Geometry-Informed Neural Networks

We strongly disagree with the reviewer and want to clarify the following:

• It is well known that topology optimization can create near-optimal designs in many engineering applications [1]. The reviewer’s claim is factually incorrect: “Most notably, there aren't many or even any good examples of where optimization alone can give rise to interesting designs.”
• Moreso, these methods start from a solid material block and do not require a good initialization as suggested by the reviewer. “Most of the time, this requires a good initialization that is already a design or the combination of user input (design) and optimization.”
• “The current submission does not have a demonstration of either 1, 2, or 3.” Steps 1 and 2 are indeed what we demonstrate, as explicitly illustrated in Fig. 3. The reviewer’s proposed approach “first generate a set of objects by optimization (e.g. furniture) and then train a generative model on this set” has been tried multiply before and is deemed impractical [2].
• The suggested reference is far-fetched as it requires data and human-feedback in the loop. Even so, it is concluded that "it cannot [..] evolve it into a richly diverse set consisting of complex and advanced shapes." Our method is data-free and explicitly produces diverse shapes.

We hope the provided information helps the reviewer reconsider their assessment and we are happy to discuss further questions.

[1]: Jihong Zhu et al., A review of topology optimization for additive manufacturing: Status and challenges, Chinese Journal of Aeronautics

[2]: Rebekka V. Woldseth et al., On the use of Artificial Neural Networks in Topology Optimisation, Springer: Structural and Multidisciplinary Optimization

First, we disagree that our arguments are subjective - we provide concrete statements from the literature that objectively contradict the reviewer's claims.

Second, we validate the proposed framework by optimizing shapes for minimal surface area, smoothness, connectedness, higher-order topological properties, and even a physics task with solution multiplicity. Regardless of the reviewer's preference for the shapes, the framework functions as intended, solving the specified design problems.

Third, our experiments are designed to clearly demonstrate key properties of the proposed machine learning framework: constraint satisfaction, objective minimization, scalability to 3D problems, generative ability recovering multiple diverse solutions with the capacity to interpolate and structure the latent space. We would invite the reviewer to judge this submission on the stated contributions instead of singling out the "abstractness" of the results.

We thank the reviewer for their comments. We would like to clarify the raised concerns and questions.

W1: No established baselines: We affirm the reviewer’s question that there are no established baselines that train a shape-generative model without data. The essence of a generative model is to map a source to a target probability distribution. In the data-free regime, classical topology optimization generates a single shape. But no prior work trains a generative model for shapes without training data. As such, we are not aware of a direct comparison that we could make.

“Does it mean that there is no related work trying to solve the engineering design problem presented in Section 4.3??”

The jet-engine bracket problem can be solved using topology optimization (TO) [1]. However, we do not consider TO an insightful baseline, as most metrics would be nearly trivial (Betti-numbers: 1, interface- and envelope-metrics: 0). On the other hand, naive TO and our results use different objectives (compliance vs surface smoothness), which also makes the comparison w.r.t. these metrics not meaningful. Lastly, the primary focus of our work is the generative aspect to produce diverse solutions. As classical TO can only produce a single solution, the diversity comparison would provide little value. We hope these arguments clarify why we omit a comparison. If the reviewer disagrees, we could provide a comparison to TO or the SimJEB dataset [2].

W2: We thank the review for their feedback on the format and structure. In Section 4 we state “we define metrics for each constraint in the main problem as detailed in Appendix C.1”. We reformulated this sentence to more clearly state that the metrics definitions are in Appendix C. As the paragraph titles “topology”, “smoothness”, “surface sampling” etc. in Section 4.1 highlight different experimental details, we would leave this structure unchanged. Please let us know if further clarification or changes are needed.

[1]: Carter et al., The GE aircraft engine bracket challenge: an experiment in crowdsourcing for mechanical design concepts, 2014 University of Texas at Austin

[2]: Whalen et al., SimJEB: Simulated Jet Engine Bracket Dataset, Computer Graphics Forum 2021

We thank the reviewer for their time and effort in reviewing our submission and the overall positive outlook. We would like to clear the raised concerns.

W1, W4, Q: Universality: The proposed approach is at least as universally applicable as any classical shape or topology optimization (TO) approach. Classical TO methods are predominantly gradient-based for which the constraints must satisfy the same properties. As such, we can reuse numerous existing works [e.g., 1-4], which cover most practically relevant constraints. Of course, some requirements or implicit know-how are hard to describe formally or efficiently, which is a known limitation of computational design.

We would also like to address the statement that “the experiments are conducted on only one workpiece”. While the jet-engine bracket is our primary experiment, we demonstrate three other geometry problems. In addition, our approach generalizes to other scientific domains where the problems are ubiquitously formalized as differential or integral equations, as we demonstrate with the reaction-diffusion system in Figure 4. These experiments echo the same main findings. This can be done with minimal hyperparameter tuning thanks to the adaptive Lagrangian method which balances the different constraints automatically further supporting universality.

Lastly, the final experiment itself is set up to illustrate generality, as it seeks general, non-extruded, asymmetric 3D parts with many common and transferable constraints.

W2: Optimization with a NN representation: The reviewer is correct that optimizing neural fields with objectives has been demonstrated before, as we also describe in related work using PINNs and TO. The primary novelty of our contribution is a framework for training shape-generative neural fields without data by leveraging design constraints and avoiding mode-collapse using a diversity constraint.

W3: Diversity in Fig. 6: The shapes in Fig. 6 are diverse and close-to-optimal w.r.t. to the defined objective of surface smoothness. As we describe in Section 4.3 and Limitations, this objective is different from physical compliance, which would produce more conventionally looking shapes, but would add computational and implementation complexity that detracts from the primary investigation.

Generation of a nut or joint bearing: Generating a nut that matches a specific bolt is fairly straight-forward in this framework. It would require defining the interface to the bolt analogous to the already demonstrated interface constraints. This can be represented either by a discrete mesh or the analytical expression of the swept thread profile known from the engineering standards. A similar procedure must be carried out for the likely desirable flat top and bottom interfaces, as well as the outer interface, which in the simplest form would be two flat and parallel surfaces to take the wrench. This can optionally be extended with angular symmetries. To connect these interfaces and make the part structurally integral, we can either reuse the presented connectedness constraint or add a mechanical stress constraint. However, since bolts are standard parts applying computational design here is probably an overkill, but we hope this explanation provides further insight and we are glad to discuss further questions.

[1] Vatanabe S. et al., Topology optimization with manufacturing constraints: A unified projection-based approach, Advances in Engineering Software, 2016.

[2] Liu J. et al., A survey of manufacturing oriented topology optimization methods, Advances in Engineering Software, 2016.

[3] Ebeling-Rump, M. et al. Topology optimization subject to additive manufacturing constraints. J.Math.Industry, 2021.

[4] Hammond A. et al., Photonic topology optimization with semiconductor-foundry design-rule constraints, Opt. Express, 2021.

We thank the reviewer for their time and effort in reviewing our submission and the overall positive outlook. We would like to clear the raised concerns.

W1: We do not claim the individual constraints as contributions. An advantage of GINNs is the ability to include different existing domain-specific constraints. For a more detailed discussion on this, we refer to our response (universality) to the reviewer 7pFZ.

W2: The conversion to explicit representations is a known limitation whenever working with implicit representations, such as in state-of-the-art topology optimization or shape-generative models. Correspondingly, there are established methods for extracting meshes, such as marching cubes in the simplest case. It is also worth mentioning that it is also possible to work and simulate directly on the implicit representations, which is an emerging trend in both academia [1] and industry [2,3].

W3: We completely agree with the reviewer that “it would be interesting to see if these methods can be used in combination with available data.” Despite our original motivation being training models on extremely sparse shape datasets, we left this for future work to focus on the key novel aspects, such as the diversity constraint and the training dynamics. We have preliminary results indicating that adding data is possible, but there are many design choices and comparisons that have to be made which we omitted to maintain the focus of this work.

Q1, W4: Hyperparameter tuning: We use the adaptive ALM algorithm (with default RMSProp parameters) to avoid hyperparameter tuning of the different loss weights. We initialize the optimizer such that the loss terms are on the order 1. We use a simple WIRE MLP with 3 hidden layers with 128 neurons each. WIRE is quite robust to hyperparameter choice, as shown in Fig. 3 of the WIRE paper [4].

Q2: In addition to the three simpler geometry problems and the one physics problem, we selected the jet-engine bracket problem as it is a publicly available and very general 3D design challenge without any exploitable symmetries. We could offer to solve another illustrative design problem using GiNNs, e.g. wheel design similar to a recent data-driven design paper [5].

[1]: Sawhney et al., Monte Carlo geometry processing: A grid-free approach to PDE-based methods on volumetric domains, ACM Transactions on Graphics 2020

[4] Saragadam et al., WIRE: Wavelet Implicit Neural Representations, CVPR 2023

[5] Jang et al., Generative design by reinforcement learning: enhancing the diversity of topology optimization designs, Computer-Aided Design 2022