Designing Mechanical Meta-Materials by Learning Equivariant Flows

Thank you very much for reviewing our paper!

$**Is the method first introduced in this paper?**$

Our proposed method, to the best of our knowledge, is first in its kind and none of the existing approaches investigate systematically the effect of pores arrangement on meta-materials mechanical responses. While current methods focused on translational symmetries of a single pore unit cell to construct cellular solids, we developed a machine learning framework that enables a systematic approach to design cellular solids with all 17 2D crystalograhic arrangement of the pores. We showed several designs beyond the reach of translational symmetry. We also modified the introduction and related works sections for more clarity.

$**Neural network architecture and continuity**$

We tested more complex neural network architectures with more hidden layers, more neurons, and different activation functions. But we observed nontrivial changes in the final designs when increasing number of hidden layers and neurons. We also found that $\tanh$ nonlinearity leads to a faster convergence of the optimization process among other activation functions we tested e.g., sine function. The simplest neural network architecture that consistently proposes meta-material designs in all 17 symmetry groups is reported in Line 319 of the original paper (fully-connected neural network with two hidden layers of size 10, with $\tanh$ nonlinearity).

We like to point that the proposed network is continuous and able to propose all designs cases we aimed for. However, there could be some other machine learning alternatives that can also work efficiently in our computational framework.

Dear Reviewer, Do you mind letting the authors know if their rebuttal has addressed your concerns and questions? Thanks! -AC

Thanks for your comments. We added some additional clarifications.

We modified the introduction and the related works sections to improve the clarity. But we briefly discuss them here: Cellular solids are porous structures with the property that the geometric features of the pores define their mechanical responses, rather than their chemical composition. Therefore, the nonlinear functionalities of cellular solids can be programmed by: i) optimizing the geometry of the pores, and/or ii) manipulating the ``arrangement" of the pores. Existing design approaches have been merely focused on the former, optimizing the pore shape of a unit cell that is used to construct a cellular solid with its translational symmetry. However, manipulating of the arrangement (that has not been studied before) can lead to uncommon mechanical behaviors (as we showed in this paper). In addition, the existing approaches use level-set method or moving morphable voids method which suffer from designs with disconnected regions and the associated numerical challenges. Our contribution is: we proposed a novel machine learning framework, the first of its kind, for learning richer classes of cellular solids, in which we not only optimize the pore shapes but also explore all possible arrangements among two-dimensional crystallographic symmetry groups. Moreover, another advantage of our proposed approach is that it ensures the solution avoids disconnected regions and associated numerical challenges via leveraging an equivariant and divergence-free neural network flow. We showed several designs from 17 2D symmetry groups that can significantly outperform designs with p1 symmetry (translational symmetry). For example, linear force-displacement responses under tension and Negative Poisson's ratio under tension and compression.

Please find the revised version of the Introduction section for more details.

$**No comparison with existing approaches**$

Our proposed method is the first of its kind, to the best of our knowledge, that enables a systematic approach to study the effect of pores ``arrangement" on mechanical responses of cellular solids. The most effective way for comparison we realized is to compare results from all wallpaper symmetry groups with cellular solids constructed with translational symmetry (p1 symmetry). This comparison is provided in all the results sections. We showed that designs with other 16 symmetry groups outperform the design with translational arrangement except in one case. This observation confirms that modifying both pore shapes and pore arrangement simultaneously enable a drastic expansion of the design space of cellular solids.

$**Typos**$

All the typo suggestions are fixed now.

$**Please compress the pdf**$

Although we added new figures to the paper, we reduced the size to $\sim$12 MB while maintaining the quality.

$**Please make it clear in Fig.1 (visually or in caption) that the columns of (b) correspond to a single sample while rows are different stages of deformation.**$

We clarified this now visually in Fig. 1.

$**Please show in the plots examples of the behaviour for cases when the target was not obtained (Design under uniaxial compression for symmetry groups other than p1, Design under uniaxial tension for p1 and$\beta=1.5$**$

We included a new section in the appendix (Appendix 8) with figures depicting designs other than p1 symmetry group that were not able to produce target force-displacements curve under uniaxial compression. And also designs for p1 with $\beta=1.5$ under uniaxial tension are added in Appendix 5.

Thank you very much for reviewing our paper!

$**2D meta-materials**$

We agree that the design space of two-dimensional (2D) cellular solids is inherently limited, whereas three-dimensional (3D) meta-materials have the potential to offer a broader range of functionalities. However, we believe for this paper 2D cellular solids were a great ansatz, where we showed several interesting nontrivial mechanical responses with our 2D designs. We would like to note that our equivariant neural network flow implementation is capable of designing 3D cellular solids. But our continuum mechanics solver have trouble at large numbers of degrees of freedom, due to the iLU factorization we perform to precondition the iterative solver. Future extensions include modern multigrid methods to scale our solver up or speeding up the simulations leveraging learned surrogate models, and neural network-based order reduction techniques.

$**Validation with real-world prototypes**$

We agree that the way we describe validation with real-world prototypes is confusing especially in the abstract. We are actually validating the simulated mechanical responses of the designed model against the behavior of fabricated meta-material counterparts in real-world experiments. to clarify this, we modified the sentence in the abstract to ``We validate simulated mechanical behaviors of these new designs against fabricated real-world prototypes."

$**Convergence study**$

Thanks for pointing this out. We included the convergence study with respect to mesh resolution in the appendix (Appendix 4) and we modified line 359 of the original version of the paper to ``there are at least $\sim$25,000 degrees of freedom in all mechanics models that found satisfactory through a mesh independence analysis described in Appendix A.4, such that simulation results were not affected by insufficient discretization resolution."

Dear Reviewer, Do you mind letting the authors know if their rebuttal has addressed your concerns and questions? Thanks! -AC

I thank the authors for addressing my comments. They were sufficiently addressed. I think that this is a good paper with valuable contributions and I have thus raised my score.

$**Is there any specific reason why such small networks were used?**$

We tested more complex neural network architectures with more hidden layers, more neurons, and different activation functions. But we observed nontrivial changes in the final designs when increasing number of hidden layers and neurons. We also found that $\tanh$ nonlinearity leads to a faster convergence of the optimization process among other activation functions we tested e.g., sine function. The simplest neural network architecture that consistently proposes meta-material designs in all 17 symmetry groups is reported in Line 319 of the original paper (fully-connected neural network with two hidden layers of size 10, with $\tanh$ nonlinearity).

Thank you for your review of our paper.

$**Not a generative model**$

We agree that there could be multiple designs that satisfy the desired properties. Our proposed method is not a generative model and we would like to elaborate on this. Generative modeling has been used to tackle challenging design problems in various engineering fields such as designing molecular structures, proteins, and mechanical CAD models. However, generative ML models may struggle to enforce physical, functional, or manufacturability constraints intrinsic to mechanical systems we are interested in our paper. When designs are well-represented as a parametric family (such as implicitly defining the design domain in this work), direct optimization is often a highly-effective way to perform rational design. This approach has a long history in computational mechanics via, e.g., the adjoint method, but advances in machine learning have dramatically increased its potential. Modern automatic differentiation tools such as JAX make it possible not only to construct differentiable physics solvers, but they also enable close coupling with ML frameworks, which can significantly enhance the design capability as shown in this paper. We note that to make sure multiple designs (if needed), we explore among all wallpaper symmetry groups.

$**Suitable for this venue?**$

In the call for papers, ICLR accepts ``applications to physical sciences (physics, chemistry, biology, etc.)". We believe this is a solid application paper with a novel setup that has demonstrated good, interesting, and unintuitive results to engineering designs. We believe we have introduced a new machine learning framework that leverages equivariant flow to design cellular meta-materials. We tried to have detailed clarification about meta-materials related terms for the unfamiliar reader as you highlighted as the strength of the paper. With regard to the impact to meta-material design community, we showed that our proposed framework drastically expand the design space of cellular solids. While existing approaches merely focused on optimizing the shape/topology of the pores, our proposed method also enables (a systematic approach of) exploring various arrangements of the pores structure. As confirmed by our results, this approach enables designs beyond the reach of existing methods. Just to clarify, the existing approach is focused on designing cellular solids constructed via translational symmetric arrangement (p1 group) of a single unit cell. However, we showed several designs with other symmetry groups that can significantly outperform designs with p1 group. For example, linear force-displacement responses under tension and Negative Poisson's ratio under tension and compression. In addition, our framework ensures the solution avoids disconnected regions and associated numerical challenges observed in existing approaches such as level-set method via implicitly defining the geometry by a neural network flow.

I thank the authors and their explanations have addressed all of my concerns. I have raised my score.

Dear Reviewer, Do you mind letting the authors know if their rebuttal has addressed your concerns and questions? Thanks! -AC

Thank you for reviewing our paper.

$**Lack of clarity**$

We modified the introduction and the related works sections to improve the clarity. But we briefly discuss them here: Cellular solids are porous structures with the property that the geometric features of the pores define their mechanical responses, rather than their chemical composition. Therefore, the nonlinear functionalities of cellular solids can be programmed by: i) optimizing the geometry of the pores, and/or ii) manipulating the ``arrangement" of the pores. Existing design approaches have been merely focused on the former, optimizing the pore shape of a unit cell that is used to construct a cellular solid with its translational symmetry. However, manipulating of the arrangement (that has not been studied before) can lead to uncommon mechanical behaviors (as we showed in this paper). In addition, the existing approaches use level-set method or moving morphable voids method which suffer from designs with disconnected regions and the associated numerical challenges. Our contribution is: we proposed a novel machine learning framework, the first of its kind, for learning richer classes of cellular solids, in which we not only optimize the pore shapes but also explore all possible arrangements among two-dimensional crystallographic symmetry groups. Moreover, another advantage of our proposed approach is that it ensures the solution avoids disconnected regions and associated numerical challenges via leveraging an equivariant and divergence-free neural network flow. We showed several designs from 17 2D symmetry groups that can significantly outperform designs with p1 symmetry (translational symmetry). For example, linear force-displacement responses under tension and Negative Poisson's ratio under tension and compression.

Please find the revised version of the Introduction section for more details.

$**No comparison with existing approaches**$

Our proposed method is the first of its kind, to the best of our knowledge, that enables a systematic approach to study the effect of pores ``arrangement" on mechanical responses of cellular solids. The most effective way for comparison we realized is to compare results from all wallpaper symmetry groups with cellular solids constructed with translational symmetry (p1 symmetry). This comparison is provided in all the results sections. We showed that designs with other 16 symmetry groups outperform the design with translational arrangement except in one case. This observation confirms that modifying both pore shapes and pore arrangement simultaneously enable a drastic expansion of the design space of cellular solids.

$**Not a data-driven approach**$

We would also like to note this approach is not a data-driven method, instead we utilized direct optimization technique as a highly-effective way to perform rational design when the design domain is well-represented parametrically (as we implicitly defined design domain via neural network flow).

Dear Reviewer, Do you mind letting the authors know if their rebuttal has addressed any of your concerns or questions? Thanks! -AC