3DID: Direct 3D Inverse Design for Aerodynamics with Physics-Aware Optimization

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.

Comment 1: "there should have been references to the works that encode both design and physical properties as well as works that do refinement."

A1: We thank the reviewer for pointing this out. We will incorporate additional references to prior works in the revised version.

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."

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.

To 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.

Table 1: Comparison of Training vs. Generated Designs

Table 2: Optimization Results on Best Training Designs

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."

“How general is the surrogation? With the 8000 training data, can one use the trained surrogates for slightly changed conditions/applications?"

A3: The surrogate model in our framework serves two main purposes:

• Computational efficiency: The surrogate model accelerates the optimization process by amortizing the training cost over multiple inference steps, reducing reliance on expensive computational solvers.

• Gradient information: It provides differentiable gradient information, enabling efficient gradient-based optimization.

Additionally, 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.

Table 3: Surrogate Model Generalization Performance

We 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].

Table 4: Pipeline Performance with Partial Surrogate Training

UNet Guidance:

GNN Optimization:

Finally, 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.

Comment 4:"The fact that the physical fields are used jointly for training the encoder, does not fit the (loose) definition of physics-aware."

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.

Instead, our framework adopts a surrogate-based strategy to integrate physics knowledge in both stages:

• 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.

• 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.

While 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.

Comment 5: "The title is a bit too broad by not including the context (car geometries and aerodynamics)"

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.

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?"

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.

Our 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.

Table5: Comparison of Optimization Strategies

Comment 7:"I had difficulty understanding the 2D projection part. Some clarification would be helpful to the paper."

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.

Comment 8: "Equation 1 is hard to parse mathematically. Please check the correctness."

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:

$\mathcal{J}(M) := J(M, F(M))$

This notation better reflects the functional dependency and avoids confusion with equality. We will update this in the revised version.

[1] Drivaernet++: A large-scale multimodal car dataset with computational fluid dynamics simulations and deep learning benchmarks. 2024

[2] Generalization capabilities of MeshGraphNets to unseen geometries for fluid dynamics. 2024

[3] TripNet: Learning Large-scale High-fidelity 3D Car Aerodynamics with Triplane Networks. 2025

[4] Continuous remeshing for inverse rendering. 2022

[5] Surrogate modeling of car drag coefficient with depth and normal renderings. 2023

[6] Interactive design of 2d car profiles with aerodynamic feedback. 2023

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.

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."

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.

Table 1: Cross-Dataset Generalization Results on AhmedML

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."

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.

We 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.

Comment 3: "In Eq (2), how to formulate "constraints, such as volume, manufacturability, and boundary conditions"

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:

$M^* = \arg\min_M J(M, F(M)) \quad \text{s.t.} \quad C(M, F(M)) = 0.$

Here, 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.

Comment 4: "In Eq (5), the = should be replaced by $\propto$"

A4: We thank the reviewer for pointing out this notational issue. We will fix this in the updated version.

Comment 5: "What does the KL divergence mean in Eq. (3)?"

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

$\mathcal{L}_{\mathrm{KL}} = \mathrm{KL}\left( q(z \mid M, F(M)) \| \mathcal{N}(0, I) \right) .$

Where q is the posterior over the latent code, and N is the Gaussian distribution.

Comment 6: "how does the hyperparameter $\gamma$ effect the performance?"

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.

Table 2: Effect of Guidance Scale on Generation Performance

Comment 7: "How to pretrain the surrogate model $f_\text{GNN}$? How does it deal with out-of-distribution prediction?"

A7: We thank the reviewer for pointing out this missing detail, and we will include the full explanation in the revised version.

We 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:

1. Noise augmentation during training by perturbing mesh node positions, which helps the model generalize better to unseen shapes.
2. Shape regularization during refinement optimization, which constrains the updated geometry to remain close to the training distribution. These strategies together ensure that both the surrogate and the final designs remain reliable and physically plausible.

Comment 8: "In Figure 4, Voxel+PCA produces non-watertight shapes. How does 3DID guarantee watertightness?"

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.

Table 3: Geometric Validity Assessment of Generated Automotive Designs

[1] AhmedML: High-Fidelity Computational Fluid Dynamics Dataset for Incompressible, Low-Speed Bluff Body Aerodynamics. 2024

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.

Comment 1: "There is no explicit enforcement of hard physical or geometric constraints during sampling."

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.

Comment 2: "How often does the sampling stage produce geometries that are physically invalid or impractical?"

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.

Table 1: Geometric Validity Assessment of Generated Automotive Designs

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.

Comment 1: "There is no explicit enforcement of hard physical or geometric constraints during sampling."

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.

Comment 2: "Formalizing the guided diffusion by referencing known results in score-based generative modeling would address theoretical concerns."

A2: In our method, we adjust the predicted noise at each denoising step by injecting the gradient of the task objective $J$

$\epsilon \prime_\phi(z_t, t) = \epsilon_\phi(z_t, t) + \gamma \nabla_{z_t} J(\hat{z}_0(z_t)).$

This 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]:

$\nabla_{z_t} \log p(z_t | J) = \nabla_{z_t} \log p(z_t) + \nabla_{z_t} \log p(J | z_t),$

where 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.

Comment 3: "A conceptual discussion of failure modes and how the method mitigates them shows theoretical awareness." Ablations for key hyperparameters are needed due to the complex workflow.

A3: The influence of hyperparameters is also concerned by Reviewer aPc8. We address this concern through two key ablation studies.

1. Ablation on Guidance Scale ($\gamma$):

We 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.

Table 1: Effect of Guidance Scale on Generation Performance

2. Ablation on Refinement Regularizers:

We 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.

Table 2: Effect of Regularization Terms on Refinement Performance

Comment 4: "It is not clear how the method compares against optimization-only pipelines such as FFD+adjoints or FFD+heuristic search." "What are the key benefits that will convince practitioners to adopt this methodology compared to existing optimization-only approaches?"

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.

Comment 5: "Demonstrating any degree of generalization or adaptability would greatly strengthen the work's impact."

"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"

"How accurate does the encoding surrogate have to be for integration in the full pipeline?"

A5: We thank the reviewer for raising these important questions regarding generalization. We address them from two perspectives:

(1) Generalization of our pipeline To 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.

Table 3: Cross-Dataset Generalization Results on AhmedML

(2) Generalization of our surrogate model

To 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.

Table 4: Pipeline Performance with Partial Surrogate Training

UNet Guidance:

GNN Optimization:

Comment 6: "The method is physics-aware only through training data; it doesn't enforce physical laws during generation." "The method relies on a surrogate trained from simulations without enforcing physical laws, limiting generalization and requiring retraining in new domains."

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.

Instead, our framework adopts a surrogate-based strategy to integrate physics knowledge in both stages:

• 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.

• 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.

While this does not enforce physical laws in the strict sense, it enables a scalable and practical way to incorporate physics into 3D inverse design.

[1] Score-Based Generative Modeling through Stochastic Differential Equations. 2021

[2] AhmedML: High-Fidelity Computational Fluid Dynamics Dataset for Incompressible, Low-Speed Bluff Body Aerodynamics. 2024

[3] Drivaernet++: A large-scale multimodal car dataset with computational fluid dynamics simulations and deep learning benchmarks. 2024