3DID: Direct 3D Inverse Design for Aerodynamics with Physics-Aware Optimization
Authors: Yuze Hao
Linchao Zhu
Yi Yang
Venue: NeurIPS 2025 poster ()
Keywords: Inverse Design
Aerodynamic Shape Optimization
Primary area: machine_learning_for_sciences

Abstract: Inverse design aims to design the input variables of a physical system to optimize
a specified objective function, typically formulated as a search or optimization
problem. However, in 3D domains, the design space grows exponentially, rendering
exhaustive grid-based searches infeasible. Recent advances in deep learning have
accelerated inverse design by providing powerful generative priors and differentiable surrogate models. Nevertheless, current methods tend to approximate the
3D design space using 2D projections or fine-tune existing 3D shapes. These
approaches sacrifice volumetric detail and constrain design exploration, preventing
true 3D design from scratch. In this paper, we propose a 3D Inverse Design (3DID)
framework that directly navigates the 3D design space by coupling a continuous
latent representation with a physics-aware optimization strategy. We first learn a
unified physics–geometry embedding that compactly captures shape and physical
field data in a continuous latent space. Then, we introduce a two-stage strategy to
perform physics-aware optimization. In the first stage, a gradient-guided diffusion
sampler explores the global latent manifold. In the second stage, an objectivedriven, topology-preserving refinement further sculpts each candidate toward the
target objective. This enables 3DID to generate high-fidelity 3D geometries, outperforming existing methods in both solution quality and design versatility.

3DID: Direct 3D Inverse Design for Aerodynamics with Physics-Aware Optimization
Yuze Hao', Linchao Zhu!?, Yi Yang!
College of Computer Science and Technology, Zhejiang University 2 The State Key Lab of Brain-Machine Intelligence, Zhejiang University
Abstract
Inverse design aims to design the input variables of physical system to optimize specified objective function, typically formulated search or optimization problem. However, in 3D domains, the design space grows exponentially, rendering exhaustive grid-based searches infeasible. Recent advances in deep learning have accelerated inverse design by providing powerful generative priors and differen- tiable surrogate models. Nevertheless, current methods tend approximate the 3D design space using 2D projections or fine-tune existing 3D shapes. These approaches sacrifice volumetric detail and constrain design exploration, preventing true 3D design from scratch. In this paper, we propose a 3D Inverse Design (3DID) framework that directly navigates the 3D design space by coupling a continuous latent representation with physics-aware optimization strategy. We first learn a unified physics- geoomerr embedding that compactly captures shape and physical field data in continuous latent space. Then, we introduce two-stage strategy perform physics-aware optimization. the first stage, a gradient- guidded diffusion sampler explores the global latent manifold. In the second stage, objective- driven,
refinement further sculpts each candidate toward the target objective. This enables 3DID to generate high-fidelit 3D geometries, out- performing existing methods in both solution quality and design versatility.
Introduction
Inverse design seeks to identify the initial variables of a physical system that, under given constraints, optimizes specified objective function. This fundamental challenge occurs across many scientific and engineering disciplines, such as materials science, mechanical engineering, aerospace design, and supports applications ranging from automotive body shaping [[] and nano-photonic device engineering 2 to mechanical materials design 4 and physics detector development [5].
Despite its broad impact, efficiently exploring the design space toward a target objective presents significant challenges. First, inverse design must contend with the inherent complexity of simulating physical systems for evaluation. These simulations are often nonlinear and high-dimensional, requir- ing fine discretizations that dominate computational resources (]. Second, the design landscape extremely large, inherently nonconvex, and riddled with local minima, making exhaustive global search infeasible [ 9] In 3D domains, where inverse design usually involves direct geometry optimization, the number of degrees of freedom grows exponentially [IO] This rapid growth in geometric complexity drives up simulation expense and intensifies the search challenge.
To tackle inverse design in the 3D domain, various techniques have been proposed [1] yet they fall short of addressing the above challenges. Traditional approaches such as adjoint-based gradient methods [13 [4 15) and Bayesian optimization [16] 17 provide broad applicability but depend
"Corresponiing author.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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repeated high-fidor fidh-fidelity simulations that incur pro- hibitive computational cost. With recent ad- vances in deep learning, pretrained surrogate models [8] 19 20 21 can efficiently approx- imate the forward physical process and support end-to-end backpropagation to update design variables, speeding up convergence by orders of magnitude. However, many prior methods adopt two simplifications. One replaces the 3D design with 2D proxies [ (multi-view ren- derings silhouettes), which removes geome- try information. The other requires an initial geometry the starting point for subsequent re- finement [22 23
3D Design Space
Optioall
Suboptimal
Ours
Random Initialization
Scarch
Provious
Space
Initial
Geometry
[24]. In practice, both assump- tions restrict design exploration and hinder thor- ough search of complex 3D design spaces (see Fig. D. As a result, they cannot support the full exploration of complex three-dimensional design spaces, limiting coverage to a narrow subset of feasible geometries.
Initial Projection
2D Projection Space
Figure I: Motivation of 3DID. Existing 3D inverse-design methods either rely on reduced- dimensional representations (2D projections or fixed parameterizations) that constrain design freedom, or require an initial geometry as start- ing point for local refinement, which highly con- strains the search space. In contrast, 3DID over- comes these limitations by directly exploring the full 3D design space from random initialization.
We identify two primary challenges in 3D inverse design. 1) The high dimensionality of 3D physics- geopetr spaces impedes exploration. Inverse design must simultaneously optimize ge- ometric structures while accurately evaluating their resulting physical properties. This coupling, combined with the continuous high-resolution na- ture of both shape and physical fields, makes the direct 3D exploration extraordinarily difficult. The lack of optimization strategies that balance trade-off. Refining a baseline geometry with surrogate model ensures
the exploration-valice constraint compliance and design validity, but it confines the search local neighborhood and can introduce adversarial artifacts when driven too far 225 26]. On the other hand, sampling candidates with a generative model offers broader exploration yet leaves results vulnerable biases in the training data (27]) Consequently, samples stay tethered to the prior and tend to imitate prevalent patterns rather than pushing toward novel optima.
To address these challenges, we introduce 3DID, a 3D inverse-design framework that explores the design space without relying on simplified parameterizations or predefined shapes. Rather than directly searching the prohibitively large, continuous physics-geomette coooetry-coupled space, we first Iearn continuous physics-geomee geomeeryy unified latent representation. This compact embedding preserves fine-grained shape and physical field variations while dramatically reducing both dimensionality and computational cost, thereby overcoming the dual obstacles of large-scale shape optimization and physics-aware simulation. Building on this latent space, we then deploy two-stage optimization pipeline tackle the opploration-validiyy trade-off. It begins with a gradient-guided diffusion sampler that traverses the manifold from pure noise to generate diverse, physics-ifformed candidates by steering sampling toward high-perit performance regions using objective gradients. Each candidate optimization, which further improves objective performance
then undergoes copology-preserve under strict mesh-quality and connectivity constraints, ensuring geometric integrity and preventing adversarial artifacts. Together, these components enable 3DID discover novel, high- high-fidelity 3D designs that reliably meet target objectives. In summary, our contributions are threcfold:
(1) We propose continuous latent embedding that jointly encodes detailed 3D geometry and high-fidelity physical fields, enabling an efficient, unified search within compact latent manifold.
(2) We develop a two-stage optimization pipeline that begins with gradient-guided diffusion sampling for global exploration and followed by oppolggy-preservin refinement, optimizing each candidate toward the desired objective while strictly maintaining structural integrity.

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(3) We validate our 3DID framework on aerodynamic shape optimization, demonstrating that it consistently generates novel geometries whose superior performance is confirmed through surrogate evaluations and high-fidelity CFD simulations, significantly surpassing baseline methods.
2 Related work
2.1 Inverse Design
Compared with the forward PDE problem, which predicts the physical response a given design
using numerical solvers or learned surrogates [28] 22)
[0)
31
32 33], inverse design seeks the
design variables that achieve a target objective under engineering constraints [26 18 [4]. Inverse
design is fundamental problem in many domains of science and engincering disciplines, including
mechanical engineering [35] 36), material science 37
38 39], chemical engineering [40], medical
Classical approaches typically combine high fidelity physics solvers with sampling- optimization methods such as the Cross Entropy Method 43 Gaussian-pree prrscesss model with Bayesian optimization [4 to explore the design space. With the advent of differentiable simulators, inverse design can be posed directly as a gradient- based optimization problem [4] 45). More recently, deep learning driven methods have shown great promise by learning surrogate models that approximate forward physics and allow end-to-end Furthermore, generative models, including variational autoencoders (46), GAN [47] 48) and diffusion models [26] 49] have been applied to inverse design. While prior methods mainly excel in 2D or oow-dimensional settings, we propose a framework that directly navigates the full 3D design space via physics-aware optimization.
engineering [11] and aerospace engineering
22
6
42
backpropagation [8 19
2.2 Aerodynamic Shape Optimization
Aerodynamic shape optimization is classical inverse design task that seeks geometries minimizing drag while satisfying constraints on lift, stability, and other performance criteria [50 51
52 53].
Generally, effective optimization critically depends on two key components: shape representation and optimization strategy. Traditional approaches typically employ simplified, low-dimensional representations such as 2D projections
[II 54 I] or spline-based parameterizations [55] 56 57] to
reduce dimensionality and computational costs. Optimization is then performed using gradient-bant based accelerate convergence, many methods optimize from a pre-selected baseline geometry [22 23 24). In contrast, we propose guided diffusion model over latent shape representation, enabling the design of unconstrained 3D
adjoint solvers for efficient local refinement 24
58 59 Additionally,
geometries directly from noise, without relying on initial shapes or 2D profiles.
Methods
In this section, we first formalize the 3D inverse design problem (Section 3.1 We then introduce our phomeesygeometry unified representation (Section 3.2 describe the gradient-guides diffusion refinement stage (Section 3.4.
sampling process (Section 3.3), and detail the Finally, we provide our implementation details (Section
3.1 Problem Formulation
We consider the problem of 3D inverse design, where the goal is to identify a solid input geometry M for physical system that optimizes specified performance objectives while satisfying geometric constraints. Formally, let M C E denote a solid geometry, and let F(M) be the corresponding steady-state physical field (c.g., pressure temperature distribution) governed by partial differential equation (PDE) or an ordinary differential equation (ODE). We define the design objective as:
[M) = (M, (MM).
which may measure quantities such as drag, lift, and structural compliance. Specifically, J depends on M in two ways: implicitly, via the resulting physical field F(M) on which J is evaluated, and explicitly, via direct geometric properties defined on M. The classical inverse design aims to solve:
M' = arg min J (M,F(M) M
S.L. C F(M) 50,
3

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Physics geomeetty Encoder
Mapping Network
Cross Attention
Salf Attention
C Self Astention
MUB
MII
Leurnable Tokens
Point Cloud
Occupancy Field
Decoder
Physical
Cross Attenion
Self Attentio
Self Attention
Leamable Tokens
XN
MLP
Reshago Sample
Latent z
Physicul Field
The overview of PG- We use transformers to encode the design geometry and its associated physical field, along with learnable tokens, into a compact triplane latent representation A decoder then upsamples the latent into high-resolution triplane feature maps, which can be reshaped into three orthogonal planes. Finally, physics geometry mapping network is applied to reconstruct both the occupancy field and the corresponding physical field from these feature maps.
Figure 2:
where the solution M is the geometry that minimizes the performance objective, C' aggregates design constraints, such as volume, manufacturability, and boundary conditions.
3.2 Physics-Geometry Unified Representation
To compactly encode 3D geometry and its physical field, we adopt triplane representation learned via our Physics-Geometry VAE (PG-VAE). As shown in Figure it includes: (1) A physics-geoment encoder that maps input geometry and physical field into a latent code. (2) latent-to-triplane
decoder that reconstructs triplane feature maps from the latent code. (3) A physics-geomenty mapping network that recovers the 3D geometry and physical field from the triplane.
Physics-Geometry Encoder. The physics-geoment encoder comprises two parallel branches: one for processing raw 3D geometry, and the other for encoding the associated physical field. Inspired by (60], each branch uses learnable tokens to capture both local and global context. For the geometry branch, given uniformly sampled point clouds Pgeo N from the geometry, where N, the number of points and C. represents features including normalized positions and surface normals, we utilize Fourier features [6]] to encode positional structure. Then, a set of learnable tokens Eg [(3xxxr)xxd queries information from these points via cross-attention and self-attention layers, resulting in geometry latent tokens 2g ((3xxx))xd, The physical-fielis branch follows the same structure, processing uniformly sampled physical-field points Pohy E R&N,xx,, where N, is the number of points and Cn is the dimension of physical-fiele features. Learnable tokens ep E ((3xrx))xd, undergo similar attention layers to produce physical-fiee -ieelll-ield latents 2p E p(3xxxr)xd, Finally, geometry and physical tokens are concatenated and passed through MI.P layers to form the unified latent representation z MLP(Concat(zg)) zp)), where E (3xrxx)xxd;
Latent-to- Triplani Decoder. After obtaining the unified physics physics-geometry latent representation, apply decoder to formulate the triplane representation. Prior to decoding, reshape the latent tokens by vertically concatenating three planes, yielding the reshaped latent tensor E (3xx(3xr)xxx following [62]. Subsequently, the latent tensor is passed through a series of convolutional layers for upsampling. The output is then reshaped into the final triplane features Tzy, Tr. Tye E ftxd, where denotes the resolution of each plane and de is the feature dimension per pixel.
Physics-Geometry Mapping Network. The mapping network serves to reconstruct 3D geometry and the associated physical field from the learned triplane representation. We utilize two parallel MLP branches: one for predicting geometric occupancy and one for estimating the physical field. Given query point E R-, we project onto the three orthogonal planes and extract features. The aggregated feature is computed as 4, Tay (Umy) + (T=)()n)) Typ (4yz). Ty=(4yz). where Qrys Urz, Qyz denote the 2D projections of q onto the respective planes. The aggregated feature by is then fed into the ML. branches to predict the occupancy field and physical field values at the corresponding point q.

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Stage 1: Initial Design Generation
Stage 2:
Topology Preserving Refinement
Control Point Wrapping
Diffission Process
Point Cloud& Physical Field
Zo
Initial Mesh Mi
A
V
1 Times
To Times
Grad
XVV
D:T
CNN
Occupanccy Ficluk Surface Field
Optimize
Retined Mesh
The optimization framework of 3DID. Starting from noise, we guide diffusion using objective gradients steer the latent toward high-performance regions. The decoded triplane then yields an initial mesh Mo and its surface physical field s, which is then refined via backpropagation free-form deformation lattice improve performance while preserving topology.
Figure 3:
End-to-End Training. Our VAE model is trained end-to-end to jointly reconstruct 3D geometry and the associated physical field. For occupancy field reconstruction, we employ the Binary Cross- Entropy (BCE) loss LICE to supervise the predicted occupancy. To reconstruct the physical field, utilize the Mean Squared Error (MSE) loss LMSE. Additionally, we incorporate a KI. divergence loss fKL to regularize the latent space. Overall, training loss be formulated as:
LPG-VAE λBCECBCE + AMSE AKL
where ABCE, AMSE, λKI. are weighting coefficients.
3.3
Objective Guided Diffusion
Once the PG- VAE is trained, it provides a compact, expressive latent code & that jointly captures 3D geometry and its physical field. We then train diffusion model 663 64 on these latents, enabling direct generation of samples on the learned manifold from pure noise. To drive inverse design, we inject gradients of the task objective T into the diffusion sampling, as shown in Figure 3
In standard diffusion sampling, each denoising step predicts noise via the learned score function:
<0(=,, - -1 ar V log)(>;),
where V., log p(2)) denotes the score function, i.e., the gradient of the log probability density of the latent variable 2g. By iteratively denoising, the model guides samples toward high- probability regions of the data manifold. In our data manifold, but we also need to incorporate optimization of I during the sampling. Therefore, inspired by [65] 66), we replace the unconditional score with the conditional score V., log (T). By Bayes' rule, we can derive:
need to consider not only guiding the noise towards the feasible
Vx, log (J) x V., logp(>;) + V P(T xt).
(5)
Here, Vz, log p(2)) corresponds to the standard score function learned by the diffusion model, while V., log P(J 2 acts as an additional guidance term that incorporates the influence of the design objective. Since Vz, log [J] z²) is unknown, approximate it by:
Vz, logp(T
2x)) = Va, logr(T {=())) a - - (,,o((()))
where 2n(2;) denotes the estimate of the clean latent code given the noisy latent 2, following [66 Accordingly, we adjust the predicted noise to incorporate the influence of the design objective J. resulting in the guided noise prediction:
<², = (Z(, +V.
where e' is the modified noise prediction, and is a scaling coefficient that controls the strength of the guidance. This objective-aware adjustment steers the sampling trajectory toward latent regions that both conform to the learned data distribution and advance the target design objective J.
5

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3.4
Refinement
After objective-guiee guided diffusion, we obtain an optimized latent code 2", which is reshaped into triplane feature maps and decoded by the tryysics-gemmaee mapping network to generate an initial 3D mesh Mo with vertex set Vo- {v, 1=) and its associated physical field 6p = { as shown in Figure [ Although guided by the design objective, the generated designs remain highly biased by the prior distribution of designs from the training data [27]. We introduce a topology-pree prrrogrriggeggngggggg controlled by gradient descent.
refinement stage based on free-form deformation (FFD) 67 68
Specifically, we first wrap Mo in a 3D lattice of control points C' ) These control points form a flexible control grid that allows smooth and structured adjustment of the mesh shape while preserving its topology. The deformation of each vertex Uj is computed as:
K
5'( - _ C..
(8)
1=1
where B,(z)) denotes the i-th trivariate Bernstein basis function 668] evaluated at the normalized parametric coordinate of vertex Uy. These basis functions provide smooth, localized influence from the control points, enabling flexible yet coherent deformation across the mesh.
At the beginning of the refinement process, the control points are unmodified, so (C) uy. The initial vertex-field field pairs {(v) (C), 9j)}} are then fed into pretrained GNN surrogate faNN which estimates the current design objective based on the mesh geometry and physical attributes:
O(C) = JGNN (1))()),))))))
With this differentiable surrogate model, we optimize the control points to improve the design objective. The overall refinement loss is defined as:
K
(C)==((C) + Ammooth Σ 1| Avol
Σ cells
Vaef
1)²
(10)
Vorig
where Ac are control-point displacements, and the term weighted by Asmooth penalizes large displacements for smooth deformations, while the term weighted by Avol penalizes cell-wise volume changes to ensure geometric consistency. Control points are updated via:
( = c (((())
(11)
We optimize using AdamW with a cosine-annealed learning rate n,. Iteration continues until an iteration count To is reached. The resulting vertices V = {of }. obtained via the FFD mapping (Eq. 8) define the refined mesh M which preserves topology and improves target performance.
3.5
Implementation Details
To train our PG-VAE we sample N₁. = NN, = 50,000 points as input and use the physics-gcomett
encoder with one cross-attention layer and 8 self-attention layers with 12 heads and d, = 64, plus
r = 64 learnable tokens of dimension de = 768, yielding a latent code of d₂ = 32. The decoder upsamples via one self-attention layer and five ResNet blocks [69] triplane with R 256 and channel d, = 64. Each branch's 's mapping network has five linear layers with a hidden dimension of 32. We train the VAE model with loss weights ABCE 10-3, AMSE 10-5, AKt. = 10- During training, we sample 50,000 points from the unit domain to supervise both occupancy and physical-field predictions. For occupancy, we adopt the semi-contimuous formulation following (60] We use a learning rate of 1e - 4, batch size of 8 per GPU, and train for 100K steps. For the diffusion model, we employ 10 layers of DiT blocks 77).. each with 16 attention heads of dimension 72. We train the diffusion model with 1000 denoising steps. For objective-guider
sampling, an auxiliary U-Net surrogate predicts the task objective directly in latent code z. To train the diffusion model, we use a learning rate of 5e 5, batch size of 4 per GPU, and train for 300K. In topology-preseee refinement, we deform candidates via a 20x6x6 control-point grid along the x, y, and axes. For the surrogate model faNN, we adopt MeshGraphNet [30 as our surrogate given its strong performance in mesh-based physical simulations. The surrogate is trained to predict aerodynamic drag from paired samples of geometry and ground-truth physical fields collected from the dataset. The model is trained with learning rate of le - 5, batch size of per GPU, and trained for 100K. All models are trained with AdamW optimizer (7I]. More training details of 3DID are included in the Appendix.

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Table 1: Quantitative comparison for aerodynamic vehicle design. The confidence interval information is detailed in the Appendix. Note that our method shows slight drop in coverage. mainly because the copology-preserving refinement pushes designs beyond the original distribution achieve better aerodynamic performance.
Method
Pred-Dragl
Sim-Drag.
Noveltyt
Coveraget
GP. Voxel
0.2997
0,4254
1.0399
0,5200
GP, VoxeI+PCA
0,3059
0,4363
0,9734
0,5850
CEM, Voxel
0.2951
0,4097
0,9792
0,4350
CEM, Voxel+PCA
0,3088
0,4393
0,9864
0,5100
CEM, TripNet
0.3154
0.4161
1.0399
0.6050
Backprop, Voxel
0,2979
0,4146
0,9860
0,4750
Backprop. Voxel+PCA
0,3061
0.4614
0.9798
0.4950
Backprop, TripNet
0.3153
0,4170
1.0294
0.5900
3DID-NoTopoRefine (ours)
0.2623
0,3766
0.9195
0,6950
3DID (ours)
0.2607
0.3536
1.1709
0,4300
Experiments
In the experiments, we aim to answer the following questions: (1) Does 3DID outperform traditional, sampling-basedd and backpropagation methods in finding high-quality designs? (2) Does our unified Mmysics-geomest triplane representation yield better objectives than alternative latent or purely geometric embeddings? (3) Does our two-stage pipeline outperform single-stage diffusion sampling and other standard optimization methods? answer these questions, we evaluate our method on the vehicle aerodynamic shape optimization task, representative example of 3D inverse design.
In the following sections, we first introduce our dataset and evaluation metrics (Section 4.1p. Next, we describe our experimental setup and compare against baseline methods (Section 4.2). Finally, we present two ablation studies: one the unified physics-geometry representation (Section and one on the two-stage optimization strategy (Section
4.1 Dataset and Evaluation Details
Dataset. We conduct our experiments on the Driv-ArNee++ dataset [72] 73), the largest available collection for acrodynamic car design, comprising over 8,000 diverse geometries paired with high- fidelity CFD simulations. For training, we use the entire dataset. We first normalize each geometry to fit within unit cube and apply the same scaling the simulation fields. We then uniformly sample surface point clouds with corresponding normals from the geometry. Finally, for both the occupancy field and the physical field, adopt the data extraction strategy of Park el al. (74), using grid resolution of 256. Further details of our data preparation are given in the Appendix.
Evaluation Metrics. We evaluate 3DID using four metrics. Pred-Drag is the drag coefficient estimated by a trained surrogate model, offering an approximation of the design objective. Sim-Drag is the drag coefficient obtained via high-fidelity CFD simulation, delivering unbiased evaluation of acrodynamic performance. Novelty computes the average nearest neighbor distance from each generated design to its closest training example, indicating how distinct the designs from existing ones. Coverage captures how well the generated designs cover the training distribution by measuring, for each training sample, the distance its nearest generated design (using k-nearest neighbor lookup) and reporting the fraction of training examples that fall within a predefined threshold. To extract features from the generated geometries, we use the pretrained PointNet model from [75 Detailed evaluation procedures and simulation parameters are provided in the Appendix.
4.2 3D Vehicle Aerodynamic Design
In this experiment, we evaluate each inverse-design method on the aerodynamic shape optimization task, where the objective I is to reduce the drag force of the designed vehicles. All methods are trained on the same collection of car geometries paired with high-fidelity CFD simulations. As base- lines, compare against the Cross-Entropy MCtM) (43. the Gaussian-procee process surrogate with
7

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1,42199
0.5067
0.4716
0.3926
0.4404
0.43044
0.3844
0.3607
Voxel
Voxel + PCA
TripNet
Ours
Figure 4: Qualitative comparisons of different representations. Each row shows four candidates with geometry (left) and simulated velocity field (right) with Sim-Drag in the top-right. Despite equal resolution, voxel methods incur higher drag and often yield non-watertight shapes (red box) due to coarse discretization. Our continuous latent representation produces watertight, smooth designs with superior aerodynamic performance, outperforming both voxel-basee and geometry-only approaches.
Table 2: Ablation study representation choices.
Method
Pred-Drag
Sim-Drage
Noveltyt
Coverage
Voxel
0.2722
0.4318
1.0683
0.3450
VoxeI+PCA
0.2720
0.4565
0.9858
0.5750
TripNet
0.2698
0.4066
1.0580
0.5500
3DID-No HoID-NoTopoRetine (ours)
0.2623
0.3766
0.9195
0.6950
3DID (ours)
0.2607
0.3536
1.1709
0.4300
Bayesian optimization GP) and the gradient-based backpropagation methood(Backprop) [8] To evaluate the impact 3D encoding, the optimizer is instantiated with three representations:
dense voxel grid [76). PCA-compressed voxel grid (VoxeI+PCA) [22]. and pure geometry triplane network (TripNet) [21]. GP with the triplane representation omitted due to its high computational cost. For fairness, generate 64 candidate designs per method and report the average performance in Table I Architectures baselines and training details are provided in the Appendix.
From T it can be observed that 3DID delivers the best drag force result for both pred-drag and sim-drag compared to all baselines. Specifically, our full 3DID model reduces simulated drag by 13.6% relative to the strongest baseline. These results demonstrate the effectiveness of our pipeline in discovering nigh-performance designs. Furthermore, our method achieves the highest novelty score (1.1709), indicating its ability explore diverse design variations. Note that the drop in coverage occurs because topology preserving refinement pushes designs beyond the training distribution to boost acrodynamic performance. A detailed ablation the cascade optimization strategy is presented in Section 4.4 More visualization results and evaluation are presented in the Appendix.
4.3 Ablation Study on
Geometry Unified Representation
In this experiment, we evaluate the performance with different representations, including Voxel 76 Voxel with PCA [22] and the pure geometry triplane (TripNet) [21]. For Voxel and Voxel with PCA, we first train a variational autoencoder (VAE) to embed the raw data into a compact latent space and then learn diffusion model within that space. Finally, we employ our two-stage optimization pipeline consisting of gradient- guided diffusion sampling followed by to generate diverse design candidates. Because the baseline representations do not include physical field information, we retrain a surrogate graph neural network that takes only geometry as input for the refinement stage. The results are reported in Table 2
refinement
As shown Table 2 our 3DID method outperforms all baselines by wide margin in both simulation drag and novelty. Compared to the best baseline TripNet, 3DID reduces Sim Drag from 0.4066 0.3536, 13.0% improvement, and lowers Pred Drag by 3.4% (from 0.2698 to 0.2607). It also increases novelty from 1.0683 to 1.1709, a 9.6% gain. In Figure our continuous latent representation consistently yields watertight smooth geometries with superior aerodynamic performance, whereas voxel-based methods suffer higher drag and non-watertight artifacts. Although TripNet embeds continuous geometry, its optimized designs remain inferior. We attribute this to the absence of physical field guidance, which weakens the optimization gradients in the refinement stage.
8

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0.3681
0.3607
0.3430
0.3342
0.3227
0.3109
0.4162
0.4102
0.3622
0.3545
0.3408
0.3387
0.3755
0.3482
0.3162
0.3122
Before Refinement
After Refinement
Before Refinement
After Refinement
Figure 5: Qualitative comparisons of hapology-preservve refinement. Each presents two design candidates comparisons with their geometry and simulated velocity field heatmaps. Sim-Drag values shown in the top-right comner of each panel. Refined designs exhibit more significant fastback profile (yellow box), reduced low-velocity recirculation zones (blue box), and stronger downward flow (green box), indicating improved aerodynamic performance.
Table 3: Ablation study on design strategies.
Method
Pred-Drag
Sim-Drag.
Noveltyt
Coveraget
CEM GD
0,3152
0.3987
1.0730
0.6800
0.3023
0.4095
1.0878
0.5800
W/O Guidance
0.2971
0.3944
0.9177
0.7104
3DID NoToppRefine (ours)
0.2623
0.3766
0.9195
0.6950
3DID (ours)
0.2607
0.3536
1.1709
0.4300
4.4 Ablation Study on Two-Stage Optimization Strategy
In this experiment, we validate the effectiveness of our two-stage optimization pipeline by comparing against two alternative design strategies, including Cross-Entropy Method (CEM) and gradient descent (GD), as well as a diffusion-only sampling approach without objective gradient guidance. All methods are based our physics-geometry unified representation.
From Table 3. we see that our full 3DID outperforms all baselines in Pred-Drag. Sim-Drag, and Novelty. Notably, when designing without guidance, diffusion model attains the highest coverage value of 0.7104, as captures the data manifold comprehensively. When generating without guidance, the diffusion model tends to mimic the distribution of the dataset, which leads coverage but lacks the targeted optimization for aerodynamic performance and novelty that our 3DID method provides. Furthermore, better validate the effectiveness of our refinement stage, we provide by-idb qualitative comparisons in Figure It can be seen that after the refinement stage, each candidate develops more significant fastback profile, with diminished low-velocity recirculation regions, and stronger downward flow patterns, all indicators of improved aerodynamic performance as confirmed by the reduced Sim-Drag values.
increase of
5 Limitations
Limited to static physical fields. Despite the fact that 3DID achieves impressive results, significant limitation its focus on static fields. The current framework does not support inverse design involving dime-dependent or dynamic physical fields. Time-dependent physical systems often involve solid geometries coupled with evolving physical properties over time. This would pose challenges for representation and optimization within our framework. Enhancing 3DID with time- aware representations and models may address these limitations, which we leave as an important direction for future work.

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Limited to single objective optimization. In 3DID, we address the inverse problem with a single objective, which may limit applicability for broader scenarios. Although is straightforward to aggregate multiple objectives into a single composite loss, this approach may overlook potential con- flicts and trade-offs between objectives. Extending 3DID support true multi-objective optimization is a promising direction for future research.
Limited to surrogate- physies awareness. We incorporate physical fields via data-driven surrogates and joint geometry-physics embeddings during generation and refinement, rather than enforeing governing laws explicitly. This guides designs toward plausible, tigh-performing regions but does not enforce hard physical constraints. Exploring hard-constraint mechanisms or tighter PDE-consistent couplings an important direction for future work.
Conclusion
In this paper, tackle the problem of 3D inverse design, which faces challenges from the high- dimensional -yrommey coupling and the exploration-validity trade-off. To represent the coupled space, propose
unified representation that preserves fine-grained shape details and physical-field information while significantly reducing dimensionality. Based on this representation, introduce two-stage physics-aware optimization strategy that first explores the latent manifold via gradient-guided diffusion sampling and then refines candidates through topology- preserving refinement. Extensive experiments demonstrate that our 3DID framework generates high-fidelity 3D models with greater versatility and superior performance on target objectives.
Acknowledgments
This work is partially supported by National Science and Technology Major Project (2022ZD01 17802). This work was also supported in part by "Pioneer" and "Leading Goose" R&D Program of Zhejiang (No. 2025C02032), the Fundamental Research Funds for the Central Universities (226-2025-0008)) and the Earth System Big Data Platform of the School of Earth Sciences, Zhejiang University.
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Appendix
In Appendix we provide additional experiments results.
In Appendix
A.I
we further provide the full results for 3D vehicle aerodynamic design.
In Appendix
A.2
provide more visualization of design results.
In Appendix
A.3
we provide more qualitative comparisons of topology preserving refine-
ment.
Appendix
B
The implementation details of baseline methods.
Appendix C
The dataset processing details.
Appendix
The implementation details of 3DID.
Appendix B The evaluation details of 3DID,
Appendix
The broader impact of 3DID.
Appendix
G
The licenses of datasets, codes, and models used in this paper.
23

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A Additional Results
A.1 Full Results for 3D Vehicle Aerodynamic Design
Here we present the full statistical results of our experiments, including 95% confidence intervals for all compared methods, shown in Table 4 A box plot of the simulation-derived drag coefficient (Sim-Drag) is shown in Figure 6 illustrating the distribution, variability, and outlier behavior across different approaches.
Table 4: Quantitative comparison for aerodynamic vehicle design.
Method
Pred-Drag
Sim-Drag
Novelty
Coveraget
GP, Voxel
0.2997=10.0436
0,4254 + 0.0351
1.0399+0005722
0,5200=:0.067
GP, VoXel+PCA
0.3059:10.0400
0.4363 + 0.0425
0.9734+000095
0.5850=10.0675
CEM. Voxel
0.2951=100022
0.4097 = 0.0279
0.9792=t0.0213
0,4350=10.0677
CEM. VoxeI+PCA
0.3088+000448
0.4393 0.0469
0.9864=+0.0250
0.5100=10.0600
CEM, TripNet
0.315410.0476
0.4161 0.0415
L.0399=0.0323
0.6050 0.0725
Backprop, Voxel
0.2979+-0.0314
0.4146 0.0244
0.9860++000004
0.4750++0.065
Backprop, VoxeI+PCA
0.3061++0.557
0.4614 0.0316
0.9798++0.0208
0.4950++0.0675
Backprop. TripNet
0.3153110.0472
0,4170 0.0444
L.229410.0290
0.590011.00700
DDID-NoTopooee (ours)
0.2623++0.0373
0.3766 0.0393
0.9195++0.0213
0.6950+10.022
3DID (ours)
0.2607+40.0331
0.3536+ 0.0313
1.1709+t0.0282
0.4300++0.0500
0.55
0.50
0.40
0.30
GR, Vosel
GR tozel-PCa
CRe, Vosel
CEN, MCrafo
cnw, triptee
Backprop, Vosel
Backprop JPre
Backprop Tripler
Dere-HoMeahitelion
Our
Figure 6: The box plot of the simulation-derived drag coefficient.
A.2
Visualization of 3DID Design
Additional visualizations of our designs provided in Figure 7 where each design is shown alongside its geometry and corresponding physical fields.
A.3
Comparisons of Tupology-Preserving Refinement
We provide additional qualitative comparisons in Figure Too demonstrate the effectiveness of our refinement stage. As shown, the design candidates consistently evolve toward a fastback profile after refinement, exhibiting reduced low-velocity recirculation regions and enhanced downward flow patterns, which indicate improved aerodynamic performance.
24

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Geometry
Velocity
Pressure
Figure 7: Qualitative results of our 3DID. Each row displays design candidate along with its
corresponding velocity and pressure field heatmaps.
25

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Before Refinement
After Refinement
Figure 8: Qualitative comparisons of aypology-preservinn refinement. Each row presents two design candidates comparisons with their geometry and simulated velocity field heatmaps.
26

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B Baseline implementation details
In our experiments, we compare our method against traditional sampling-bos and backpropagation- based approaches using various design representations. As baselines, we include the Cross-Entrop (CEM) 43 the Gaussian-prowe prrscesss surrogate with Bayesian optimization(GP) 44 and the gradient- gradient-based backpropagation method method(Backpingg (Backprop) [8]] For representations, the optimizer instantiated with three representations: a dense voxel grid 76]. PCA "CA-ccmpressedd voxel grid (Voxel+PCA) (22], and pure geometry triplane network (TripNet) [2T]. For each representation, we train VAE model [77] to compress the high-dimensional geometry into compact latent code, which serves as the optimization space for inverse design.
B.1 Representation Baseline
Voxel. We train voxel VAE 777 model directly on dense voxelized geometry learn latent
embedding, as demonstrated in Figure To train the model, we utilize the entire Driv AerNet++ [72] dataset, and voxelize the provided geometry with 256 resolution. For the Encoder, we leverage sequence of 3D convolution layers followed by batch normalization and LeakyReLU to encode the voxel grid into compact latent 2voxel- For voxel decoder, the latent vector 2voxel is first projected to singh-dimensonall feature space and reshaped into a 3D tensor. A sequence of 3D transposed convolutional layers then applied to reconstruct the voxel grid from this intermediate representation. Additionally, a separate drag prediction head, implemented as multi-layer perceptron (MLP), is applied estimate the target drag coefficient. We train the VAE model with reconstruction loss Crecom KL loss LKL. and the drag coefficient prediction loss Lang. The hyperparameters of the model and training are provided Table 5
Encoder
Decoder
Voxel
Voxel
Predictor
Drag
Figure 9: The overview of Voxel- VAE.
Voxel-PCA. Our Voxel-PCA representation is based on the representation proposed by [22] with modifications. In contrast to the Voxel- VAE, which directly uses voxel grids as input, the Voxel-PCA model first applies a dimensionality reduction step before downstream processing, as shown in Figure 10 Specifically, given the voxel data, we perform PCA 78 obtain compact representation of each geometry. Then, with this representation, we leverage series of MLPs to encode the reduced features into latent code 2voxel-pen- For reconstruction, an MI.P decoder first applied to reconstruct the PCA features from the latent code, which are then projected back to the voxel grid using the inverse PCA transformation. For drag prediction, similar to Boxel-VAE. separate drag prediction head is applied to estimate the target drag coefficient. Our Voxel-PCA model is also trained with reconstruction loss Cresmn KL loss CKL. and drag prediction loss Ldrag- The hyperparameters of the model and training provided in Table
TripNet. Our TripNet representation a pure geometry- based triplane representation, similar the one proposed in [21]. where it was used for forward prediction. The training procedure mirrors that of our unified phesicr-goometry framework, but excludes the physical field prediction branch, as illustrated in Figure [1] To obtain the representation, we utilize transformers with learnable tokens to extract features from the input point cloud. These features are then decoded using a transformer- based decoder and geometry mapping network to predict the occupancy field of the design geometry. We
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PCA
PCA
Encoder
Decoder
Recon
Voxel
Voxel
Predictor
Drag
Figure 10: The overview of Voxel-PCA-VAE.
utilize the Binary Cross Entsopy loss LBCE and KL loss LKL to supervise the training of VAE. To further predict the drag coefficient, adopt the same U-Net architecture used in objective-guided diffusion model. The TripNet-VAL architecture adopts the same hyperparameter configuration as the geometry branch of PG-VAE. More training hyperparameters provided in Table 7
Point Cloud
Cross Attentionn
Attention
Attention
X
Decoder
Self
Occupancy
Goometry Encoder
Reshape Sample
Geometry Mapping Network
Learablee
Tokens
Latent
Figure 11: The overview of TripNet-VAL VAF.
B.2
Optimization Baseline
CEM. Cross Entropy Method 43 traditional sampling-based optimization method widely used in classical inverse design problems. starts with an initial distribution, and each iteration, samples multiple candidates from the current distribution. Then, these candidates are evaluated against the target objective function to select a subset of clite samples with the best performance. The distribution parameters are updated based on these elite samples. A smoothing coefficient controls the rate of distribution updates between iterations. This process continues until convergence or maximum number of iterations is reached. our experiment, we utilize Gaussian distribution derived from encoded randomly selected samples as the initial distribution to provide valid starting point.
GP. Gaussian- surrogate with Bayesian optimization a classical optimization method for black-box optimization [6]] 17. Bayesian optimization (BO) operates by constructing a probabilistic surrogate model, commonly a Gaussian-process model, to approximate the objective function based on past observations. uncertain regions and exploitation of promising areas, guiding the selection of the next evaluation point. This strategy enables efficient optimization in high-cost or sample-limited scenarios by focusing evaluations on the most informative regions of the design space. While this method is effective in low-dimensional settings, constructing an accurate GP model becomes computationally expensive and challenging as the dimensionality of the design space increases. Therefore, GP-based Bayesian optimization is typically limited small-scale or low-dimensional problems, where surrogate can be reliably trained. In our experiment, our Gaussian process employs a Matérn kernel with constant and white noise components model the objective function. At each iteration, Expected Improvement (El) is used the acquisition function.
each iteration, an acquisition function is used balance exploration of
28

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Backprop. With the trained surrogate models, end-to-end backpropagation enables efficient gradient- based optimization of the design, leveraging the differentiability of the surrogate guide updates [T8 19]. In our experiments, we use the trained drag predictor as the surrogate and update the latent code using the Adam optimizer.
C Dataset processing details.
In this work, we conduct experiments on DrivAerNet++ (72), which is the largest aerodynamic car design dataset, comprising diverse car designs with corresponding CFD simulations. To train our model, the dataset with 8085 car designs to extract the point cloud and physical field. We first normalize each geometry of cars to fit within a unit cube, then uniformly sample 50,000 points with corresponding normals from the geometry surface. For the physical field, we apply the same scaling factor to ensure alignment with the normalized geometry. Subsequently, we randomly sample 50,000 points within the unit cube and interpolate the physical field values each location. These points serve as the input of our VAE. For supervision, we additionally sample another 50,000 points, each annotated with both occupancy values and physical field data. In this work, we focus on the pressure and velocity fields for the physical field representation, as wall shear stress is defined only the surface of the geometry and is thus not suitable for volumetric sampling. During physical field interpolation, since some Driv we map each sampled point its symmetric counterpart when necessary. For the U-Net and GNN surrogate models used in guided diffusion sampling and topology-preservingg preserving optimization, we employ the drag coefficient values provided by the DrivAerNet++ dataset as ground truth supervision during training.
samples are simulated using only half of the geometry,
D Implementation details.
Our framework consists of three key components: the Physics-Geometry VAE (PG- PAE), Objective- Guided Diffusion, and Topology-Preserving Refinement. Below, we provide detailed implementation descriptions for each component.
PG-VAE. The PG-VAE serves to compress both the design geometry and the corresponding physical field into a unified latent representation. We sample N, Np 50,000 points for the geometry and physical field branches, respectively. The encoder consists of one cross-attention layer and eight self-attention layers, each with 12 attention heads and an embedding dimension of d; = 64. We use = 64 for learnable tokens, and each with a channel dimension of de 768, to enhance representation expressiveness. The latent code dimension is set to d' = 32. The decoder architecture consists of one self-attention layer followed by five ResNet blocks [69]. which upsample the latent vector into a triplane representation with resolution output triplane is then queried using a mapping network composed of five fully connected layers with a hidden size of 32 per branch. We adopt semi-continuous occupancy formulation [60] and supervise both occupancy and physical field predictions using 50,000 sampled points within the normalized unit cube. We optimize the VAE using a combination of three loss terms: binary cross-entropy loss (ABCE 10 10-3) 30 mean squared error for field regression (AMSE 10-5), and KL divergence (λKι = 10 10-6). Training is performed using the Adam W optimizer [71] with a learning rate of 1 x 10-4, batch size 8 per GPU, for 100,000 steps. We use four NVIDIA RTX A6000 GPUs to train the model.
256 and channel dimension d, = 64. The
Objective-Guided Diffusion. To explore the latent design space efficiently, we employ a latent-space diffusion model composed of 10 DiT blocks [70). each containing 16 attention heads with a head dimension of 72. The difffusion process includes 1,000 denoising steps. During inference, an auxiliary U-Net surrogate network is used to predict the task objective directly from the latent code z, thereby guiding the sampling process toward optimal designs. The diffusion model is trained using learning rate of 5 x 10-6, batch size of 4 per GPU, for 300,000 steps with the Adam W optimizer. We use four NVIDIA RTX A6000 GPUs to train the diffusion model.
Refinement. To refine the initial design candidates while maintaining mesh topology, we apply a Free-Form Deformation (FFD) grid with 20 x 6 x 6 control points along the x, y, and z axes, respectively. The deformation is guided by a surrogate model based on Mesh- GraphNet [30], which comprises 8 message- passing blocks and operates on the surface mesh. The
Sapology-Preservee
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MeshGraphNet is trained to predict the drag force from deformed mesh serving as differentiable objective function during refinement. This model is trained with learning rate of 10 10-5, batch size of 8 per GPU, for 100,000 steps, using AdamW as the optimizer. We use two NVIDIA RTX A6000 GPUs to train the MeshGraphNer.
E Evaluation details.
In experiments, evaluate the design candidates using four metrics: predicted drag force (Pred-Drag), simulated drag force (Sim-Drag), novelty, and coverage.
Pred-Drag. We use the pretrained surrogate model to estimate the drag force of each candidate mesh. Given the mesh of designed candidates M our surrogate model directly predict the objective drag force J which can be formalized as:
J == J surrogate (M*),
(12)
where surrogate denotes the learned mapping from 3D mesh geometry the predicted drag coefficient. For our surrogate model, we adopt a MeshGraphNet 30) with 8 message passing blocks as the surrogate model. Unlike the model used in
preseeyyi refinement stage, this predictor operates solely on geometry, without requiring the associated physical field. To train the model, we use the entire AerNe+ttt [72 dataset. Given that different representations may produce varying topological structures, we apply remeshing and simplification to all candidates for fair comparison.
Sim-Drag. To obtain unbiased evaluation of the generated designs, perform figh-fideiity Computational Fluid Dynamics (CFD) simulations and compute the corresponding drag coefficients. Following Driv AeiNe+++++++ (72), employ the OpenFOAM@VII [7] to conduct steady incompressible simulation using the k - w SST turbulence model, based on Menter's 's formulation (80). We performed series of quality checks ensure the generated geometries and properly aligned within the CFD domain. During simulation, considering the computation cost, set the maximum local cells to 10 million and the maximum global cells 50 million in snappyHexMesh. The simulation iterates for 1000s, and we use the final 30% simulation data to calculate the average drag coefficient. The hyperparameters of our simulation are provided in Table X
simulation-ready
Novelty. To quantitatively assess how different the generated designs are from the training data, we measure the novelty of each candidate. Specifically, novelty is computed as the average distance from each generated design to its nearest neighbor the training set, reflecting how distinct the generated designs are from existing ones. Let {g. ) denote the set of generated designs and {t5} } denote the set of training designs the feature space. The novelty defined as:
N₂ Σ min d(g., tj), Ng 1=1
Novelty =
(13)
where d(::-) denotes the distance between feature embeddings, computed using the pretrained PointNet encoder (75)
Coverage. The coverage metric (also known as recall) evaluates how well the generated designs cover the training distribution by measuring, for each training sample, the distance to its nearest generated design (using a k-nearest neighbor lookup) and reporting the fraction training examples that fall within a predefined threshold. Let ) denote the set of generated designs and { denote the of training designs in the feature space. The coverage defined as:
1 N N
Coverage
Σ 1=1
imiin d(tz.g.) S <]]
(14)
where d(.,) .) a distance metric, 7 is predefined threshold, and [[-] the indicator function that equals if the condition is true and 0 otherwise.
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Broader Impacts
Academic Impact. 3DID's methodology, which enables direct navigation through 3D physics- geometry space, simplifies the 3D inverse design process. With the unified shysics-geometry representation, the computation gap between 3D and lower-dimensional inverse design is narrowed, allowing researchers to focus more on exploring cutting-edge inverse design strategies rather than being constrained by computational limitations. With the two-stage optimization strategy, our method balances between exploration and validity, offering researchers effective approach for inverse design involving 3D geometry.
The proposed 3D Inverse Design (3DID) framework extends the scope of geometry- driven design by enabling direct optimization of full 3D structures from scratch. By combining unified physics-geometry representations with physics- more efficient, automated design workflows in fields such aerospace engineering, biomedicine, additive manufacturing, and nanophotonies. In particular, 3DID can be applied to complex design tasks that traditionally rely on expert-crafted initial geometries and time-consuming simulation- based evaluations. In mechanical engineering, it can be used to optimize structural components for strength, weight, and thermal performance without manual trial-and-error. the medical field, 3DID enables the fabrication of patient-specific implants by automatically generating geometries tailored to individual physiological and functional requirements.
Social Impact.
optimization, our method opens the door to
31

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G License
The code will be publicly accessible. We use standard licenses from the community. We include the following licenses for the codes, datasets, and models we used in this paper.
1. Dataset
DrivAerNet++
72 CC BY-NC 4.0
2. Codes
NVIDIA PhysicsNcMM:
Apache License 2.0
3. Evaluation
OpenFOAM 79
GNU General Public License
Table 5: Hyperparameters for Voxel-VAL
Hyperparameter name
Value
Hyperparameters for Voxel- VAE architecture:
Input shape
[8, 256, 256, 256]
Output shape
[8, 256, 256, 256]
Number of 3D convolution layer
5
Dimension of latent Zvoxel
$12
Number of 3D transposed convolutional layer
5
Number MLPs drag predictor
Batch size
8
Dimension of encoder
(1, 64, 128, 256, 512) (512, 256, 128, 64, 32, 1)
Dimension of voxel decoder
Dimension of drag predictor
(512, 256, 128, 64, 32, 1)
Hy perrparraseerss for Voxel- VAE training:
Optimizer Learning rate
AdamW
le - 4
Learning steps Learning rate adjustment strategy
100K
Cosine
Warm-up steps
5K 10-3
Grecon weight
LK.. weight
10-4
Lana weight
10-3
32

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Table 6: Hyperparameters for Voxel-PCA-VAE
Hyperparameter name
Value
Hyperparameters for VoxcI-PCA-VAE architecture:
PCA output dimension
400
Number of MLP layers in encoder
Dimension of latent Evoxel-pea
64
Number of MLP layers in decoder Number of MLPs in drag predictor Batch size
2
32
Dimension of encoder
(400, 256, 128, 64, 64) (64, 64, 128, 256, 400) (64, 32, 1)
Dimension of PCA decoder
Dimension of drag predictor
Hyperparameters for Voxel-PCA-VAE training:
Optimizer
AdamW
Learning rate
5e 4
Learning steps Learning rate adjustment strategy
100K
Cosine
Warm-up steps
5K
Lreoo weight
10-2
CKL weight
10-4
Larag weight
10-3
Table 7: Hyperparameters for TripNet VAE
Hyperparameter name
Value
Hyperparameters for TripNet-VAL VAE training:
Batch size Optimizer Learning rate
8
AdamW
le 4
Learning steps
100K
Learning rate adjustment strategy
Cosine
Warm-up steps
5K
LECE weight
10 3
LKL weight
10-6
33

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Table 8: CFD Simulation Parameters for OpenFOAM
Value
Parameter name
Solver Configuration:
OPAM version
v11 incompressiblelludd
Solver
Algorithm Turbulence model
SIMPLE
k-w-SST
Simulation type
Steady-state RANS
Flow Conditions:
Flow velocity (ux))
30 m/s
Kinematic viscosity (V)
1.56 x 10-5 m²/s
Air density (p)
1.184 kg/m² 0.375 m²/s² 1.78 s
Turbulent kinetic energy (k)
Specific dissipation rate (w)
Computational Domain:
Domain dimensions
44x8x6.4 m
Inlet distance
12 m upstream 32 m downstream
Outlet distance
Solver Tolerances:
Pressure absolute tolerance Pressure relative tolerance
1 x 10-6
3 x 10
Velocity absolute tolerance Velocity relative tolerance Turbulence absolute tolerance
1 10
x 10-3
1 10-8
Turbulence relative tolerance
1 X 10
Potential solver absolute tolerance
X 10-7
Potential solver relative tolerance
1 x 10-2
Mesh Refinement:
Surface refinement level
3-4
Feature refinement level
4
Regional refinement level
2
Wake refinement level
2
Boundary layers
5 layers
Layer expansion ratio
1.2
Final layer thickness
0.5
Force Calculation:
Reference length (Iref) Reference area (Aref) Reference center
4.777
2.0 m²
(0, 0, 0)
Drag direction
(1, 0, 0)
Lift direction
(0, 0, 1)
Simulation Control:
End time
1000 s
Time step
s 100 steps
Write interval
Force coeffs write interval
10 steps
34


Reviews:
[{"review_id": "gVkOVhehTx", "reviewer": "Reviewer_egpt", "summary": "The paper introduces 3DID, a framework for direct 3D inverse design that addresses the limitations of prior approaches constrained by 2D projections or initial geometry assumptions. Instead of operating in the raw high-dimensional space, 3DID first learns a compact, unified latent representation that couples geometry and physics through a triplane-based VAE. It then performs a two-stage optimization: (1) a gradient-guided diffusion sampler that generates diverse candidate designs in latent space with objective-aware steering, and (2) a topology-preserving refinement stage based on free-form deformation that further improves the design while maintaining geometric integrity. Applied to aerodynamic shape optimization, 3DID outperforms existing methods in objective quality, novelty, and simulation fidelity.", "questions": "While the diffusion process is guided by gradients of the objective, there is no explicit enforcement of hard physical or geometric constraints during sampling. This raises the question of how often does the sampling stage produce geometries that are physically invalid or impractical?", "limitations": "Yes.", "rating": 5, "confidence": 1, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 2, "strengths_and_weaknesses": "The methodology is well presented and evaluated. Results on aerodynamic shape optimization show improvements in simulated drag and design novelty over baselines. The proposed physics-geometry triplane embedding and the gradient-guided diffusion and topology-preserving refinement look both novel.\nClarity is generally good, with clear figures. The work targets a challenging and relevant problem in scientific design and makes clear progress compared to existing methods. Overall, I find this to be a solid and well-executed paper with meaningful contributions, and I lean toward acceptance.", "was_revised": true, "final_justification": "Based on the overall quality of the submission and the authors’ responses to my comments in the rebuttal, I’m inclined to recommend acceptance for this paper.", "rebuttal": "We thank the reviewer for the positive feedback. We appreciate that the reviewer thinks our work is novel in the method and makes clear progress compared to existing methods. Below, we address the reviewer's concerns one by one.\n\n> **Comment 1: \"There is no explicit enforcement of hard physical or geometric constraints during sampling.\"**\n\n**A1:** We sincerely thank the reviewer for pointing out this important aspect, which indeed opens a valuable direction for future work. However, we would like to respectfully note that the gradient-guided mechanism already helps steer the generative process toward regions of the design space that implicitly satisfy geometric constraints and exhibit improved physical performance. Specifically, for **geometric constraints**, our diffusion model is trained on a large amount of automotive shapes, which enables it to learn the underlying structure and generate topologically coherent geometries (see Comment 2). For **physical constraints**, since inverse design lacks direct PDE-based formulations as in forward simulations, we instead rely on gradient guidance from the performance predictor to bias sampling toward designs with better aerodynamic properties. This soft guidance effectively incorporates physical awareness without requiring explicit constraint formulations.\n\n> **Comment 2: \"How often does the sampling stage produce geometries that are physically invalid or impractical?\"**\n\n**A2:** Thanks for suggesting such an important experiment. To validate the plausibility of the generated geometries, we sampled 500 designs using our method and evaluated them for physical and geometric validity (including watertightness, self-intersection, average volume, and average surface area). The table below compares these metrics across three stages: the original training data, our method's output after sampling, and the final results after the refinement stage. From the table, we can see that 100% of the sampled shapes are watertight and free from self-intersections. This robustness is largely attributed to the proposed continuous latent embedding, which, when combined with the marching cubes algorithm, ensures that the decoded shapes remain topologically coherent and geometrically well-formed. After the optimization, we observe that the geometry completeness remains unchanged, as our topology-preserving optimization does not alter the mesh connectivity or introduce artifacts.\n\n**Table 1: Geometric Validity Assessment of Generated Automotive Designs**\n\n| Method                | Watertight Ratio | Self-Intersection Free Ratio | Average Volume | Average Surface Area |\n| --------------------| :----------------: | :----------------------------:| :--------------: | :--------------------: |\n| Data                  | 100%             | 100%                         | 5.10           | 22.27                |\n| Ours (w/o refinement) | 100%             | 100%                         | 6.10           | 24.45                |\n| Ours                  | 100%             | 100%                         | 5.45           | 22.56                |"}, {"review_id": "Eio77guhIM", "reviewer": "Reviewer_Vqus", "summary": "The authors learn a unified latent representation that compactly encodes both the 3D shape geometry and its associated physical field (such as flow pressure). This latent space significantly reduces the dimensionality of the design problem while preserving fine geometric details and high-fidelity physics information. Building on this representation, 3DID employs a two-stage optimization pipeline: (1) a gradient-guided diffusion sampler that generates diverse candidate shapes from scratch by guiding a 3D diffusion model with the gradient of the target objective, thus biasing generation toward high-performance designs, and (2) a topology-preserving refinement stage that fine-tunes each candidate via physics-aware gradient descent (using a differentiable surrogate model) to further improve performance while maintaining the shape’s structural integrity.", "questions": "Demonstrating any degree of generalization or adaptability would greatly strengthen the work’s impact. The authors are requested to test two conditions: 1) Cluster Drivaernet++ using parameters and completely hold out one of the clusters as the test set. 2) Test the method on a different dataset (e.g., ShapeNet geometries) to demonstrate the degree of generalization.\n\nHyperparameter tuning: The paper has a complex workflow, and ablations for more key design choices are needed to understand the outcome. For example, an ablation on the effect of the objective guidance strength (the coefficient in diffusion guidance) would show sensitivity to that hyperparameter and help answer, “How does the strength of this guidance impact the downstream optimization results?” Another ablation could remove or vary the regularizers in the refinement stage (smoothness and volume preservation) to demonstrate their effect on the design outcome.\n\nWhile the approach is “physics-aware” in that it uses training data from simulations, it doesn’t enforce the physical laws during generation – it relies on the surrogate’s learned approximation. If one wanted to apply 3DID to a new domain, one would have to retrain the entire model and ensure the surrogate is accurate, which could be a barrier. It will help if the authors discuss this issue.\n\nHow accurate does the encoding surrogate have to be for integration in the full pipeline?", "limitations": "A current limitation is the narrow scope of testing. A rigorous study would test generalization in multiple ways. One suggestion is cross-generalization: train the model on one subset of shapes or one objective, and test its performance on a slightly different scenario. How well does the method work on out-of-distribution geometries?", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "A core contribution is the learned latent space that jointly encodes geometry and physics (e.g., flow fields) in a compact form. The paper introduces a clever optimization strategy that balances global exploration with local refinement. The first stage, a gradient-guided diffusion sampler, leverages the generative power of diffusion models and steers them using objective gradients. \n\nWhile this work is largely empirical, adding theoretical underpinnings where possible would improve its rigor. For example, providing a convergence analysis for the refinement stage (even if simplified, like proving that the FFD gradient descent will locally improve the objective under certain smoothness assumptions) would lend more credibility. Similarly, formalizing the guided diffusion by referencing known results in score-based generative modeling would address theoretical concerns. If exact analysis is intractable, even a conceptual discussion of failure modes (e.g., how too-strong guidance can lead to off-manifold samples) and how the method mitigates them (via the chosen hyperparameter) shows theoretical awareness. Grounding each algorithmic choice in either theory or established literature would dispel any impression of ad-hoc design.\n\nAblations to establish the impact of different hyperparameters are needed due to the complex workflow.\n\nIt is not clear how the method compares against a well-formulated optimization-only pipeline (FFD+adjoints or FFD+heuristic search), which avoids data collection, training, and the limited generalizability challenge. These methods are commonly used in aerospace and vehicle design. What are the key benefits that will convince practitioners to adopt this methodology compared to existing optimization-only approaches?", "was_revised": true, "final_justification": "I believe the author's response has addressed a few of the issues I raised regarding hyperparameters and generalizability.", "rebuttal": "We thank the reviewer for the detailed feedback and helpful suggestions. We are pleased that the reviewer appreciates our method's ability to balance global exploration with local refinement. Below, we address the reviewer's comments one by one. \n\n> **Comment 1: \"There is no explicit enforcement of hard physical or geometric constraints during sampling.\"**\n\n**A1:** We thank the reviewer for this valuable suggestion. In our current pipeline, we observed that increasing the number of refinement steps does not always lead to improved performance, which we attribute to the accumulation of surrogate bias during optimization. To mitigate this issue, we introduce two regularization terms that help constrain the optimization within the data manifold and reduce the risk of generating out-of-distribution designs. As shown in our response to **Comment 3**, these regularizers are empirically effective in stabilizing the refinement process. We acknowledge this as a limitation of our method and plan to explore more principled convergence strategies in future work. \n\n\n> **Comment 2: \"Formalizing the guided diffusion by referencing known results in score-based generative modeling would address theoretical concerns.\"**\n\n**A2:** In our method, we adjust the predicted noise at each denoising step by injecting the gradient of the task objective $J$\n\n$$\\epsilon \\prime_\\phi(z_t, t) = \\epsilon_\\phi(z_t, t) + \\gamma \\nabla_{z_t} J(\\hat{z}_0(z_t)).$$\n\nThis objective-aware guidance can be interpreted as replacing the unconditional score $\\nabla_{z_t} \\log p(z_t)$ in score-based models with the conditional score[1]:\n\n$$\\nabla_{z_t} \\log p(z_t | J) = \\nabla_{z_t} \\log p(z_t) + \\nabla_{z_t} \\log p(J | z_t),$$\n\nwhere the second term is approximated by $-\\nabla_{z_t} J(\\hat{z}_0(z_t))$. This aligns with prior work viewing the reverse diffusion process as a gradient-based optimization over the data distribution. Thus, our method can be understood as performing optimization over a composite log-likelihood objective $\\log p(z_t | J)$, integrating both data fidelity and design performance into a unified sampling process. We will incorporate this theoretical clarification into the revised version.\n\n> **Comment 3: \"A conceptual discussion of failure modes and how the method mitigates them shows theoretical awareness.\"**\n> **Ablations for key hyperparameters are needed due to the complex workflow.**\n\n**A3:** The influence of hyperparameters is also concerned by Reviewer aPc8. We address this concern through two key ablation studies.\n\n**1. Ablation on Guidance Scale ($\\gamma$):**\n\nWe evaluate how varying the guidance strength $\\gamma$ would affects generation quality. As shown in the Table 1, increasing $\\gamma$ initially improves performance by enhancing the effect of the target-driven gradient, resulting in lower predicted and simulated drag. However, excessively large $\\gamma$ (e.g., 10 or 20) causes the generation to drift off the learned data manifold, leading to worse simulation performance and a sharp drop in coverage. In our paper, we choose $\\gamma$ based on the best evaluation performance.\n\n**Table 1: Effect of Guidance Scale on Generation Performance**\n\n| Guidance Scale ($\\gamma$) | Pred-Drag | Sim-Drag | Novelty | Coverage |\n|:-------------------:|:-----------:|:----------:|:---------:|:----------:|\n| 0 (no guidance) | 0.2971 | 0.3944 | 0.9177 | **0.7104** |\n| 0.1 | 0.2930 | 0.3956 | 0.9182 | 0.7101 |\n| 0.5 | 0.2812 | 0.3809 | 0.9118 | 0.7089 |\n| 1 | 0.2715 | 0.3798 | 0.9114 | 0.7027 |\n| 2 | 0.2673 | 0.3787 | 0.9167 | 0.7014 |\n| 5 | **0.2623** | **0.3766** | 0.9195 | 0.6950 |\n| 10 | 0.2823 | 0.4122 | 0.9227 | 0.5923 |\n| 20 | 0.3204 | 0.5125 | **1.4072** | 0.4512 |\n\n**2. Ablation on Refinement Regularizers:**\n\nWe conducted additional experiments to assess the effect of regularization terms in the topology preserving refinement stage, namely volume preservation and control point smoothness. As shown in the Table 2, removing the volume constraint results in a noticeable drop in performance, particularly in simulated drag and coverage, due to uncontrolled volume shrinkage during refinement. Removing the smoothness regularizer leads to degraded drag performance, indicating that the lack of regularization results in less effective shape refinement.\n\n**Table 2: Effect of Regularization Terms on Refinement Performance**\n\n| Method | Pred-Drag | Sim-Drag | Novelty | Coverage | Volume |\n|--------|:-----------:|:----------:|:---------:|:----------:|:---------:|\n| With both regularizers | 0.2607 | 0.3536 | 1.1709 | 0.4300 | 5.9578 |\n| Without volume | 0.2765 | 0.4190 | 1.3472 | 0.2375 | 4.7624 |\n| Without smooth regularizer | 0.2715 | 0.4070 | 1.2762 | 0.3711 | 5.4654 |\n\n> **Comment 4: \"It is not clear how the method compares against optimization-only pipelines such as FFD+adjoints or FFD+heuristic search.\"**\n> **\"What are the key benefits that will convince practitioners to adopt this methodology compared to existing optimization-only approaches?\"**\n\n**A4:** Compared to traditional optimization-only pipelines such as FFD+adjoints or FFD+heuristic search, our method offers two key advantages. First, our diffusion-based sampling stage enables automatic exploration of diverse and high-quality initial designs, removing the need for manually selected initial shapes, which are often critical in conventional pipelines. Second, while our refinement stage shares similarities with methods like FFD+adjoints in using gradient-based updates, it leverages a pre-trained surrogate model, allowing efficient optimization without requiring access to differentiable solvers.\n\n> **Comment 5: \"Demonstrating any degree of generalization or adaptability would greatly strengthen the work's impact.\"**\n>\n> **\"Please test generalization by (1) Cluster DrivAerNet++ using parameters and completely hold out one of the clusters as the test set (2) evaluating on a different dataset\"**\n> \n> **\"How accurate does the encoding surrogate have to be for integration in the full pipeline?\"**\n\n**A5:** We thank the reviewer for raising these important questions regarding generalization. We address them from two perspectives:\n\n**(1) Generalization of our pipeline**\nTo evaluate whether our method generalizes beyond car geometries, we conducted additional experiments on the **AhmedML** dataset[2], which contains 500 diverse 3D bluff body geometries with simulated aerodynamic fields. This dataset differs significantly from DrivAerNet++[3] in shape distribution. As illustrated in Table 3, our method still demonstrates effective performance, indicating that our pipeline is not overfitted to a specific dataset and can generalize to new shape families.\n\n**Table 3: Cross-Dataset Generalization Results on AhmedML**\n\n| Method | Pred-Drag | Sim-Drag |\n|--------|:-----------:|:----------:|\n| W/O Guidance | 3.1181 | 3.9549 |\n| 3DID–NoTopoRefine | 2.9987 | 3.7874 |\n| 3DID | **2.8554** | **3.6017** |\n\n**(2) Generalization of our surrogate model**\n\nTo evaluate the generalization of our surrogate model, we retrained it on the DrivAerNet++[3] dataset while holding out the Fastback category. We then integrated the partially trained model into our 3DID pipeline. As shown in Table 4, although the guidance is weaker than with full-data training, it still improves performance over unguided generation.\n\n**Table 4: Pipeline Performance with Partial Surrogate Training**\n\n**UNet Guidance:**\n| Method | Pred-Drag | Sim-Drag |\n|--------|:-----------:|:----------:|\n| No guidance | 0.2964 | 0.4080 |\n| UNet (All) | 0.2733 | 0.3698 |\n| UNet (Partial) | 0.2825 | 0.3930 |\n\n**GNN Optimization:**\n| Method | Pred-Drag | Sim-Drag |\n|--------|:-----------:|:----------:|\n| No optimization | 0.2838 | 0.3958 |\n| GNN (All) | 0.2644 | 0.3628 |\n| GNN (Partial) | 0.2776 | 0.3822 |\n\n\n> **Comment 6: \"The method is physics-aware only through training data; it doesn't enforce physical laws during generation.\"**\n> **\"The method relies on a surrogate trained from simulations without enforcing physical laws, limiting generalization and requiring retraining in new domains.\"**\n\n**A6:** We thank the reviewer for highlighting this important point, which is also raised by Reviewer ht5R. In our context, **\"physics-aware\"** refers to incorporating physics information into both stages of our optimization pipeline, rather than relying solely on geometry for guiding the design process. While explicit enforcement of physical laws via projection operators is feasible in some 2D or low-dimensional inverse design tasks, such formulations become intractable in high-dimensional 3D settings. \n\nInstead, our framework adopts a surrogate-based strategy to integrate physics knowledge in both stages:\n\n- **In the generation phase**: We jointly embed physical fields and geometric shapes into a unified latent space, guiding the design generation toward physically plausible and performance-oriented solutions.\n\n- **In the topology optimization phase**: Our MeshGraphNet-based surrogate model takes both geometry and physical field information as input, providing gradients that guide the refinement process toward improved performance.\n\nWhile this does not enforce physical laws in the strict sense, it enables a scalable and practical way to incorporate physics into 3D inverse design. \n\n[1] Score-Based Generative Modeling through Stochastic Differential Equations. 2021\n\n[2] AhmedML: High-Fidelity Computational Fluid Dynamics Dataset for Incompressible, Low-Speed Bluff Body Aerodynamics. 2024\n\n[3] Drivaernet++: A large-scale multimodal car dataset with computational fluid dynamics simulations and deep learning benchmarks. 2024"}, {"review_id": "7hKlX2GPnH", "reviewer": "Reviewer_aPc8", "summary": "This paper proposes a new 3D inverse design method with physics-aware optimization, differentiating from prior work in two aspects: (1) avoiding 2D projection; (2) abstaining from fine-tuning existing 3D shapes. The 3DID framework employs a unified latent representation within a VAE to encode both shape geometry and induced physical fields. It performs two-stage generation: (1) direct latent space sampling via guided diffusion models; (2) topology-preserving mesh refinement under constraint. Results on vehicle aerodynamic shape design validate 3DID’s effectiveness.", "questions": "Two critical questions:\n\n(1) should provide one more task for evaluation.\n\n(2) should clarify the necessity of the topology preserving refinement stage.\n\nOther questions:\n\n(1) In Eq (2), how to formulate \"constraints, such as volume, manufacturability, and boundary conditions\"?\n\n(2) In Eq (5), the = should be replaced by $\\propto$\n\n(3) What does the KL divergence mean in Eq. (3)?\n\n(4) how does the hyperparameter $\\gamma$ effect the performance? Too large $\\gamma$ may make generated shape deviated from the manifold and adversarial artifacts, and too small $\\gamma$  has little effect on optimizing the objective.\n\n(5) How to pretrain the surrogate model $f_\\text{GNN}$? How does it deal with out-of-distribution prediction?\n\n(6) In Figure 4, Voxel+PCA produces non-watertight shapes. How doese 3DID guarantee watertightness?", "limitations": "yes", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 3, "originality": 2, "strengths_and_weaknesses": "1. Strengths\n\n   (1) The proposed method is novel.  The Physics-Geometry VAE unifies shape and physical field encoding in a latent space, ensuring physical consistency during design. The two-stage generation process combines objective-guided diffusion with topology-preserving refinement, addressing prior limitations.\n\n   (2) The empirical results on vehicle aerodynamic shape optimization is significant, as shown in Table 1/2/3 and Figure 4/5.\n\n2. Weaknesses\n\n   (1) As a general 3D inverse design method, the current validation is limited to a single task (vehicle aerodynamics). To establish generality, at least one additional engineering task is required—e.g., 3D aircraft design, a well-studied problem in physics-based engineering design.\n\n   (2) The motivation for the topology preserving refinement stage is ambiguously stated as \"biases from training data prior,\" lacking clear theoretical grounding. The authors should clarify: Why is refinement needed after objective-guided sampling? A visual/intuitive illustration of the refinement process (e.g., fine-grid visualizations) is essential for understanding.", "was_revised": true, "final_justification": "I appreciate the paper and the rebuttal process. Based on its current quality and additional experiments/clarification, I would like to provide a \"4: Borderline accept\" rating. The reason why I do not provide a \"5: Accept\" rating is that its 3D experiments results are not strong enough.", "rebuttal": "We sincerely thank the reviewer for their constructive feedback and for recognizing the novelty of our method. Below, we provide detailed responses to each of the concerns raised.\n\n> **Comment 1: \"As a general 3D inverse design method, the current validation is limited to a single task (vehicle aerodynamics). To establish generality, at least one additional engineering task is requirede.g., 3D aircraft design, a well-studied problem in physics-based engineering design.\"**\n\n\n**A1:** We appreciate the reviewer’s suggestion, which is also concerned by Reviewer Vqus.  Due to the lack of high-quality, open 3D aircraft simulation datasets with paired physical fields, we instead conducted additional experiments on the AhmedML dataset[1], which includes 500 diverse 3D bluff body geometries for aerodynamic evaluation.  Demonstrate that our method generalizes beyond car geometries and performs effectively across diverse shape distributions.\n\n**Table 1: Cross-Dataset Generalization Results on AhmedML**\n| |  Pred-Drag | Sim-Drag |\n|:--------------:|:-----------:|:-----------:|\n| W/O Guidance| 3.1181| 3.9549|\n| 3DID–NoTopoRefine| 2.9987| 3.7874| \n| 3DID |**2.85543**| **3.6017**| \n\n\n> **Comment 2: \"The authors should clarify: Why is refinement needed after objective-guided sampling? A visual/intuitive illustration of the refinement process (e.g., fine-grid visualizations) is essential for understanding.\"**\n\n**A2:** The refinement stage is introduced to further improve performance beyond what can be achieved by the diffusion model alone. While the objective-guided diffusion model incorporates target-driven gradients during generation, the resulting designs remain heavily influenced by the training data’s prior distribution. To overcome this limitation, our refinement phase performs physics-aware optimization using a differentiable surrogate model. This enables fine-grained, continuous adjustments that enhance the design’s performance while preserving mesh-level topology.\n\nWe agree that visualizations would greatly aid understanding. However, rebuttal guidelines do not permit the inclusion of new figures. We will provide detailed visual comparisons of pre- and post-refinement results in the revised manuscript to clearly illustrate the necessity and effectiveness of this stage.\n\n>**Comment 3: \"In Eq (2), how to formulate \"constraints, such as volume, manufacturability, and boundary conditions\"**\n\n**A3:** We thank the reviewer for pointing out the ambiguity in the formulation of constraints in Eq. (2). To clarify this, we will revise the manuscript to explicitly state that the constrained optimization problem can be formulated as:\n\n$$M^* = \\arg\\min_M J(M, F(M)) \\quad \\text{s.t.} \\quad C(M, F(M)) = 0.$$\n\nHere, M denotes the 3D shape, $F(M)$ is the induced physical field, $J$ is the design objective (e.g., drag), and $C(M,F(M))=0$ encodes geometric and physical constraints.\n\n\n> **Comment 4: \"In Eq (5), the = should be replaced by $\\propto$\"**\n\n**A4:** We thank the reviewer for pointing out this notational issue. We will fix this in the updated version.\n\n> **Comment 5: \"What does the KL divergence mean in Eq. (3)?\"**\n\n**A5:** To prevent excessive variance in the physics–geometry latent representation,  the KL divergence loss is applied in the training of PG-VAE. It can be formulated as \n\n$$ \\mathcal{L}_{\\mathrm{KL}} = \\mathrm{KL}\\left( q(z \\mid M, F(M)) \\| \\mathcal{N}(0, I) \\right) .$$\n\nWhere q is the posterior over the latent code, and N is the Gaussian distribution. \n\n\n> **Comment 6: \"how does the hyperparameter  $\\gamma$  effect the performance?\"**\n\n**A6:** This is a very important question for the evaluation of our method. It is also concerned by Reviewer Vqus. We have performed additional experiments to test the influence of $\\gamma$. The results are listed in the following table.  From the table, we can see that our method remains stable within a broad range of $\\gamma$. As $\\gamma$  increases, the guidance becomes stronger, leading to lower predicted drag and simulated drag, but if too large (e.g., $\\gamma$ = 10 or 20), it causes off-manifold generations, resulting in degraded simulation performance and poor coverage. In our paper, we choose $\\gamma$ based on the best evaluation performance. \n\n**Table 2: Effect of Guidance Scale on Generation Performance**\n| Guidance Method | Pred-Drag | Sim-Drag | Novelty | Coverage |\n|:-----------------:|:-----------:|:----------:|:---------:|:----------:|\n| 0               | 0.2971    | 0.3944   | 0.9177  | **0.7104** |\n| 0.1             | 0.2930    | 0.3956   | 0.9182  | 0.7101   |\n| 0.5             | 0.2812    | 0.3809   | 0.9118  | 0.7089   |\n| 1               | 0.2715    | 0.3798   | 0.9114  | 0.7027   |\n| 2               | 0.2673    | 0.3787   | 0.9167  | 0.7014   |\n| 5               | **0.2623**| **0.3766** | 0.9195  | 0.6950   |\n| 10              | 0.2823    | 0.4122   | 0.9227  | 0.5923   |\n| 20              | 0.3204    | 0.5125   | **1.4072** | 0.4512   |\n\n\n> **Comment 7: \"How to pretrain the surrogate model $f_\\text{GNN}$? How does it deal with out-of-distribution prediction?\"**\n\n\n**A7:** We thank the reviewer for pointing out this missing detail, and we will include the full explanation in the revised version.\n\nWe adopt MeshGraphNet as our surrogate model $f_\\text{GNN}$, given its strong performance in mesh-based physical simulations. The surrogate is trained to predict aerodynamic drag from paired samples of geometry and ground-truth physical fields collected from the dataset. To improve robustness and mitigate out-of-distribution (OOD) issues, we adopt two strategies:\n\n1. Noise augmentation during training by perturbing mesh node positions, which helps the model generalize better to unseen shapes.\n2. Shape regularization during refinement optimization, which constrains the updated geometry to remain close to the training distribution.\nThese strategies together ensure that both the surrogate and the final designs remain reliable and physically plausible.\n\n> **Comment 8: \"In Figure 4, Voxel+PCA produces non-watertight shapes. How does 3DID guarantee watertightness?\"**\n\n**A8:** Unlike voxel-based representations that operate on discrete grids and often result in artifacts such as non-watertight surfaces, 3DID leverages a continuous triplane-based latent representation that jointly encodes geometry and physical fields. This continuous encoding enables smooth and coherent surface reconstruction using the Marching Cubes algorithm. To further validate the generated geometry quality of our method, we sampled 1000 designs using our method and evaluated them for physical and geometric validity (including watertightness, self-intersection, average volume, and average surface area).  The table below compares these metrics across three stages: the original training data, our method's output  after diffusion sampling, and the final results after the refinement stage. From the table, we can see that 100% of the sampled shapes are watertight and free from self-intersections. After the optimization, we observe that the geometry completeness remains unchanged, as our topology-preserving optimization does not alter the mesh connectivity or introduce artifacts.\n\n**Table 3: Geometric Validity Assessment of Generated Automotive Designs**\n| Method | watertight Ratio  | Self-Intersection Free Ratio | Average Volume  | Average Surface Area |\n|:-----------------:|:-----------:|:----------:|:---------:|:----------:|\n| Data              | 100%    | 100%  | 5.10  | 22.27 |\n| Ours(Wo-refinement)| 100%    | 100%  | 6.10  | 24.45  |\n| Ours          | 100%    | 100%   | 5.45  | 22.56   |\n\n[1] AhmedML: High-Fidelity Computational Fluid Dynamics Dataset for Incompressible, Low-Speed Bluff Body Aerodynamics. 2024"}, {"review_id": "s8PsLCPhOe", "reviewer": "Reviewer_ht5R", "summary": "This paper performs inverse design of car geometries for optimizing aerodynamics properties, e.g., drag. The paper creates a latent space that encodes both the geometry of the cars and the simulated physical fields around the car geometry. The paper then uses this latent space for optimizing for a new shape (in the latent space). As a final post processing step, the paper further optimizes the obtained design through a shape optimization that preserves the topology. The paper compares its method to two other inverse design method and reports superiority. It also studies some of its own design decision through ablations.", "questions": "- On the amortization note above, how general the surrogation is? With the 8000 training data, can one use the trained surrogates for slightly changed conditions/applications? \n\n- How much topology variation exist in the training data and how much the topology of optimized designs is different from the training data? Are we talking about mesh-topology or the more macro-scale car topology? \n\n- I had difficulty understanding 2D projection part. Some clarification would be helpful to the paper. \n\n- Equation 1 is hard to parse mathematically. Please check the correctness.", "limitations": "Not much discussed in the main paper. Some of the points raised in Strengths/Weakness section above could potentially be used.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths: \n- The choice of the problem: this problem is inherently difficult and underrepresented in ML. \n\n- Validation is relatively good and the paper has done the right thing by including how a ground-truth (and not only surrogate) rates the optimized design, through 'Sim-Drag' metric. \n\nWeaknesses: \n- The two main contributions of the paper could have been more discussed. For example, there should have been references to the works that encode both design and physical properties as well as works that do refinement. \n\n- An important missing evaluation that acts as a sanity check is to compare the performance of the optimized design with the best performing design in the training dataset. \n\n- The whole point of surrogation is to amortize its training during inference. Is this the case in this paper? The paper shows the best performing shape through surrogate but what if we optimize for 1000 different drags and show the statistics? Ideally, this must be coupled with other objectives and have a multi-objective optimization. \n\n- I am not sure why this method is called a physics-aware optimization. The fact that the physical fields are used jointly for training the encoder, does not fit the (loose) definition of physics-aware. \n\n- Minor, but the title is a bit too broad by not including the context (car geometries and aerodynamics).", "was_revised": true, "final_justification": "I think this is a good paper and the rebuttal helped with some clarifications. I am still not sure how my main concern will be reflected in the paper which I believe is quite fundamental: \n\"The authors performed a suggested experiment where they check whether any generated design (using their method) can beat the best performing design in the training set. And the answer is no (I command them for their transparency). As I mentioned in the review, this is a sanity check. This means that the 'interpolated' designs cannot achieve better performance. In other words, the method is not capable of giving us better designs but only different designs.\"", "rebuttal": "We thank the reviewer for the detailed feedback. We appreciate the recognition of our effort in validating results with simulated drag and in tackling a difficult but underrepresented problem. Below, we address each comment point-by-point.\n\n> **Comment 1: \"there should have been references to the works that encode both design and physical properties as well as works that do refinement.\"**\n\n**A1:** We thank the reviewer for pointing this out. We will incorporate additional references to prior works in the revised version. \n\n> **Comment 2: \"An important missing evaluation that acts as a sanity check is to compare the performance of the optimized design with the best performing design in the training dataset.\"**\n\n**A2:** We conducted a comparison between the best-performing designs in the training set and our generated samples. Among the 100 generated designs, none directly outperformed the best-performing design from the training dataset, as shown in Table 1. This occurs because our diffusion model aims to generate diverse geometries across the entire data distribution rather than specifically targeting the global optimum.\n\nTo further validate the effectiveness of our optimization pipeline, we performed an additional experiment: we took the Top-1 and Top-10 best-performing training shapes, encoded them into our latent space with added noise, and then optimized them using our generation and refinement pipeline. The optimized shapes achieved lower drag values compared to their original versions. This suggests that our method not only generalizes across the design space but is also capable of fine-tuning within high-performance regions.\n\n**Table 1: Comparison of Training vs. Generated Designs**\n\n| Design | Sim-Drag (Training) | Sim-Drag (Ours) |\n|--------|-------------------|------------------|\n| Top-1 | **0.2857** | 0.2944 |\n| Top-10 | **0.3017** | 0.3125 |\n\n**Table 2: Optimization Results on Best Training Designs**\n\n| Design | Sim-Drag (Before) | Sim-Drag (After) |\n|--------|------------------|------------------|\n| Top-1 | **0.2857** | 0.2912 |\n| Top-10 | 0.3017 | **0.2996** |\n\n\n>**Comment 3:\"The whole point of surrogation is to amortize its training during inference. Is this the case in this paper? The paper shows the best performing shape through surrogate but what if we optimize for 1000 different drags and show the statistics? Ideally, this must be coupled with other objectives and have a multi-objective optimization.\"**\n>\n> **“How general is the surrogation? With the 8000 training data, can one use the trained surrogates for slightly changed conditions/applications?\"**\n\n**A3:** The surrogate model in our framework serves two main purposes:\n\n- **Computational efficiency**: The surrogate model accelerates the optimization process by amortizing the training cost over multiple inference steps, reducing reliance on expensive computational solvers.\n\n- **Gradient information**: It provides differentiable gradient information, enabling efficient gradient-based optimization.\n\nAdditionally, our surrogate model demonstrates a certain degree of generalization capability, extending its application beyond the training domain. To validate this, we conducted an experiment using the DrivAerNet++ dataset [1], which contains car geometries categorized into three types: **Estateback, Fastback, and Notchback**. Specifically, we retrained the surrogate models by completely excluding the **Fastback** category from training. The performance results are shown in Tabel 3.\n\n**Table 3: Surrogate Model Generalization Performance**\n\n| Surrogate Model | Relative Accuracy |\n|-----------------|-------------------|\n| UNet with Fastback | 2.87% |\n| UNet without Fastback | 7.61% |\n| GNN with Fastback | 2.09% |\n| GNN without Fastback | 9.34% |\n\nWe further integrated these partially trained surrogate models into the 3DID pipeline for inverse design, the results are illustratred in Table2. These results indicate that our surrogate models maintain reasonable accuracy and yield meaningful optimization improvements even when evaluated on geometries from categories excluded during training. This observation aligns well with prior works that investigate the surrogate model's generalization capabilities[2,3].\n\n**Table 4: Pipeline Performance with Partial Surrogate Training**\n\n**UNet Guidance:**\n| Method | Pred-Drag | Sim-Drag |\n|--------|-----------|----------|\n| No guidance | 0.2964 | 0.4080 |\n| UNet (All) | 0.2733 | 0.3698 |\n| UNet (Partial) | 0.2825 | 0.3930 |\n\n**GNN Optimization:**\n| Method | Pred-Drag | Sim-Drag |\n|--------|-----------|----------|\n| No optimization | 0.2838 | 0.3958 |\n| GNN (All) | 0.2644 | 0.3628 |\n| GNN (Partial) | 0.2776 | 0.3822 |\n\nFinally, regarding multi-objective optimization, we agree that extending our surrogate-based framework to accommodate simultaneous optimization of multiple design criteria is a valuable direction, and we plan to pursue this in future work.\n\n> **Comment 4:\"The fact that the physical fields are used jointly for training the encoder, does not fit the (loose) definition of physics-aware.\"**\n\n**A4:** We thank the reviewer for raising this important point, which is also concerned by Reviewer Vqus. In our context, \"physics-aware\" refers specifically to incorporating physics information in our two-stage optimization pipeline, rather than relying solely on geometry for optimizing the numerical objective. While some 2D or low-dimensional inverse design tasks can enforce physical constraints explicitly via projection or analytical formulations, this becomes significantly difficult in high-dimensional 3D inverse design problems. \n\nInstead, our framework adopts a surrogate-based strategy to integrate physics knowledge in both stages:\n\n- **In the generation phase**: We jointly embed physical fields and geometric shapes into a unified latent space, guiding the design generation toward physically plausible and performance-oriented solutions.\n\n- **In the topology optimization phase**: Our MeshGraphNet-based surrogate model takes both geometry and physical field information as input, providing gradients that guide the refinement process toward improved performance.\n\nWhile not enforcing physical laws explicitly, our approach offers a practical and generalizable way to incorporate physics into 3D inverse design. We will clarify this distinction in the revised manuscript.\n\n> **Comment 5: \"The title is a bit too broad by not including the context (car geometries and aerodynamics)\"**\n\n**A5:** While our method does not explicitly enforce physical laws, it incorporates physics via joint geometry-field encoding and surrogate-based optimization. Since the surrogate learns from data, the approach is not strictly limited to PDE-based systems. However, in this paper, we focus on aerodynamics due to the available datasets. To better reflect this scope, we decided to update the title to: **\"3DID: Direct 3D Inverse Design for Aerodynamics with Physics-Aware Optimization.\"** We will explore broader applications in other physics-constrained domains as part of future work.\n\n> **Comment 6:\"How much topology variation exist in the training data and how much the topology of optimized designs is different from the training data? Are we talking about mesh-topology or the more macro-scale car topology?\"**\n\n**A6:** The DrivAerNet++ dataset[1] includes three distinct car typologies: **Notchback, Fastback, and Estateback**, representing meaningful macro-topological differences. Our generation process stays within these topological families while producing refined geometric variations, as shown in Figure 5.\n\nOur topology-preserving optimization uses Free-Form Deformation (FFD) to maintain mesh-level topology during shape refinement. We validated FFD's effectiveness by sampling 50 instances with different optimization strategies. As demonstrated in the table below, compared to directly updating the vertex position and mesh connectivity following method proposed in [4], FFD preserves watertightness and structural integrity more effectively, avoiding issues like mesh tearing or self-intersections.\n\n**Table5: Comparison of Optimization Strategies**\n\n| Method | Watertight Ratio | Self-Intersection Free Ratio |\n|--------|------------------|------------------------------|\n| FFD | 100% | 100% |\n| Continuous remeshing | 12% | 8% |\n\n> **Comment 7:\"I had difficulty understanding the 2D projection part. Some clarification would be helpful to the paper.\"**\n\n**A7:** We thank the reviewer for noting this. The 2D projection refers to prior works that simplify the 3D inverse design task by utilizing 2D views[5] or silhouette[6] to represent the 3D geometries. These methods assume that optimizing over 2D representations will result in high-performing 3D designs, which is often inaccurate due to the loss of spatial and geometric detail. Our method avoids this limitation by working directly in 3D space. We will clarify this in the revised manuscript.\n\n> **Comment 8: \"Equation 1 is hard to parse mathematically. Please check the correctness.\"**\n\n**A8:** We thank the reviewer for pointing out this ambiguity. We clarify that the objective function $\\mathcal{J}(M)$ depends on both the geometry $M$ and its physical response $F(M)$, and will revise the equation to:\n\n$$\\mathcal{J}(M) := J(M, F(M))$$\n\nThis notation better reflects the functional dependency and avoids confusion with equality. We will update this in the revised version.\n\n[1] Drivaernet++: A large-scale multimodal car dataset with computational fluid dynamics simulations and deep learning benchmarks. 2024\n\n[2] Generalization capabilities of MeshGraphNets to unseen geometries for fluid dynamics. 2024\n\n[3] TripNet: Learning Large-scale High-fidelity 3D Car Aerodynamics with Triplane Networks. 2025\n\n[4] Continuous remeshing for inverse rendering. 2022\n\n[5] Surrogate modeling of car drag coefficient with depth and normal renderings. 2023\n\n[6] Interactive design of 2d car profiles with aerodynamic feedback. 2023"}]

Author rebuttal:

Meta-review: This paper presents 3DID, a framework for direct 3D inverse design that learns a joint physics–geometry latent representation and performs a two-stage optimization with gradient-guided diffusion sampling and topology-preserving refinement. The method addresses the limitations of prior 2D or fine-tuning-based approaches and achieves strong results on aerodynamic shape optimization, validated with both surrogate predictions and ground-truth simulations. Reviewers raised questions on constraint handling, generalization, refinement necessity, and baseline comparisons, but the authors provided thorough clarifications and additional experiments, including ablations, cross-dataset tests, and optimization-only baselines. The paper is technically solid, clearly presented, and tackles a difficult and underrepresented problem with meaningful contributions. Overall, this is a clear accept.
Decision: Accept (poster)