticeably poorer transi- tion quality. Figure clearly shows the transitions in latent
Training Strategies. We also compared two training strategics: 1. Training the latent representation using the Heaviside decoder and then fine-tuning the Reconstruction decoder on the trained latent representation. 2. Training the latent representation using the Reconstruction decoder

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Feature-Mapping Topolegy Optimization with Neural Hcaviside Signed Distance Functions
space between different geometric features using the Heav- iside decoder and the Reconstruction decoder.
to demonstrate that even with less material, better compli- ance values can be achieved. Refer the comparison in Table 2.
6.2. Topology Optimization
Additionally, we present comparative results from NTopo (Zehnder et al., 2021) and TopoDiff (Mazé & Ahmee, 2023). Both methods are free-form topology optimization approaches that use neural models to predict optimal topol- ogy solutions. NTopo employs deep learning technique to parameterize the density and displacement fields within the optimization process. TopoDiff uses a conditional dif- fusion model as a surrogate guidance strategy that actively favors designs with low compliance. The TopoDiff method is implemented only for square design domains. Therefore, we introduced two new tasks (Tasks and 6), which repli- cate Tasks and 3 on square domains. Although NTopo supports domain modifications, defining boundary condi- tions is challenging, so we did not apply it to these tasks. TreeTOp is implemented exclusively for rectangular do- mains. Consequently, the comparison for the Bracket prob- lem (Task 2) includes only the SIMP and Ellipses FMTO approaches. The final topology results for all methods are presented in Figure 7, and the von Mises stress distribution plots are provided in Appendix H.
We present the results of utilizing the Neural Heaviside SDF for feature mapping in topology optimization across several classic cases. Detailed descriptions and examples of solutions obtained using the SIMP method are provided in Figure 6 and Appendix D.
0.96
Task 0.56 C- 125
Daok
P
PL
Task 1 vE- 0.264 C- 3.78 10
Tack
C 1.04 10
VI - 0.34
vf 0.41 Task
2.21 10'
Figure 6. Classical topology optimization tasks, and solutions ob- tained using the SIMP method.
For FMTO methods, the metric values obtained by SIMP and NToop are unattainable because they can create small, locally conditioned edges through free-form approach, which our methods counteract to ensure manufacturability. So SIMP method as added to estimate the lower bound of the compliance metric. However, it is worth noting that our method closely matches the SIMP solution when the topol- ogy is simpler, as observed in Task 1.
Table 2. Comparison of different topology optimization methods via Compliance metric, the final volume fraction (yf) is given in brackets.
Task x10
Task 2
Task 3
Task x10 4
Task x10*
Task 6
Method
x10 33
x x10 3
x10*
3.78
1.42
1,25
1.04
1.51
2.21
Free- Form
SIMP
(0.254)
(0.36)
(0.44)
(0.34)
(0.41)
(0.41)
1.70
2.44
TopoDiff
N/A
N/A
N/A
N/A
(0.424)
(0.415)
Table 2 shows that among all FMTO methods (TreeTTOp, Ellipses, and NeuralHeavisideeSDF) our approach achieves the best compliance metric while using less material. Addi- tionally, in some experiments our method is comparable to NTopo (see Task 3) and outperforms TopoDiff on square- domain tasks.
4.08 (0.255)
1.63
1.07
1.60
2.31
NTopo
N/A
(0.438)
(0.339)
(0.412)
(0.41)
4.39
2.22
1.74
1.55
1,58
2.45
Ellipses (0.297)
(0.368)
(0.449)
(0.345)
(0.426)
(0.416)
FMTO
5.85 TreeTOp (0.266)
3.73
1,62
1.57
3.09
N/A
(0.455)
(0.356)
(0.449) (0.439)
NHSDF 3,91
1.98
1.63
1.52
1.55
2.34
(Ours)
(0.255)
(0.354)
(0.437) (0.337)
(0.409) (0.401)
7. Liiittttions and Future Work
Among the various Feature Mapping methods, we imple- mented an approach similar to (Kumar & Saxena, 2022), where all geometric features are represented by ellipses. Additionally, we use the official Tree'TOp implementation (Padhy et al., 2025) for comparison, where geometric fea- tures are represented as polygons constructed from set of half-spaces. Utilizing ellipses as geometric features signifi- cantly limits the value of the parameter Vmax In our experi- ments, we used the minimum volume fraction (vf) required to obtain a valid solution. Here, the volume fraction is de- fined as the ratio of the volume of solid elements to the total volume of the design domain. Specifically, with our method, we intentionally reduced the target value of Vinux
In the previous section, we demonstrated successful inte- gration of Neural Heaviside Signed Distance Function for Feature Mapping topology optimization. Nevertheless, sev- eral limitations must be addressed, which also offer av- enues for future research.
Poor transitions between different geometric feature types. We observed that shape transitions are largely cor- rupted by the Reconstruction decoder, whereas the Heav- iside decoder produces accurate contours for intermediate shapes.
8

---

Feature-Mapping Topology Optimization with Neural Heaviside Signed Distance Functions
Task 2
C 2.22 10-3
0 368
Tisk
vf 0.449
Task
0
4242
Tusk vf 0.438
C 1.6-10 10-2
Task 5
vf 0.412
C= 1.6 102
P
IP
P
0.297
Task 4 V2
0.416
vf 0.255
Task 4
Task
vf
4.39
1.56 10
1.07
(a) Ellipses (Kumar & Saxena, 2022)
(b) NTopo (Zehnder et al., 2021)
Task 2
C 1.96-100 10-3
D 364
Task vf 0.437
Tusk
0 409
155
Task
vf 0.456
C 3.73-10-
Task 5
C 157-100 10
vf 0.449
A
P
IP
P
3.91
Task 4 3 1.52 10-4
0.337
2.34
vf 0.266
10-5 Task
C 1.62 10
vf 0.439
(c) Neural Heaviside SDF (OUR)
(d) TreeTOp (Padhy et al., 2025)
Figure 7. Classical topology optimization tasks and corresponding solutions obtained using different methods: (a) Ellipses, (b) NTopo, (c) Neural Heaviside SDE, and (d) Tree TOp.
ditions. Future work will explore improved initialization
P
strategies and expand our geometric feature set.
We performed t-test assess the statistical significance of our model approximator's performance relative other models, and present the results in Appendix F. In fu- ture journal paper, we will provide a detailed analysis of our model's robustness.
P
8. Conclusion
Task vf 0.424
Task
vf = 0.415
In this work, have demonstrated several new approaches that can significantly expand the capabilities of FM topol- ogy optimization: generalization of parameters of various geometric features through latent space; mechanism for training the latent representation via a Heaviside decoder, which allows focusing on the geometry of features; a refac- toring mechanism that avoids the issue of the latent rep- resentation exceeding the permissible domain and enables control over individual high-level feature parameters. Our experiments demonstrate that this approach can be applied to Feature Mapping topology optimization, thereby ensur- ing the technological feasibility of the optimized solution, as FM approaches are always technologically feasible. We anticipate that the proposed approaches can be extended to more complex tasks, involving a greater variety of geomet- rie features and ftree-dimensional (3D) problems.
C 1.7 10a
C = 2.44 10a
Figure 8. Solutions ohtained using the TopoDiff method (Mazé & Ahmed, 2023) Square domain tasks.
Presence of a large number of conflicting variables dur- ing optimization. This issue necessitates alternating the optimized variables or periodically decreasing their learn- ing rate. Furthermore, the optimization process struggles to stabilize the offset variables, leading to oscillatory move- ments of geometric figures at the end of the optimization. Currently, this problem is effectively resolved by artificially slowing down the optimization speed.
As noted in (Wein et al., 2020), and as we also empha- size, topology optimization outcomes depend initial con-

---

Feature-Mapping Topolegy Optimization with Neural Hcaviside Signed Distance Functions
Impact Statement
Chibane, J., Mir, A., and Pons-Moll, G. Neural un- signed distance fields for implicit function learning. In Advances in Neural Information Processing Systems (NeurIPS), December 2020.
This paper presents research aimed at advancing the field of Feature Mapping Topology Optimization (FMTO) through the application of neural network methodologies. While there several potential societal implications of our work, we do not consider any of them to warrant detailed discus- sion within the scope of this manuscript.
Deng, H. and To, A. A parametric level set method for topology optimization based on deep neural net- work. Journal of Mechanical Design, (330))091702, 03 2021. ISSN 1050-0472. doi: 10.1 10.1115/1.4050005. 4050105. URL https / /twp//diii///////// org/ /ttgg/0..0 1110/1111111111111 1011111111111110.. 4050105.
Acknowledgments
The work was supported by the grant for research centers in the field of Al provided by the Ministry of Economic Development of the Russian Federation in accordance with the agreement 000000C313925P4F0000 and the agreement with Skoltech N0139-1--202--033.
Dunning, P. D. Minimum length-scale constraints for pa-
rameterized implicit function based topology optimiza- tion. Structural and Multidisciplinary Optimization, 58
(1):155-169, July 2018. ISSN 1615-147X.
Galerkin, B. G. Series occurring in various questions con- cerning the elastic equilibrium of rods and plates. Vest- nik Inchenerov i Tekhnikov (Engineers and Technologists Bulletin), 19:897-908, 1915.
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Feature-Mapping Topolegy Optimization with Neural Heaviside Signed Distance Functions
A. Notations and Definitions
Table 3. Notations and Definitions (Part I)
Notation
Definition
K
Global stiffness matrix obtained by assembling all element stiffness matrices K
K.
Element stiffness matrix for element e. The size of the matrix is equal to the size matrix K.
ke
Local element stiffness matrix (for element before assembly. It is equavelent the matrix K2 but without the zero rows and columns. Nodal displacement vector the FEM discretization.
f
External force vector.
Index of element in the FEM mesh.
N
Total number of elements in the FEM mesh.
Pe
Pscudo-density variable for element with Pe. E [pmin, 1].
Vector of pscudo-density variables for all elements the FEM mesh.
Prain
Minimum allowable pseudo-density (to avoid singular stiffness matrices).
Ve
Volume of element e.
Vector of volumes for all elements in the FEM mesh.
Vanx
Maximum allowable volume in the design (volume constraint).
p
Penalization factor in the SIMP interpolation (pE). with Compliance of the structure, defined as u or u Ku.
C(p) or ((s)
Set of high-level design variables defined the set of geometric features.
8m
Subset parameters defining geometric feature m.
M
Total number of geometric features used in the optimization.
sdf (sm. E)
Signed distance function for geometric feature m at point E.
A point in the design domain.
((mm)
Domain occupied by geometric feature m determined by Sm.
08(sm)
Boundary of the geometric feature m.
d(s, (s)))
Euclidean distance from point & the boundary a((sm)
Hsm: 5)
Approximated Hcaviside of the signed distance function for feature m at point & using sigmoid function.
8
Steepness parameter in the sigmoid approximation of the Heaviside function. Typical value is f = 20.
[s E)
Neural approximation latent representation of the feature parameters 8m, mapped using shape encoder.
Heaviside signed distance function feature m point E using
NN NNBevviside (2m. {)
Neural network approximator for the Heaviside of the signed distance function for feature m at point ç, taking latent vector 4m as input.
Zim
Latent vector associated with geometric feature m, obtained via the shape encoder.
Xon
Shape code for geometric feature (the encoded representation of its parameters).
NNone
Shape Encoder network that maps Xon to the latent vector 2m.
Xm
Reconstructed shape code for geometric feature using the reconstruction decoder.
NNNecon
Reconstruction Decoder that maps latent vector 2n back to a reconstructed shape code Xmn
13

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Feature-Mapping Topology Optimization with Neural Heaviside Signed Distance Functions
Table 4. Notations and Definitions (Part
Notation
Definition
B
Batch size used during training (in the loss computations).
Batch shape codes for geometrie features.
Batch of latent vectors associated with all geometric features, obtained via the shape encoder.
DRL.
Kullback-Leibler divergence function.
AKL
Regularization parameter for the KL divergence.
(x)))
Encoder distribution for the latent vector & given the shape code Xt.
p(z)
Prior distribution over the latent vector 2;-
Avol
Penalty factor for volume constraint violations the objective function.
&e
A center point of element e.
te.m
Transformed coordinate for point & with respect feature m; defined
f Wm (E- bm.) R(amm).
Wm
Scale factor for geometric feature m used to adjust its size in topology optimization.
bm
Offset (translation) for geometric feature
O'm
Rotation angle for geometric feature m,
R(am)
Rotation matrix corresponding to the angle
YKS
Smoothing parameter in the Beaesellmeir--teenhaaaee function used smooth maximum approxi- mation. Typical value is "Kk = 40.
Sminj:: Smax,
the design variables 8m.j- Since for proposed method, the 8m defined as 8m = (wm, 6m, Om, ), ), 8min, = Wminn 8min,2 = bmin 8 Omnnn and 8min, and similarly for the upper bounds.
Lower upper bounds
8m.)
Unconstrained (logit) representation of design variable Sm.j before sigmoid reparameterization.
Smth
Smoothness metric quantifying the noise in the neural Hcaviside approximation:
Nr
A (NNNHen (NNNHavvittde (Zmi {.) (Xxmxx)))
Smth
N
Np
Number of sensor points used compute the smoothness metric.
Nasde
Total number of nodes in the FEM mesh.
A,
Lamé constants used in the elasticity (Hooke) matrix construction.
dofsrree
Set of degrees of freedom with unknown displacements (free DOFs).
dofsricedMovall
Set of degrees of freedom with prescribed (fixed) displacements.
14

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Feature-Mapping Topolegy Optimization with Neural Heaviside Signed Distance Functions
B. Assembling of the Global Stiffness Matrix
f
0
fly
11.8
0
122
:
Ug,
f
Us,
Fr.y
f5.2
Ug,r
U8.8
18..
Kg Kavi Keber
k1xx
22
K1
k3.0
3.
kt.y
k
k
k2.3
23.x
3.
2
2
63.x
3.y
x2
kie
K
k
k2.v
k-
k33xx
k2.y
2.
K3.9
k
k3
k3
k3-y
K3
K3.y
k3.y
k2.x
k.yy
k3
k3.
dolsFixedMoved
[0 1, 2, 3
dofsFree [, 5,
13
Figure 9, Example of building the element stiffiness matrix for a triangular element for domain discretized with nodes in the FEM method.
In this section, we do not provide the full derivation of the system of linear equations from the 2D elasticity differential equations. Instead, present only assembling of the element stiffness matrix. Comprehensive explanations can be found in (Cenanovic & Jonsson).
The discretized displacement field for the domain Ω with Nmade nodes is represented by the vector
U1,2, U1,g U2,35 U2. Mi,r, Miys N UN..tte
(17)
where U; and uyyy are the displacement components along the x and y axes, respectively, and denotes the node index.
Consider a local element e within the discretized domain, for which a local stiffness matrix k constructed using the Galerkin method and Mendel notation (Galerkin, 1915). For a triangular element, the matrix dimensions are 3dof x 3dof, where dof represents the number of degrees of freedom per node. In the 2D case, dof making k; matrix. From
15

---

Feature-Mappinn Topology Optimization with Neural Hcaviside Signed Distance Functions
this matrix, element stiffness matrix Kg E R2N x2Naade is constructed such that all values except for those corresponding the degrees of freedom of the nodes belonging element An example of assembling the stiffness matrix for a domain discretized with 8 nodes can be found in Figure 9.
To assemble the global stiffness matrix K E R22 mode x xndu it is necessary to sum all the element stiffness matrices K,
N
K
K
(18)
During the topology optimization process, pseudo-density variables Pe. are used to indicate the solidity of each element
N
K =
K
(19)
< Pmin S S Pmax
Implementation of boundary conditions. Let doffspieeMevedd denote the set of degrees of freedom with known displace- ments, and dofspre denote the set of degrees of freedom with unknown displacements (see Fig. 9). The global stiffness
matrix used for the solution is obtained by removing the rows and columns corresponding to dofSsrixeMMoved and incorporat- ing the known displacements into the right-hand side of the system of equations
[ddofsrree dofspree) u [doffFree] f[dofSrree]
K [dofsree dofsriseaMoeal)
u [ddofsriseMMoeel
(20)
During optimization, whenever an update to u required, the system in Equation 20 is solved, and the entire vector u updated with the solution values.
16

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Feature-Mapping Topology Optimization with Neural Hcaviside Signed Distance Functions
C. Model Architecture
Parameter
Defined
Value
Heaviside Reconstr. Shape decoder (sym.) decoddr
encoder
dinput
dx
h
dim.
BNorm + Leak Rel.
Linear: A, + 2h
BNorm + Leak
Lincar: 2/. -3 4h
BNorw LeakReLUU
BNorm + Leak U
BNorm + LeakReLU
Linear: h. + doutput
daurgus
dz
Linear: 2h, -
Parameter
Defined
Linenr: desput-
Lincar: 4h,
Aaput
d
h
hidden dim. dy
128
daurqus
16
Parameter
Defined as
Value
deupot
d, 2
17
h
hidden dim.
128
douipus
(a) Architecture used for the Encoder, Reconstruction Decoder, and symmetric version of the Hcaviside Decoder
Lincar: 512
Linear: 512, + 512
Linear: 512, - 512
512, 495
Linear: 512, -> 512
Lincar: 512, - 512
Linear: 512, - 512
Linear: 512. + 512
BN NNomm ReL.
BNom ReLU
BNorm + ReLU
BNorm + ReLU
BN orm ReLU
BN ReLU
BN BNorm ReLU
BN BNorm ReLU
BNorm ReLU
Linear: 512, +
Tanh
(b) Architecture of the Heaviside Decoder, using the DeepSDF architecture
Figure 10. Schematics and parameters of the model's architecture.
Figure 10(a) shows the architecture used for the Encoder, Reconstruction Decoder, and the symmetric version of the Heaviside Decoder. The Encoder architecture consists of block of fully-connected layers with batch normalization and LeakyReLU activation functions. The Reconstruction Decoder mirrors the Encoder's architecture. Figure 10(b) illustrates the architecture the Heaviside Decoder, which utilizes the DeepSDF decoder architecture (Park et al., 2019).
D. Topology Optimization Schemes
The detailed schemes of the tasks presented in Figure 11.
Task 1: Cantilever Beam. The design domain is fixed on the left side, while a load of P -0.0025 N is applied at the midpoint of the right side. The rectangular domain measures 0.25 m x 0.60 m.
Task 2: Bracket. This task has an L-shaped form with dimensions 0.15 m x 0.15m. The top part is fixed, and uniform load -0.0025 N is distributed on the top part the protrusion.
Task 3: MBB Beam Half Design Domain. This domain typically represents a symmetric half of the MBB beam. The design domain measures 1.2 m x 0.4m. A vertical rolling condition is applied to the left side, and a horizontal rolling condition is applied to the bottom right corner. A load of P = -0.0025 N is applied at the upper left corner.
Task 4: Cantilever Beam with distributed load. The domain of the task is a rectangle with dimensions 1 m x 0.50 m. A distributed load of P 0.0025 N is applied to the upper part of the domain, and the left side is fixed.
Task 5: Square Cantilever Beam. This task replicates Task with the design domain modified to a square measuring
17

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Feapure-Mapping Topology Optimization with Neural Heaviside Signed Distance Functions
Task 2
Task 3
Task 5
P
Square
MBB Beam Half
Contilever
Beam
Contilever Beam
P
P
P
P
Beam
Square Beam
Distributed Lood
Brecket
P
Task 1
Task 4
Task 6
Figure 11. Schematics and solutions of the tasks.
1 1m X 1m.
Task 6: Square Beam. This task replicates a previous task (Task 3), but the design domain has been modified square measuring mmm 1..
For all cases, the Poisson ratio and Young's ' modulus set to 0.3 and E Pa. Schematics of these cases shown in Fig. 6, along with the solutions obtained using the SIMP method.
E. Datasets
Dataset for training the Heaviside Decoder consists of 5k samples for each type of geometric feature (ellipse, triangle, quadrilateral) with varying rounding radii. As illustrated in Fig. 3, each sample is designed include the minimal number of variables for each feature. For ellipses, one axis is fixed at 0.5, and the axis ratio is used as the variable in x. For triangles and quadrilaterals, two vertices are fixed at coordinates (-0.5, -0.25) and (0.5, -0.25), respectively, while the coordinates of the remaining vertices are used in along with the rounding radii. For each geometric feature, 000 random points are generated within the square [- 1, x [-1, 1, The approximated Heaviside function is computed using Equation
Dataset for training the Reconstruction Decoder contains only 5 million samples of X for each type of geometric feature.
Dataset for testing the Heaviside Decoder is generated similarly to the training dataset, but it contains 500 samples for each type of geometric feature.
Dataset for testing the Reconstruction Decoder is generated similarly to the training dataset, but it contains 10k samples for each type geometric feature.
The surface test dataset contains 100 geometric features with 1225 points per feature, generated grid format. This dataset is used to compute the smoothness metric.
F. T-test for Comparison of Different Model Architectures Note:
The test conducted metric evaluations after 20 model training runs. All seeds not fixed.
The p-value is shown in the graphs on the right, relative the best method for each specific metric.
18

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Peature-Mappinn Topolegy Optimization with Neural Heaviside Signed Distance Functions
"stl" means the first strategy, where the latent space is trained via the Heaviside decoder. "st2" means the second strategy, where the latent space is trained via the Reconstruction decoder.
MSE
st2_vAE_ LeE___eeeepS
4.76-09
SA2_VEE
6.3e-09
VAE DeepSDF
5.0e-00
s22 MMD VAE
1.5e.07
Strategy Model
s22 AE_ DeepSDDF
4.4e-08
81 VAE DeepsoF
Prvalue wrt
st2_AE
2.1e-07
BL__VA_DDDepppp DeepSDF
s11 VAE
7.6e.08
st1_ MMD VAE DeepSDF
1.5e-08
st1 VAE
3.1e:07
st1, AE DeepSDF
8.30-08
AE
15e-06
0.00025
0.00050
0.00075
0.00100
0.00125
0.00150
0.00175
0.00200
0.00225
MSEer
Smooth
st2 VAE DeepSDF
+ o
7.3e-12
st2 VAE
13e-12
#22MMD_VAL VAE_DeepSDF
Be-09
st2 MMDMMDDWW VAE
9.1e-12
Strategy Model
st2 AE DeepSDF
4.7e-11
81 VAE DeepSDF
p-value wrt
1.6e-3
DeepDDD
SAL
3.6e-13
St1 MMD VAE DeepSDF
4.5e-05
st3 AE DeepSDF
(+
0.074
_E
9.9e-12
0.006
0.008
0.010
0.012
0.014
0.016
0.018
Smooth
MSE,
sAE_ VAE DeepSDF
#t2
0.14
st2 VAE
922 VAE DeepSDF
32 MMD VAE
7.6e-10
Strategy Model
s22 AE DeDDSDD
8.7e-07
pvvalue wrt
AE
16c-07
S22 VAE
911, BLE_VVEDDeeeeeee DeepSDF
17e-07
st1 VAE
19e-05
#11, MMD VAE DeepSDF
2.9c-08
st1
st1, AE, DeepSSDD
1.ee06
2.9e-09
0.00005
0.00010
0.00015
0.00020
0.00025
0.00033
MSE,
Figure 12. Boxplot for all training strategies and models.
19

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Feature-Mapping Topolegy Optimization with Neural Heaviside Signed Distance Functions
G. Kreisselmeier-Steinhaan (KS) Function, Implementation Details
important for us to maintain sensitivity with respect to the objective function; however, to ensure the method does not fail, we are forced to clamp the p parameter because the KS function can produce values greater than Therefore, in addition to equation (11), we use the following formula to scale p
In EM S + Pmiin
Pe
K Saax
Puiin
K Kmin
K S max
NKS
(21)
where K Kmiin is the minimum value of the KS function, which is equal to
In P K
(22)
7KS
and KmSmax is the maximum value of the KS function, which is equal to
KS In(Pexp(TKs))
(23)
7KS
where is the expected maximum number of geometric primitives intersecting at one point (by default, this 2).
Therefore, the combined o will exceed the range [pmin. 1] only at intersections where more than P geometric primitives overlap. In these cases, p is clamped the range and becomes insensitive the objective function.
In our case, the primary topology shape is non-overlapping primitive in which void must form. For large values of P but small values of YKS. the value of p inside the primitive becomes considerably higher than Pmin This behavior is evident from the minimum p values (min(p)) reported Table 5 for Task 3: MBB beam half, where no primitives intersect. Therefore, we choose S to be as large as possible while ensuring that computations remain practical and that excessively large values of exp(Tкs) are avoided.
Changing 7KS above 10 has little impact on convergence or final topology, whereas lower values worsen convergence because Pe remains too high for void formation.
We conducted experiments with different Ykt values for Example 3: Bracket. The results are shown in Table 6 and Figure 13.
Table 5. Experiments with different The values for Task 3: MBB beam half
7kk
10
20
30
40
50
60
70
80
vf
0.454
0.471
0.487
0.437
0.510
0.460
0.451
0.465
C
0,00182
0.00169
0.00151
0.00163
0.00141
0.00155
0.00156
0.00153
min(p)
0.06997
0.03533
0.02378
0.01800
0.01454
0.01223
0.01058
0.00934
max(p) (p)
0.93014
0.96479
0.97634
0.98212
0.98559
0.98790
0.98955 0.99078
Table Experiments with different Tes values for Task Bracket
Tks
10
20
30
40
50
60
70
80
vf
0.373
0.358
0.347
0.355
0.347
0.344
0.357
0.354
C
0.00238
0.00212
0.00209
0.00207
0.00210
0.00213
0.00203
0.00198
min(p)
0.00067
0.00105
0.00504
0.00089
0.00116 6 0.00163
0.00623
0.00103
max(p)
(p)
0.87418
0.93682
0.95769
0.96813
0.97439
0.97857
0.98155
0.98379
20

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Feature-Mapping Topology Optimization with Neural Heaviside Signed Distance Functions
Gamma_Ke LS: 10
Gamma KS:
vf. 0.373
vt: 0.347
compliance: 0.00238 max stress: 0.632 H_ min: 0.00067
compliance: 0.00210 max stress: 0.881
H. mini 0.00116
H_max: 0.87418
H_max: 0.97439
Ven Woue 1977
Gamma_KS: 20
Gamma KS: 60
vf: 0.358
vf: 0.344
compliance: 0.00212 max stress: 0.406
compliance: 0.00213 max stress: 0.900
H. min: 0.00105
H_min: 0.00163
II_ max: 0.93682
H_max: 0.97857
Wh DVI
Ien MBNS
Gamma_KS 30
Gamma KS: 70
vf: 0.347
vf: 0.357
compliance: 0.00209
compliance: 0.00203 max stress: 0.571
max _ax__stres:: 0.511
H_ min: 0.00504
min: 0.00623
H m_xax: 0.95769
H_max: 0.98155
We
Me Mora
Gamma KS: 40
Gamma KS: 80
vf: 0.355
vř: 0.354
compliance: 0.00207 max stress: 0 746 min: H. max: 0.96813
compliance: 0.00198 max stress: 0.581
H. H_max: 0.98379
(a) The 10...40
(b) 7hk 50...80
Figure 13, Breeket results for different %ha ranges.
21

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Feature-Mappinn Topolegy Optimization with Neural Heaviside Signed Distance Functions
H. Method Comparison
The task schemes are presented in Figure 11.
Notes:
The presented von Mises plot is used here for a visual assessment of the quality of the internal stress distribution in the structure. Using its maximum values as a metric for comparing methods impractical in this case.
The specified Vmaa value in FMTO is chosen to demonstrate that even with the smallest amount of material, method achieves better compliance metric values.
TOP is implemented only for rectangular domains. Although the NTopo method allows for domain modifica- tions, specifying the boundary conditions is complicated, and therefore did not implement it. Consequently, the comparison for the "Bracket" problem includes only SIMP and Elliptical FMTO.
23.67
Von Mises stress (Pa)
Von Mises stress (Pa)
13:39
SIMP
53.3
SIMP
0.41
21.9
vf: 0.41
C: 15.1
6.20
vonMiseSmari 28.8
vonMiseesmur 132
11.5
3.11
23.22
Von Mises stress (Pa)
Von Mises stress (Pa)
TopoDiff
33.1
TopoDiff
3.43
0.0048
vf: 0.415 = 0.00417
C: 17 = 0.368
C: 24.4 = 0.602
vonMiseSmari 64.5
16.5
vonMises 133 5.45
320
11.3
Von Mises stress (Pa)
105.3
Von Mises stress (Pa)
12.04
NH - SDFF(OO))))
52.7
NH SDFSSD(Ours)
vf: 0.409
vf: 0.401 C: 23.4
15.5
3.66
vonMisesmari 30
15.8
vanMises 132
0.07
0.0
Task Square Cantilever Beam
Task Square Beam
Figure 14, Solutions obtained using SIMP, TopoDiff and Neural Heaviside SDF for Task 5 and Task 6.
22

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Peapure-Mapping Topolegy Optimization with Neural Heaviside Signed Distance Functions
NeuralHeavisidesoo
vonMiseSmax 0.607
0.522
vonMisesmux 6.08
0.00373
vf: 0 0.449 C: 0 0.00174
C: 0.00163
TreeTOp
vf: 0 00.45
Ellipses
vf: 0. 0.437
FMTO methods
Von KIDS STRSS
Vor Nives 2te5s
Figure 15, MBB Beam Half
vonMisesmuni 0.483
vanMisesmaii 2.52
C: 0.00125
C: 0.00163
vf: 0.438
vf O 44
NTopo
SIMP
Hig Saa
Free-form methods
Vón Wab stress
23

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Feature-Mapping Topolegy Optimization with Neural Heaviside Signed Distance Functions
NeuralHeeavisideSDE
vonMiseSmare 0.674
vonMiseSman 0.575
vonMiseSmax 43.8
C: 0.000152
C. 0.000155
C: 0.000155
TreeTOp
vf: 0. D.3557
Ellipses
vf: 0.345
ví. 0.
FMTO methods
Ven Noses siress
Figure 16. Beam Distributed Load
vonMiseSmaze 0.194
vonMiseSmar' 0.349
C: 0.000104
C: 0. 0.000107
vf: 0.339
vf: 0.34
NTopo
SIMP
Free-form for-form methods
Von Mives stress
Von
24

---

Feature-Mapping Topology Optimization with Neural Heaviside Signed Distance Functions
Free-form methods
FMTO methods
Von Mises stress
Von Mises stress
0016R:
11510
3076
stress (Pa)
SIMP
Von Mises (Po)
TreeTOp
oucees
vf: 0.24
3002
vf: 0.266
C: 4.51e-05
C: 5.85e-05
vonMiseSmax 0.117
vonMisesmax 0.259
Von Mises stress
Von Mises stress
001332
NTopo
Ellipses
Mises stress
0.2284
vf: 0.241
vf: 0.297
C: 4.26e 05
C: 4.39e-05
vonMisesmax' 0.135
vonMisesmax 0.125
Von Mises stress
584236
NeuralHeavisideSDF
vf: 0.255
C: 3.91e 05
vonMisesmax' 0.142
040222
043002
Figure 17. Cantilever Beam
25

---

Peature-Mapping Topolegy Optimization with Neural Heaviside Signed Distance Functions
NeuralHeawisideSDDt
vonMiseS 0.956
vonMises 0.442
0.00222
C: 0.0017
Ellipses
vf: 0 6.368
vf: 0. 0.354
FMTO methods
bi mrre (P9)
Sio Nivm MlDAA
Wo Nocs 2015s
Figure 18. Bracket
vonMisesmari 0.41
C: 0.00142
vf: 0.33
SIMP
Free-form methods
Van Sharr XITEe
26


Reviews:
[{"review_id": "MWpBXPkk6i", "reviewer": "Reviewer_rbNQ", "summary": "In this work, the authors work on topology optimization, specifically, they propose a deep learning method to simulate Feature-Mapping Topology Optimization (FMTO) (and not SIMP). They propose two decoders, one for the reconstruction and another to approximate the heaviside function. They show results on variations of autoencoder architecture types and training strategies.\n\n## update after rebuttal\nAfter the rebuttal, I am increasing my score. From the experiments, the proposed work brings significant improvements compared to existing work such as free-form methods NTopo and TopoDiff, and other FMTO methods. Using standard shapes eases manufacturing, and in the rebuttal for other reviewers, the authors discuss how they can improve boundary fitting further.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your thorough and insightful review!! We deeply appreciate the time and effort you invested in evaluating our work. \n\n***Comparison with other methods*** Our method follows the FMTO approach by creating topology with geometric primitives. Unlike traditional free-form optimization that focuses only on compliance, FMTO constrains voids to specific shapes. Thus, comparing with other FMTO methods is most appropriate, even though many lack available code and full reproducibility details.\n\nNevertheless, we can still provide more comparisons with four methods, including two new frameworks: the first competitor is a free-form method based on deep learning, $\\texttt{NTopo}$ [1], and one  recently published work $\\texttt{TreeTOp}$ [2], which directly relates to FMTO. These comparisons will be added to the manuscript.\n\n| Method | Method type | $\\text{vonMises}_{max}$ |  Compliance | Volume Fraction |\n| :--- | :---: | :---: | :---: | :---: |\n| SIMP | Free-form | $\\textbf{0.483}$ | $\\textbf{0.00125}$ | 0.44 |\n| NTopo | Free-form | 2.52 | 0.00163 | 0.438 |\n| TreeTOp | FMTO | 6.08 | 0.00373 | 0.455 |\n| Ellipses | FMTO | 0.607 | 0.00174 | 0.449 |\n| NeuralHeavisideSDF | FMTO | $\\textbf{0.522}$ | $\\textbf{0.00163}$ | 0.437 |\n\nTable: Methods comparison for Example 3: MBB beam half\n\n| Method | Method type | $\\text{vonMises}_{max}$ |  Compliance | Volume Fraction |\n| :--- | :---: | :---: | :---: | :---: |\n| SIMP | Free-form | $\\textbf{0.194}$ | $\\textbf{0.000104}$ | 0.34 |\n| NTopo | Free-form | 0.349 | 0.000107 | 0.339 |\n| TreeTOp | FMTO | 43.8 | 0.000155 | 0.357 |\n| Ellipses | FMTO | 0.674 | 0.000155 | 0.345 |\n| NeuralHeavisideSDF | FMTO | $\\textbf{0.575}$ | $\\textbf{0.000152}$ | 0.337 |\n\nTable: Methods comparison for Example 4: Beam Distributed Load\n\n$\\textbf{Note:}$ Since the NTopo method is quite limited in altering the parameters of the initial conditions, we had to adjust the parameters of the other methods to match those of the NTopo method. Therefore, the metric results differ from those in the original manuscript.\n\nOur approach achieves the best Compliance metric values among all FMTO methods while using less material. Additionally, in some experiments, our method is comparable to the NTopo method (see Example 3).\n  \n1. J. Zehnder, Y. Li, S. Coros, and B. Thomaszewski. NTopo: mesh-free topology optimization using implicit neural representations. 2021.  \n\n2. K. Padhy, R., Thombre, P., Suresh, K. et al. Treetop: topology optimization using constructive solid geometry trees. 2025.\n\n***Training and inference*** During training, we optimize our model to best approximate the Heaviside SDF for various geometric primitives, independent of topology optimization. At inference, we use the frozen Heaviside decoder for SDF evaluation in FMTO and for geometry reconstruction during post‐processing.\n\nConstraints $V_{max}$, $s_{max}$, and $s_{min}$ are enforced at inference—$V_{max}$ controls the design domain volume via the Lagrangian, while $s_{max}$/$s_{min}$ bound the shape variables via a sigmoid.\n\nWe have clarified our method by distinctly separating training from inference. These changes improve clarity and reproducibility.\n\n***Table 1, Bolding*** Best metric values in Table 1 are bolded. Due to layout limitations, the table was split; we apologize for the confusion and will restore the original format.\n\n***Statistical Significance*** We conducted a t-test on 20 independent runs, comparing each model to the best-performing model. The results are presented in the table below and and be provided in the revised appendix, confirming our choice of VAE architecture and training strategy based on $MSE_{sdf}$.\n\n|  | AE | MMD-VAE | VAE | AE | MMD-VAE | VAE | AE | MMD-VAE | VAE | AE | MMD-VAE | VAE |\n|---|---|---|---|---|---|---|---|---|---|---|---|---|\n| Strategy | st1 | st1 | st1 | st1 | st1 | st1 | st2 | st2 | st2 | st2 | st2 | st2 |\n| Decoder | Symm. | Symm. | Symm. | DeepSDF | DeepSDF | DeepSDF | Symm. | Symm. | Symm. | DeepSDF | DeepSDF | DeepSDF |\n| mean | 0.0014 | 0.00134 | 0.00128 | 0.000346 | 0.000368 | **0.000277** | 0.00155 | 0.0015 | 0.00146 | 0.000796 | 0.00059 | 0.000475 |\n| std | 0.000268 | 0.000233 | 0.000201 | 1.81e-05 | 1.95e-05 | **1.07e-05** | 0.000257 | 0.00023 | 0.000327 | 0.000101 | 5.42e-05 | 3.51e-05 |\n| p-value | 1.4e-11 | 3.9e-12 | 9.4e-13 | 4.9e-13 | 8.3e-15 | -- | 1.1e-12 | 3.3e-13 | 1.2e-10 | 4.1e-13 | 2.9e-14 | 8.8e-15 |\n\nTable: Metric: $MSE_{sdf}$ (p-values computed relative to st1\\_VAE\\_DeepSDF)\n\nIn the case of Table 2, which presents the comparison results with other methods, we did not perform a t-test because these methods are deterministic with respect to the input parameters.\n\n***Notations*** We apologize for the missing definitions. Inconsistencies in the method description have been corrected, and an updated version—including details on the shape code $\\chi$—will appear in the main text, as well as a revised Figure 4. Additionally, a table of notations will be added to the appendix.", "claims_and_evidence": "See Experiments section", "methods_and_evaluation_criteria": "See Experiments section", "theoretical_claims": "N/A", "experimental_designs_or_analyses": "- In this work, the authors propose using a Heaviside Decoder for the topology-optimization task. They have two decoders, different options for each decoder (DeepSDF or Symmetric) and different training strategies (train reconstruction before heaviside or vice-versa). The experiments compare their method with SIMP solver and FMTO (Ellipse) which are non-deep learning methods. While the proposed method doesn’t reach the levels of SIMP, it performs better than FMTO.\n- The authors did extensive ablation study on different decoder types and strategies (Table 1) and compared against SIMP and FMTO in Table 2. However, the authors do not compare with any other deep learning based method. Specifically, in L31 and L36 they discuss diffusion-based topology-optimization methods by Maze & Ahmed’22 and Mohseni & Khodaygan’24. The authors do not provide any discussion of why existing methods cannot be compared to theirs. Hence, such baselines should be included. \n- The model is trained as a reconstruction task, and the authors provide the training objective functions. However, how are the constraints like $V_{max}$, $s_{min}$, $s_{max}$ enforced during inference? They seem to be used only during the training objective function to penalize the outputs, and it is unclear how they are incorporated during inference/test.\n- In Table 1, authors should bold the best value under each metric. Currently, they omit bolding when their proposed method doesn’t achieve the best performance.\n- In Table 1 and Table 2, the authors should perform t-test to determine if the performance improvement is statistically significant or not.\n- I appreciate the code release in the supplementary", "relation_to_broader_scientific_literature": "The work is important because most CAD-integrated design requires human intervention in the post-processing stage. This work aims to minimize human intervention.", "essential_references_not_discussed": "As mentioned above, the authors need to compare with existing deep learning based methods. The authors could also consider comparing with the following:\n- Zhang, Zeyu, et al. \"Topology optimization via implicit neural representations.\" Computer Methods in Applied Mechanics and Engineering 411 (2023): 116052.\n- Zelickman, Yakov, and James K. Guest. \"Introducing a general polygonal primitive for feature mapping-based topology optimization.\"\n- Chi, Heng, et al. \"Universal machine learning for topology optimization.\" Computer Methods in Applied Mechanics and Engineering 375 (2021): 112739.\n- Sosnovik, Ivan, and Ivan Oseledets. \"Neural networks for topology optimization.\" Russian Journal of Numerical Analysis and Mathematical Modelling 34.4 (2019): 215-223.", "other_strengths_and_weaknesses": "The presentation of the paper needs a lot of work. The authors tend to use notations without defining them, which makes the paper very difficult to read; the reviewer has to guess themselves what the notation means. I list a few below, but there were several such issues:\n- Equation 7 uses $\\Omega$ but never defines it. It is likely the boundary of the feature.\n- What is H in the left side of Figure 3? It is not defined in text. Only $\\tilde{H}$ and $\\tilde{H}_{true}$ are defined in the text, but not H.\n- No clear separation between the training and inference procedure.\n- $X$ is not explicitly defined in the text. It is first used in L208, and the reader has to go to Fig. 4 to understand what it represents.", "other_comments_or_suggestions": "See above.", "questions_for_authors": "I would re-consider my score after the authors' responses to my comments, and after discussion with other reviewers.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "0fO9gi2dXL", "reviewer": "Reviewer_ZNau", "summary": "The authors propose a novel neural approximation framework based on a variational autoencoder (VAE) model to approximate the Heaviside function of the Signed Distance Function (SDF), enabling a unified representation of diverse geometric features in a single latent space. This approach integrates machine learning with traditional topology optimization (TO) to overcome the limitations of existing TO methods. The key contribution includes an improved optimization framework that incorporates volume constraints into the objective function like the Kreisselmeier-Steinhauser (KS) function for smooth maximum function approximation, and efficient gradient computation and sensitivity analysis using adjoint differentiation.", "questions": "", "limitations": "", "rating": 2, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your comprehensive and constructive feedback on our work. We value your insights on the VAE encode-decode structure, the Neural Heaviside SDF representation, and the potential challenges related to geometric approximation and boundary precision.\n\n**Novelty of the Proposed Method.** We acknowledge that VAEs have been widely used for learning shape distributions. However, our work differentiates itself by integrating a learned neural approximation of the Heaviside SDF function within Feature Mapping Topology Optimization (FMTO) tasks. Conventional FMTO approaches rely on explicitly defined SDFs, restricting the geometric diversity that can be effectively represented. Our approach, in contrast, enables a unified latent space where multiple geometric primitives can coexist and transform smoothly, facilitating shape evolution in FMTO beyond conventional methods.\n\nMoreover, although sigmoid and Heaviside functions are commonly used for defining geometric boundaries, our method employs a learned neural surrogate. This allows for an adaptive, data-driven boundary representation, enhancing both optimization flexibility and saving manufacturability.\n\n**Smooth Transitions Between Geometric Features.** Your concern regarding smooth transitions, particularly between ellipses and polygons, is well-taken. \n\nWhile our dataset does not explicitly include gradual shape interpolations between such forms, our model demonstrates an ability to approximate smooth deformations even beyond the trained shape classes. As illustrated in Figure 6 of the main text, the learned latent space permits meaningful shape variations, even if exact interpolations between distinct classes are not explicitly encoded. Nevertheless, we recognize that the latent space could be further refined to better support smooth morphing between different geometric classes. Future work may explore improved training strategies, such as incorporating intermediate transition shapes or using additional regularization to better structure the latent space for smooth interpolations.\n\n**SDF Precision and Boundary Fitting.** We understand the importance of accurately capturing high-curvature regions and sharp edges in SDF representations. To reduce inaccuracies in boundary fitting, we have enhanced our dataset generation strategy as follows:\n\nEach shape type in our dataset now includes 5,000 instances, sampled to ensure comprehensive coverage of geometric variations.\n\nAdditionally, we have modified the point generation method. Out of 10,000 points per shape, one third are now generated directly along the shape boundaries, improving precision and ensuring that the trained SDF captures intricate features more effectively. The remaining points are evenly distributed between those drawn from a Gaussian distribution centered around the shape and those concentrated near vertices and edges.\n\nThese improvements have significantly enhanced boundary fidelity, minimizing discrepancies between original and reconstructed shapes. The updated dataset and training approach result in more accurate SDF modeling, especially in areas with high curvature and sharp transitions.\n\n**Discussion on Related Work.** We appreciate the reviewer’s suggestion to better position our contributions within the existing literature. While previous works [1,2,3] have showcased the use of VAEs for shape learning, our study uniquely incorporates this framework into FMTO with a learned Heaviside SDF representation. We will expand our discussion in the manuscript to clearly contrast our method with prior approaches and emphasize our contributions more explicitly.\n\nOnce again, we sincerely thank the reviewer for their valuable feedback. We believe that our revisions effectively address the concerns raised and further strengthen our work.", "claims_and_evidence": "Yes.", "methods_and_evaluation_criteria": "Yes.", "theoretical_claims": "There is no theoretical claim.", "experimental_designs_or_analyses": "Yes.", "relation_to_broader_scientific_literature": "In my opinion, this work is more suitable for industrial or material journals or conferences.", "essential_references_not_discussed": "Yes. Please refer to the weaknesses part.", "other_strengths_and_weaknesses": "**Strengths**:\n\n- The authors conducted several experiments using a set of reasonable metrics to evaluate the accuracy of predictions (MSE) and measure the noise of the gradient on grid points (Smoothness Metrics).\n\n- The authors implemented an approach similar to an ellipse-based method that achieves superior compliance values. While the SIMP method creates small, locally conditioned edges through a free-form approach, the proposed method avoids and ensures manufacturability.\n\n**Weaknesses**\n- VAE encode-decode structure: The use of VAE to learn geometric feature distributions has been extensively explored [1, 2], even in 3D scenes [3]. Learning the shape distribution is a commonly used approach that does not introduce efficient improvements or novel insights.\n\n- Neural Heaviside SDF: SDF, Heaviside, and sigmoid representations are the most basic techniques. Although they express geometric boundaries and allow for training of discrete spatial variables with continuous representations, similar functions like sigmoid and Heavide also widely improved and not taken as sufficient innovations. \n\n- Noise in training and geometric approximation: Does the latent space representation derived from the VAE struggle to capture smooth transitions between certain geometric features (e.g., ellipses to polygons)? And considering the SDF precision, does it perform well in resolving sharp edges or high-curvature regions? How to avoid or decrease inaccuracies in boundary fitting.\n\n[1] Chadebec, C., & Allassonnière, S. (2022). A geometric perspective on variational autoencoders. Advances in Neural Information Processing Systems, 35, 19618-19630.\n\n[2] Vadgama, S., Tomczak, J. M., & Bekkers, E. J. (2022, November). Kendall shape-VAE: Learning shapes in a generative framework. In NeurIPS 2022 Workshop on Symmetry and Geometry in Neural Representations.\n\n[3] Kosiorek, A. R., Strathmann, H., Zoran, D., Moreno, P., Schneider, R., Mokrá, S., & Rezende, D. J. (2021, July). Nerf-vae: A geometry aware 3d scene generative model. In International conference on machine learning (pp. 5742-5752). PMLR.", "other_comments_or_suggestions": "Please see the above weaknesses part.", "questions_for_authors": "Please see the above weaknesses part.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "JTxaxQG0N0", "reviewer": "Reviewer_UvMW", "summary": "This paper designs a new learning-based approach for feature mapping based topology optimization. The major advantage of the proposed method against previous works is that the generated voids are guaranteed to be directly manufacturable, thus circumventing cumbersome post-processing procedures. Technically, the integration of Neural Heaviside SDFs with structured latent spaces addresses the limitation of traditional feature-mapping methods by enabling diverse geometric features, thus improving manufacturability. In general, this characteristic well aligns with actual industrial needs.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you very much for taking the time to provide such a detailed and insightful review. Your comment has been extremely valuable in enabling us to carefully analyze various aspects of our method's implementation, including the integration of the Kreisselmeier-Steinhauser (KS) function.\n\n***Using Sigmoid instead of Heaviside.*** The use of the term \"sigmoid\" is, of course, acceptable. However, the Heaviside function is a more traditional term in the context of topology optimization. We, like many other authors in this field, prefer to retain the possibility of using various smooth approximations of the Heaviside function without changing the name of the method.\n\n***Higher Degrees of Complexity.*** Adding new primitives forces an increase in latent space dimensions to keep accuracy, which can harm convergence. However, since our latent space captures overall contours rather than exact parameters, we believe that there's a threshold beyond which more dimensions are unnecessary. We will investigate this further.\n\n***Kreisselmeier-Steinhauser (KS) function.*** The current implementation includes several additional details that are not covered in the main text.\n\nIt is important for us to maintain sensitivity with respect to the objective function; however, to ensure the method does not fail, we are forced to clamp the $\\rho$ parameter because the KS function can produce values greater than 1. Therefore, in addition to equation (11), we use the following formula to scale $\\rho$:\n\n$$\n\\\\rho_e = \\\\frac{1 - \\\\rho_{\\\\min}}{KS_{\\\\max} - KS_{\\\\min}} \\\\left(KS_{\\\\max} - \\\\frac{\\\\ln \\\\sum_{m=1}^{M} \\\\exp(\\\\gamma_{KS} \\\\widetilde{H_{m,e}})}{\\\\gamma_{KS}}\\\\right) + \\\\rho_{\\\\min}\n$$\n\nwhere $ KS_{\\min} $ is the minimum value of the KS function, which is equal to\n\n$$\n    KS_{\\min} = \\\\frac{\\\\ln P}{\\\\gamma_{KS}}\n$$\n\nand $ KS_{\\max} $ is the maximum value of the KS function, which is equal to\n\n$$\n    KS_{\\\\max} = \\\\frac{\\\\ln (P\\\\exp(\\\\gamma_{KS}))}{\\\\gamma_{KS}}\n$$\n\nwhere $P$ is the expected maximum number of geometric primitives intersecting at one point (by default, this is 2).\n\nTherefore, the combined $\\rho$ will exceed the range $[\\rho_{min}, 1]$ only at intersections where more than $P$ geometric primitives overlap. In these cases, $\\rho$ is clamped to the range and becomes insensitive to the objective function.\n\nIn our case, the primary shape for topology is a non-overlapping primitive within which a void must form. For large values of P but small values of $\\gamma_{KS}$, the value of $\\rho$ inside the primitive becomes considerably higher than $\\rho_{min}$. Therefore, we choose $\\gamma_{KS}$ to be as large as possible, while ensuring that computations do not become impractical due to excessively large values of $\\exp(\\gamma_{KS})$.\n\nChanging $\\gamma_{KS}$ above 10 has little impact on convergence or final topology, whereas lower values worsen convergence because $\\rho_e$ remains too high for void formation.\n\n$\\gamma_{ks}$ | 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80\n----- | ----- | ----- | ----- | ----- | ----- | ----- | ----- | -----\nvf | 0.373 | 0.358 | 0.347 | 0.355 | 0.347 | 0.344 | 0.357 | 0.354\nC | 0.00238 | 0.00212 | 0.00209 | 0.00207 | 0.00210 | 0.00213 | 0.00203 | **0.00198**\n$min(\\rho)$ | 0.00067 | 0.00105 | 0.00504 | 0.00089 | 0.00116 | 0.00163 | 0.00623 | 0.00103\n$max(\\rho)$ | 0.87418 | 0.93682 | 0.95769 | 0.96813 | 0.97439 | 0.97857 | 0.98155 | 0.98379\n\nTable: experiments with different $\\gamma_{ks}$ values for Example 3: Bracket\n\n***B-spline and Bézier-based Feature-Mapping Methods.*** Unfortunately, most published works in topology optimization come without accompanying code and do not always provide enough details for reproduction. However, we have the opportunity to compare our method with a recently published approach [1], where the main geometric feature is a polygon constructed from half-spaces. These comparisons will be added to the manuscript.\n\n[1] K. Padhy, R., Thombre, P., Suresh, K. et al. Treetop: topology optimization using constructive solid geometry trees.\n\n| Method | Method type | $\\text{vonMises}_{max}$ |  Compliance | Volume Fraction |\n| :--- | :---: | :---: | :---: | :---: |\n| SIMP | Free-form | $\\textbf{0.483}$ | $\\textbf{0.00125}$ | 0.44 |\n| NTopo | Free-form | 2.52 | 0.00163 | 0.438 |\n| TreeTOp | FMTO | 6.08 | 0.00373 | 0.455 |\n| Ellipses | FMTO | 0.607 | 0.00174 | 0.449 |\n| NeuralHeavisideSDF | FMTO | $\\textbf{0.522}$ | $\\textbf{0.00163}$ | 0.437 |\n\nTable: Methods comparison for Example 3: MBB beam half\n\n$\\textbf{Note: }$ Since the NTopo method is quite limited in altering the parameters of the initial conditions, we had to adjust the parameters of the other methods to match those of the NTopo method. Therefore, the metric results differ from those in the original manuscript.\n\nOur approach achieves the best Compliance metric values among all FMTO methods while using less material. Additionally, in some experiments, our method is comparable to the NTopo method (see Table with Example 3).", "claims_and_evidence": "Partially.", "methods_and_evaluation_criteria": "Yes.", "theoretical_claims": "Yes.", "experimental_designs_or_analyses": "Yes.", "relation_to_broader_scientific_literature": "Well related to prior studies on topology optimization and geometry processing.", "essential_references_not_discussed": "No.", "other_strengths_and_weaknesses": "As presented in Eq. (8), the Sigmoid function tuned by \\beta is used as a soft approximation of the Heaviside function. I think throughout the paper it would be more straightforward to directly use the description of \"Sigmoid\", instead of \"Heaviside\".\n\nEmpirically, the authors experimented with examples of ellipses. triangles, and quadrilaterals. I am wondering if more types of geometric primitives with higher degrees of complexities can be included for evaluation.\n\nEq. (12) introduces Kreisselmeier-Steinhauser (KS) function for smooth max approximation, but sensitivity to the smoothing parameter γ_KS is not analyzed. How does γ_KS affect convergence?\n\nThe ellipse-based baseline is limited. Why not compare against B-spline or Bézier-based feature-mapping methods?", "other_comments_or_suggestions": "N/A", "questions_for_authors": "N/A", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "DQ6aVNJPzb", "reviewer": "Reviewer_ZcDR", "summary": "This paper presents a novel deep constrained topology optimization algorithm. Using the SIMP formalism, the authors propose learning an encoding of a space of fabricable shapes and solve the optimization problem on this space using gradient-based methods.\n\n## update after rebuttal\nIn light of these comparisons and clarifications (and on the assumption that they all are included in the updated manuscript), I am raising my score. I hope this paper gets presented at ICML!", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your thorough and insightful review! We are grateful for your detailed comments, which have not only highlighted the strengths of our methodology but also pointed out key areas for improvement, ultimately helping us enhance the clarity and impact of our research.\n\n***Refactoring mechanism.*** \nDuring topology optimization, reconstructed geometry can significantly deviate from the Heaviside function boundaries because the latent representation $ Z $ extends beyond the learned shape distributions. To address this, we initially reprojected latent vectors into the correct regions using an additional objective function, but this hindered convergence. Our refactoring mechanism instead steers latent vectors back to their learned distribution and potentially allows for geometric corrections during optimization, such as enforcing minimum radii or correcting overlapping arcs.\n\nOf course, we did not try every possible variant that might address the shortcomings of our method. In the future, we plan to explore alternative approaches to improve the algorithm.\n\n***Heaviside-“SDF” representation.*** We chose to use the Heaviside-SDF representation instead of a direct SDF during model training for the following reasons:\n\n1. The Heaviside transformation produces a sharper, more distinct boundary—exactly what we need for our application.\n\n2. This transformation enhances FMTO's robustness to neural network noise. Using a direct SDF requires applying the Heaviside function later, which significantly amplifies boundary noise occurring in the model.\n\n***Motivation for the method.*** \nOur method is motivated by the common industry practice of creating parts by extruding sketches—typically polygons with rounded corners—for milling and casting. We developed a framework that allows polygons to evolve from an initial circle during optimization, producing more natural and manufacturable designs. Additionally, we plan to extend the framework to include other common geometric features, such as polygons with arc segments.\n\n***Essential References.*** Thank you for the note. These directions are indeed closely related to our work, and we would be happy to include them in our review.\n\n***Comparison with other methods.*** We can provide more comparisons with other methods, including  one recently published work $\\texttt{TreeTOp}$ [1], which directly relates to FMTO.\n\n1. K. Padhy, R., Thombre, P., Suresh, K. et al. Treetop: topology optimization using constructive solid geometry trees. 2025.\n\n| Method | Method type | $\\text{vonMises}_{max}$ |  Compliance | Volume Fraction |\n| :--- | :---: | :---: | :---: | :---: |\n| SIMP | Free-form | $\\textbf{0.483}$ | $\\textbf{0.00125}$ | 0.44 |\n| NTopo | Free-form | 2.52 | 0.00163 | 0.438 |\n| TreeTOp | FMTO | 6.08 | 0.00373 | 0.455 |\n| Ellipses | FMTO | 0.607 | 0.00174 | 0.449 |\n| NeuralHeavisideSDF | FMTO | $\\textbf{0.522}$ | $\\textbf{0.00163}$ | 0.437 |\n\nTable: Methods comparison for Example 3: MBB beam half\n\n| Method | Method type | $\\text{vonMises}_{max}$ |  Compliance | Volume Fraction |\n| :--- | :---: | :---: | :---: | :---: |\n| SIMP | Free-form | $\\textbf{0.194}$ | $\\textbf{0.000104}$ | 0.34 |\n| NTopo | Free-form | 0.349 | 0.000107 | 0.339 |\n| TreeTOp | FMTO | 43.8 | 0.000155 | 0.357 |\n| Ellipses | FMTO | 0.674 | 0.000155 | 0.345 |\n| NeuralHeavisideSDF | FMTO | $\\textbf{0.575}$ | $\\textbf{0.000152}$ | 0.337 |\n\nTable: Methods comparison for Example 4: Beam Distributed Load\n\n$\\textbf{Note:}$ Since the NTopo method is quite limited in altering the parameters of the initial conditions, we had to adjust the parameters of the other methods to match those of the NTopo method. Therefore, the metric results differ from those in the original manuscript.\n\nOur approach achieves the best Compliance metric values among all FMTO methods while using less material. Additionally, in some experiments, our method is comparable to the NTopo method (see Example 3).\n\n***“Cantilever” is misspelled, incorrect citation, “linear elasticity” keyword.*** Thank you for pointing this out. We will correct the spelling error, add the “linear elasticity” keyword, and fix the citation.\n\n***What is $\\xi_e$ and how does it differ from $\\xi$ in Eqs. (7-8).*** $\\xi$ represents any point within the design domain and is used to train the model. In contrast, $\\xi_e$ denotes the center point of element $e$.\n\n***Why are there 6 entries in $\\xi$ in Fig. 3? Isn’t $\\xi$ a (x,y,z) coordinate?*** We apologize for the confusion caused by our previous depiction of a batch size of 3. We have updated the scheme to display a batch size of 1. Our experiments focus on 2D problems, so only two coordinates (x, y) are used.\n\n***Shape code $X$.*** We apologize for the missing definitions. Inconsistencies in the method description have been corrected, and an updated version—including details on the shape code $\\chi$—will appear in the main text, as well as a revised Figure 4. Additionally, a table of notations will be added to the appendix.", "claims_and_evidence": "All claims made in the paper are thoroughly supported. It should be noted that these claims are measured: for example, the authors limit themselves to exploring several possible new frameworks for topology optimization, without necessarily claiming its superiority with respect to other works.", "methods_and_evaluation_criteria": "The authors evaluate their method on a broad range of examples, which are described in the supplemental. They even include code in their supplemental material that reproduces most results.\n\nIn general, I was surprised that the authors did not evaluate and ablate their algorithmic choices more thoroughly. For example, this paper relies on encoding the shape code X into a (smaller) latent space Z, and then optimizes the code Z according to the physical problem using projected gradient descent, effectively ensuring that the optimal Z is the image through the encoding of a valid X. A question that may occur to a reader is: why is this encoding necessary? Can’t one simply do projected gradient descent on X, projecting (or “refactoring”, as said in the paper) X into the set of possible X after every 5 iterations? Given the encoding adds a lot of the complexity in the algorithm, I would have expected to have a clear answer to this question in the text or experiments, but I missed it.\n\nSimilarly, the paper uses a Heaviside-“SDF” representation for the primitives. This choice feels somewhat ad-hoc, justified by saying “[it] is convenient for the optimization process and for training the neural network”. I wished the authors either elaborated on this, or compared to other strategies like learning the non-Heaviside SDF directly (in fact, isn’t the Heaviside loss in Eq. 9 the same as an SDF loss with anisotropic sampling of B?).", "theoretical_claims": "See above", "experimental_designs_or_analyses": "See above", "relation_to_broader_scientific_literature": "The paper does not (nor does it claim or intend to) include a full survey of work in topology optimization, a large field of research with decades of work. Instead, it focuses on deep feature-mapping methods for topology optimization and, as far as I can tell, does a good job of covering these.\n\nNonetheless, I am somewhat disappointed that the manuscript does not attempt to place this algorithm in context with other topology optimization strategies. Given that this is a problem considered in different fields of research and with many interested practicioners, I would have expected this paper to clearly answer the question “how is this algorithm better or worse than existing topology optimization methods?”, i.e., “if I were a manufacturing company, when if ever do I wish to use this method as opposed to others?” This can be answered in text or in experimentation.", "essential_references_not_discussed": "Like I said above, the authors are not expected to perform a full survey of this topic, but I would have appreciated including works from the engineering community (e.g., “Multiscale structural topology optimization with an approximate constitutive model for local material microstructure”) or the graphics one (e.g., “Two-Scale Topology Optimization with Microstructures”).", "other_strengths_and_weaknesses": "The main strength of the paper is its novelty: using learned latent spaces of shapes as a way of guiding topology optimization to fabricable shapes is a good idea, and this is a relatively good execution of it.\n\nAs said above, I believe the main weakness in this work is the lack of comparison to prior art (especially for a task as crowded as this), as well as the lack of experiments evaluating the algorithmic choices. These are the only reasons I am not more excited about accepting this work, although I will carefully read and review the authors’ response and other reviewer comments.", "other_comments_or_suggestions": "“Cantilever” is misspelled in C.1, and the citation “Maz’e” should be “Mazé”.", "questions_for_authors": "- Can the authors please clarify the dependencies of each variable on rho in Sec 3.2.?\n- What is \\xi_e and how does it differ from \\xi in Eqs. (7-8)\n- Why are there 6 entries in \\xi in Fig. 3? Isn’t \\xi a (x,y,z) coordinate?\n- The shape code X is never introduced, defined or explained beyond Figure 4 (which appears a full page after X is used in the text). Perhaps front loading the definition of X would be useful to a reader?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: This paper proposes a new method to simulate Feature-Mapping Topology Optimization (FMTO). Through incorporating a VAE-based heaviside function decoder, the method achieves superior practical solution compared with existing FMTO-type methods. Most reviewers are positive about the paper. Although some concerns were raised regarding novelty of the approach, the introduced techniques seem to be well-tailored for the application in a nontrivial way, thus could be considered a valid method contribution. Different concerns regarding writing, experiments, comparison with free-form methods, approximation of fine-detailed boundaries were raised, but were addressed satisfactorily during the rebuttal.
Decision: Accept (poster)