Designing Mechanical Meta-Materials by Learning Equivariant Flows
Authors: Mehran Mirramezani
Anne S. Meeussen
Katia Bertoldi
Peter Orbanz
Ryan P Adams
Venue: ICLR 2025 Poster (2025)
Keywords: neural network equivariant flow
inverse design
differentiable simulator
crystallographic symmetries
cellular solids
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)

Abstract: Mechanical meta-materials are solids whose geometric structure results in exotic nonlinear behaviors that are not typically achievable via homogeneous materials. We show how to drastically expand the design space of a class of mechanical meta-materials known as $\textit{cellular solids}$, by generalizing beyond translational symmetry. This is made possible by transforming a reference geometry according to a divergence free flow that is parameterized by a neural network and equivariant under the relevant symmetry group. We show how to construct flows equivariant to the space groups, despite the fact that these groups are not compact. Coupling this flow with a differentiable nonlinear mechanics simulator allows us to represent a much richer set of cellular solids than was previously possible. These materials can be optimized to exhibit desirable mechanical properties such as negative Poisson's ratios or to match target stress-strain curves. We validate these new designs in simulation and by fabricating real-world prototypes. We find that designs with higher-order symmetries can exhibit a wider range of behaviors.

Published as a conference paper at ICLR 2025
DESIGNING MECHANICAL META-MATERIAN BY LEARNING EQUIVARIANT FLOWS
Mehran Mirramezani
Anne S. Meeussen & Katia Bertoldi
Department of Computer Science
School of Engineering and Applied Sciences
Princeton University
Harvard University
m. mi mmirramezani@eeeeeeeee. edu
(meeussen, bertoldi @seas. harvard.
Peter Orbanz
Ryan P. Adams
Gatsby Computational Neuroscience Unit University College London
Department of Computer Science
Princeton University rpa rpa@princeton.. edu
p. orrbanzuucl.ae
ABSTRACT
Mechanical veta-materiias porous solids whose geometric structure results in exotic nonlinear mechanical behaviors that are not typically achievable via homogeneous materials. We show how drastically expand the design space of a class of mechanical meta-materials known as cellular solids, by generalizing beyond translational symmetry of unit pore cell. This is made possible by transforming a reference geometry according to a divergence free flow that is parameterized by neural network and equivariant under the relevant symmetry group. We show how to construct flows equivariant to the space groups, despite the fact that these groups are not compact. Coupling this flow with a differentiable nonlinear mechanics simulator allows us to represent much richer set of cellular solids than was previously possible. These materials can be optimized to exhibit desirable mechanical properties such negative Poisson's ratios to match target stress-strain curves. We validate simulated mechanical behaviors of these new designs against fabricated higher-order symmetries can exhibit a wider range of behaviors.
prototypes. We find that designs with
INTRODUCTION
Mechanical meta-materials engineered porous structures in which geometric features of the pores lead uncommon mechanical behaviors not achievable from natural materials (Bertoldi et al., 2017) with various applications in soft robotics (Khajehtouriaan & Kochmann, 2021). biomedical devices (Wang et al., 2023), etc. One promising class mechanical mata-iaterials are cellular solids, characterized by an array of pores, that exhibit exotic nonlinear properties such auxeticity (expanding under tension) (Bertoldi et al., 2010) and reversible shape morphing (Wenz et al., 2021). 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 approaches cellular solids design have been focused on the former, optimizing the shape (Overvelde & Bertoldi, 2014; Xue & Mao, 2022; Medina et al., 2023) or topology (Wang al., 2014: Gao al., 2019) of the pores of a cellular solid constructed by translational symmetric arrangement of a single pore unit cell. The design of such cellular solids has also begun to attract substantial attention from the machine learning community, e.g., Beatson et al. (2020); Ma al. (2020); Kollmann et al. (2020); Ha et al. (2023). Figure 1(a) shows examples of such cellular solids designs from Overvelde et al. (2012). However, the space of potential symmetric arrangement the pores is not limited only to translation, and exploiting other spatial patterns of pores can potentially unlock new cellular solids with unexpected mechanical responses. Representing and optimizing this larger space of cellular solids is challenging, but recent developments in machine learning of crystallographhiclly inavaiiiallly unconstrained parameterizations of the symmetric arrangement of the pores that can be used for
invariant functions (Adams & Orbanz, 2023) make it possible to construct

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140 odd
trunte tramelattinn conlaationna symmetts
differom: of
(a)
(c)
(d)
Figure 1: (a) Example of three shape functions s: S C R + {0, } constructed by translational
symmetric arrangement of different unit pore cells, from Overvelde et al. (2012). (b) Real-world experiment demonstrating different nonlinear behaviors of fabricated cellular solids specified by the shape functions in (a), from Overvelde et al. (2012). (c) A vector field defined by a flow equivariant under the space group p4, constructed in Section 3. (d) A design based level set of the p4-invariant function in (c). Note that it is disconnected.
shape optimization. In this work, we leverage these models and introduce novel framework for learning richer classes of cellular solids, in which we not only optimize pore shapes but also explore all possible arrangements of the pores from two-dimensional crystallographic symmetry groups.
The approach we take to the problem to define the geometry implicitly via a neural network flow from a reference configuration. By carefully constructing this flow to preserve group invariance, that enables all 17 crystallographically symmetric arrangement of the pores including translation, simple symmetric cellular structures can be transformed parametrically into complex cellular structures, as illustrated in Figure 3. Another advantage of this approach is that it ensures the solution avoids disconnected regions and associated numerical challenges observed in existing techniques such as level set method (Wang et al., 201 4) shown Figure (d) or moving morphable voids method (Du al., 2024). We additionally show that such neural flows can be made divergence-freee which makes it possible to avoid modifying the total volume of the structure, allowing the optimization to focus purely on geometry as the means of solving the task.
The paper is structured as follows. We first introduce cellular solids, framed in terms of a design space that can exhibit symmetries. Section then develops constrained optimization problem for learning shapes which are symmetric with respect to space group while achieving target volume fraction. Our approach uses a novel formalism based on equivariant neural network flows. Next we provide overview of our implementation, including details of the neural network, the differentiable mechanics solver, and the mechanical material model used. We then demonstrate the application of proposed approach in designing cellular solids with nontrivial functionalities in both simulation and real experiments. We conclude with related work and a discussion on limitations and future steps.
2 CELLULAR SOLIDS AND SYMMETRIC SHAPES
An example of a cellular solid is illustrated in Figure 1(a). It consists of a homogeneous solid material, interrupted by empty regions that are called pores. Once the material chosen, the solid is entirely characterized by its geometry. This geometry can be specified by function f -> {0. 1). where is the entire volume occupied by the solid, including pores. This function is called a shape. A value s(x) 1 specifies there is material at location æ, value s(x) = 0 that is in a pore. We are particularly interested in cellular solids that are-like the example in Figure 1(a) caapccllleelly determined by -io-diiensional cross- section. We can then reduce the shape
two-dimensional function Ω C IR² + {0, 1}. where is now the area of the cross-section. The methods developed here can be used construct shape functions Ω IRth for both and real-world experiments assume n 2.
and 3. Our simulations
Periodic designs as tilings. Clearly visible in Figure 1(a) is the periodic structure. This periodicity can be formalized as a tiling: If IT R convex polytope, and a group of translations of R. then T is said to tile the space with IT if the images ΦII of the polytope under elements & T cover the entire space and only their boundaries overlap-formalle if
U कT TET
n
and
ΦT [ I
empty set or face of Øf
(1)
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for all o, T. II is, for example, axis-aligned square in R,, the group T would consist of all horizontal and vertical shifts by multiples of the edge length of A shape s is periodic if there is a function Ø {0, } such that
s(x) = a(br)
x E TT for some & E ET.
In words, specifies a pattern of pores on a small area II, and this pattern is then replicated over Ω by shifts. To avoid dealing with the boundaries of Q explicitly, we can define s as function s 2 + {0. 1 on the entire plane, and restrict it to Q where necessary (see Section 4 for details); observe that (2) indeed specifies s on all of R², if tiles R² with II. We also note that satisfies (2) if and only if it is invariant under T.
s(Øz)
s(x)
for all T and R²
in which case a is the restriction to II.
Expanding the design space. Our point of departure from existing designs is the observation that tilings cannot only be generated by translations. In general, one may substitute T by group G of isometries of I.... Such a group is said to tile R with convex polytope II if (1) holds with G substituted for T. Any group of isometries that tiles Iº" with a convex polytope II is called space group or crystallographic group on R We are particularly interested in the case n = 2 that are also known as wallpaper groups. addition to translations, space groups may also contain rotations, reflections, and other isometries. Our approach is to construct cellular solids with shape functions that satisfy the generalized periodicity
s(gz)
sxx) for all Φ E G and E R.?
for space group G. We call such shape symmetric with respect G.
Properties space groups. The two-dimensional space groups are completely classified: Two space groups G and G2 are isomorphic if one can be transformed into the other by affine deformation of the underlying space, i.e., if there is an affine, order-pressen prder-ppes bijection that o E (2 if and only if cvver E Gg. Up isomorphism, there are only finitely many space groups for each dimension n. For n = 2, there are 17 such groups. These are illustrated in Appendix A. Some properties of space groups relevant in the following are:
R" - R" such
(i) Each space group G is countably infinite.
(ii) Since each element & of G is an isometry, is of the form dæ Agæ + bωs where Ag is an orthogonal matrix and by E R
(iii) G contains countably infinite subgroup of pure translations, namely
Ag = identity) {Φ E d(x) = x + by for some by E }"). This group is always infinite, and generated by n linearly independent shifts: There exist n linearly independent vectors b ........ b, E I such that
Tg == fo
Φ E To
<<x) + Σ I c,(O)b; for some (o)..... (Ø) Z.
We refer to Vinberg & Shvartsman (1993); Ratcliffe (2006) for more on the geometry of such groups, and Hahn et al. (1983) for their crystallographic properties.
SYMMETRIC SHAPES FROM SYMMETRY-PRESERVING FLOWS
We select class of symmetries by fixing a space group G. Our goal is to solve two problems:
Constructively define a set S = (5o 0 E ) of shapes that satisfy (3), indexed by parame- ters in some parameter space θ
II. Given loss function & shape - IR, solve 0* == arg mingee ((sg).
The loss represents how well the shape achieves some physical property, such as the Poisson's 's ratio illustrated in Figure 5. To solve (I), proceed follows:
We start with a symmetric reference shape sp. This is a G-invariant function so R - {0, 10 that satisfies (3).
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pl
pg
p4gm
p3m1
p6
Figure 2: Examples of the reference shape 8o for different space groups. The labels pl, pg, etc. refer to the crystallographic naming standard for such groups see Appendix A.
We then construct class { E ) of flows that are symmetry- and volume-preserving. Each shape Se is defined as the deformation of 50 by Fg- That is, fix "terminal time" max > 0, and define
sa(x) Fo mmax))
These functions are shapes on IR" that satisfy (3), and can be restricted to shapes on Ω.
The remainder of this section explains this construction in detail. This solution of (I) can then be combined with standard differentiable mechanics simulator to solve (II), as described in Section 4.
3.1 SYMMETRY- AND VOLUME-PRESERTE FLOWS
flow is function F RR x RR. + Th that satisfies
Fzz,0) 0) = and F(z,s+t) = F(F(z,s),t)
for x E R,, and s, E R
For our purposes, can restrict the second argument of F an interval / [0, max]] We call flow gemmetry-preservee for space group G if is G-equivariant pointwise in time,
do F(x,t) F(dx,t) for all ò G and (z,t) R" I.
that is the case, then 8g is G-invariant, since 86 G-invariant and so
sœ(σz) = (oFFo( tmax) = so(oFg(I, max)) = so max)) = sp(I). We also require F to be volume-preserving. continuous function g R" ER" is volume- preserving if vol(g(It)) = vol(It for every (measurable) set I C R For flow F, this means vol(R) for every (measurable) set R R and I. This ensures that the ratio of volume occupied by pores does not change shape deformed using F.
vol(F(R,t))
is well known that a flow can be constructed as the solution of a differential equation. If H _ x Rn is (sufficiently smooth) function, the constrained equation
F( t) H(F((o, t),t) subject to F( 0) = Zo
for t E 1 E R
(4)
has unique solution F, and this solution a flow. Fach smooth function H hence determines flow and we can specify a parameterized class of flows Fg by specifying parameterized class of smooth functions Hq. Theorem below shows that the flow F is volume- and symmetry-presenvee H is volume-preserve and satisfies
H Hób f) AzH(I, t)
for ø E G,T
(5)
To construct such function H, define symmetrization operator I' and operator A that turns functions into volume-preserving functions.
3.2 VOLUME PRESERVATION
To define A, we can draw on existing work: continuously differentiable function g is volume- preserving if and only if it is divergence-
div (xx) (V (x) for all R
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It was noted in Richter-Powee al. (2022) that, if g R (p 11/2 is function that is twice continuously differentiable then
0
(29)(x)
(1)2(x)
0n (x)
(x) -x
0
Baga (x)
" 02n-
(x))
(2gn(x)
(6)
- 11 (xx)
- -0agan-at (x) - -0g8sn-6(4)
divergence-free function Ag IR" + R².
3.3 SYMMETRIZATION
For finite groups, invariant or equívariant functions are typically constructed by the "summation trick", which starts with function and sums over all compositions f with group elements. This is not possible for space groups, since these groups are countably infinite. We can, however, decompose the symmetrization into steps. This decomposition based on the following property.
Theorem 1. Let & be space group IR., For each & G, denote by c,(0) the linear coefficient of by with respect to the shift basis vector by that is, o(x) = A&I + Eign c((0)b, Then
G = {Ø E G G....))..... E 10,
finite subset of G. For each o G, there unique elements & E G and Ts Te such that & ② + T If R + R is invariant under the shift subgroup Te of G, the fanction
(T)) = 1
Eoee A5'f(xz)
satisfies Ax(Tf) (I')) o for each @ E G.
Proof. See Appendix A.
To construct the Tg-invariant function required by the result, we use maximal invariants: Define an n X n-matrix B as
= matrix whose columns are the lattice basis vectors by....,b,.
This matrix describes change of basis from the cartesian basis to bi...., bn The set {B- E To therefore consists of all shifts by vectors with integer entries, or in other words, B B-TT dn. Define a function p : R= + 222n as
p(x) p( B 2) where
Ç(u) (cos(2 (2os() sin (o sin sin(2ru,)))
This function maps R= onto the 2n-dimeensionll torus. It is well known that ý is a maximal invariant for the shift group z". This means that a function g on IR" is Z" invariiant if and only if it can be represented as g g' for some function g' on IR. 2m, It follows that p is maximal invariant for Tç-
3.4 FLOW CONSTRUCTION
We can now combine Г, A and p to construct divergence- function H that satisfies (5) from a smooth function h, the next result shows. We therefore neural network with parameter vector 0 to represent class {hg]0 E {} of such functions. For each 0, we then obtain a unique flow Fg that preserves volume and symmetry.
Theorem 2. Let G be space group E.. Choose function h 2 x - [Rn(n-1)/2 that is Lipschitz-continnuut and (k: + 1)-times continuously differentiable on for some E Then
(x,,t) = I(A(h(p(2),t)))
for (x,t) R x
is divergence-fee function Tnth x I - RR. that satisfies (5). For this function H, the differential equation (4) has unique solution F. The function F' is a flow, is volume colume-preserving and is symmetry- preserving for G. It is continuous in its first argument, and k times continuously differentiable in the second.
Proof. See Appendix A.2.2.
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cellular mntea-iattrial
flowed model w symı.
neural network flow (
initial model
symmetry volume- preserving
pulling
differentiable mechanics solver
/target shape
final design
Figure An overview of our modeling framework implemented JAX. An equivariant neural network parameterized by 0 works in tandem with a differentiable mechanics solver to flow an initial geometry learn cellular solids with desired functionalities from rich set of volume- and pyrseery geometries. The upper left shape is su and the upper right shape is 8g-
4 IMPLEMENTATION
To design cellular solids, we combine the shape representation defined above with a mechanics simulator. We proceed as follows:
1) Start with reference shape $p- This is generated by selecting space group G and tiling convex polytope IT for this group. We then define a set of pores on II and tile with the images øIl under group elements Φ E G such that the entire region Ω is covered.
2) The region {a so(x) = 1} represented by triangular mesh using pygmsh, an open-source python package for geometry and mesh processing. This yields discretization of the shape in terms of N points R.2.
3) The image the mesh points under a flow F, is constructed as in Section to define symmetric shape covering the same volume So for each parameter value 6.
Using mechanies simulator, simulate the behavior of 8, under physical effects, such lateral deformation. We define cost function that measures the response to given physical effect, in such a way that small values of l encode desired behavior.
5) We minimize with respect to using gradient descent, as described more detail below.
The overall setup illustrated in Figure 3.
To solve the optimization problem, we must backpropagate gradients through both the mechanics solver and the flow representation. This involves solving static nonlinear elasticity problem, described in Appendix A.3. To do so, we use and-to-end differentiable continuum mechanics solver (Oktay, 2024). This solver is based on variant of the finite element method called isogeometric analysis (IGA) (Hughes et al., 2005; Hughes, 2012). IGA represents both geometry and solutions in the same smooth B-spline basis functions. This allows us to directly construct differentiable maps from (i) geometry parameters to the solution, and (ii) to specified loss function defined by the solution. The optimizer implemented in JAX (Bradbury et al., 2018), and uses automatic differentiation and adjoint methods.
Differentiable mechanics simulator.
Neural network ansatz. The function ho in Section is represented by a fully-connected neural network with two hidden layers of size 10, with tanh nonlinearity. To optimize these parameters, we use ADAM, with learning rate of 0.001. Each optimization is performed several times with different initialization of the neural network parameters.

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Boundary conditions. In Section 3, shapes are constructed functions I", here for 2. To represent cellular solids, the function must be restricted to the finite volume Ω2 and we must ensure this restriction respects boundary conditions. To ensure preservation of the boundaries regardless of the parameters 0, we use an envelope function which takes H (x, () -> 0 at the edges of f2.
5 APPLICATIONS
Our experiments design novel cellular solids with two types of nonlinear behaviors: 1) nontrivial diccedisplacement responses, and 2) effective Poisson's 's ratios' under tension and compression. These types of behaviors important in engineering applications such soft roboties and programmable actuations. Here we explore examples of such behavior that would be challenging or impossible to achieve with homogeneous materials.
In uniaxial loading we apply a displacement uniformly to the top edge of the material while fixing the bottom edge; the left and right boundaries are free move. The force-displacement response is usually described by a nominal stress-strain curve, in which nominal stresses (S) in our experiments are computed as the resulting vertical reaction forces per thickness (in the bottom boundary) normalized by the width of the cellular solids. Strain (e) values are the applied displacements per meta-material height (H) that in this study we use a displacement of equal to 10% of the height of the structure resulting in the final applied strain of 0.1.
The effective Poisson's 's ratios Vef are approximated using horizontal displacement in region of size H/2 in the middle of left and right boundaries (S2, and S, see Figure 5 top left) as in Medina al. (2023):
[22 / undss
Vef HE 14
ds
When targeting dinelaiiplac responses we use curves that are obtained at equally spaced displacement loading increments applied at the top edge to minimize loss -= - (c)) with respect to a target 5 response. We also design cellular solids with target Poisson's 's ratios 0.1.
The mechanical behavior of cellular solids in this study is captured with a nearly incompressible neo-Hookean model with strain energy density function given in (13) that has material properties of Young's 's modulus E = 50 MPa and Poisson's 's ratio y = 0.46. In our experiments, cellular solids are of size 7 with initial material volume fraction of 50% that is preserved during designs.
In all force-displacement experiments, neural network parameters are optimized until a loss value less than 0.0001 is achieved, where for Poisson's 's ratio designs loss value less than 0.01 was used for stopping simulations. Using single NVIDIA GeForce RTX 2080 Ti GPU, each optimization step (which requires forward simulation and gradient computation) requires approximately 60 seconds; there are at least 25, 000 degrees of freedom in all mechanics models that found satisfactory through a mesh independence analysis described Appendix A.4, such that simulation results were not affected by insufficient discretization resolution.
5.1 DESIGNING UNDER UNIAXIAL TENSION
Cellular solid structures built from hyperelastic materials usually exhibit nonlinear force-displacement responses. Here, we aim at proposing cellular solids with linear responses under uniaxially pulling experiments i.e., ('(c) C' such that (' 0.1) 3S, where 3 changes from 0.1 to 1.5 with an interval of 0.1 for a given normalized nominal stress Sg. We find that for all considered 17 symmetry groups, p2gg group indicated the best performance by being able to learn cellular solids with linear force-eisplacement responses for all B values except | = 1.5. The learned meta-materials with their linear responses depicted in Figure 4 for 3 0.1, 0.5 and 1.4. While we able to achieve wide range of linear responses with p2gg group, with p1 translational symmetry we can design meta-materials that have mechanical responses with 8 up to 1.1 (see Appendix A.5).
Informmlly, Poisson's 's ratio is how much material expands laterally when squeezed from the top, contracts when pulled. Almost all materials have positive Poisson's 's ratio.
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1.450
stress
stargt
2
0.387
3
0.181
strain
Figure 4: Undeformed configurations (i.e., é = 0) of three cellular solids designs with 50% volume fraction and pore shapes respecting p2gq symmetry group, and their linear stress-strain responses with the corresponding target curves during a uniaxial tension of the top edge up to e = 0.1.
Next consider designing auxetic meta-materials with negative Poisson's 's ratios. We explored cellular solids that can exhibit large Vef = -0.5 during tension. With p1 symmetry, the best design only achieve Vef -005,, whereas pg symmetry Our framework was able to learn a cellular solid with p4 symmetry that has Ves = -0.45 that is very close to the target negative Poisson's 's ratio. The eight best-performing cellular solids designs among the 17 group-constrained designs in achieving Pef -0.5 are shown in Table 1. This indicates the benefit of exploring larger space of cellular solids than those exhibiting simple translational symmetries. In this case, six symmetry groups have better performances than pl symmetry. All final eight designs are depicted in Appendix A.6.
improve this to Vef -0.12 (Figure 5).
Real-world experiments: To demonstrate transfer to the real world, we manufactured the final designs with pq and p4 symmetry groups and quantitatively verified their behavior under uniaxial tension test (detailed in Appendix A.7). As predicted by our simulations in Figure 5 (second column) and confirmed with experimental observations (third column), the two proposed designs have negative Poisson's 's ratios. The experimental measurements Vez = -0.14 14 and V.z -0.49 under pulling for pg and p4 models, respectively, are in strong agreement with simulation results. The video footage of the pg cellular solid sample during pulling experiment with the corresponding simulation result are included in the supplementary matcrials.
5.2 DESIGNING UNDER UNIAXIAL COMPRESSION
The design of cellular solids under uniaxial compression is more challenging task because of nonlinear buckling instabilities observed in such structures (Overvelde & Bertoldi, 2014). In such cases, the nominal stress-strain responses are characterized by a critical strain value (e.r) which mechanical responses vary. Here, we seek to design cellular solids that behave linearly before an cor while having constant nominal stress as the applied strain increases the final value of 0.1:
S(c) -
[Ce
(
Ceer
if if 0.10.1
Figure illustrates mechanical responses of three cellular solid designs under uniaxial compression. We were able to design materials with various compression responses by controlling the plateau forces for a fixed eor (designs and 4) and tuning both the plateau force and the location of e,r (design 2) using p1 symmetry, which found to be a challenging design factor in previous works (Medina et al., 2023). Interestingly, the only symmetry group that enabled such designs was pl. Examples of designs with other symmetries that force-displacement responses do not match target values are depicted in Appendix A.S.
Next, focus tailoring cellular solids with negative Poisson's 's ratios under compression. Here we aim to identify structures with Vef = -0.2, i.e., requiring the material to shrink. To ensure this
Table 1: Poisson's 's ratios of cellular solids from eight symmetry groups with top performances when
aiming for Ves = 0.5.
groups
pl
p2
pg
p2mg
p2gg
p4
p3
p6
Vef 0.05
- 0.04
-0.12
-0. -0.40
-0.10
-0.45
00.11
- 00.14
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undeformed
pulled (simulation)
pulled (experiment)
undeformed
pg
Q
Vef
0.12
1/2
0.14
p4
Ves 0.45
Per 0049 1: 0.01
Figure 5: Undeformed configurations (i.e., 0) of two cellular solids designs that have negative Poisson's 's ratios with 50% volume fraction and pore shapes respecting pg and p4 symmetry groups (first column), and their deformed configurations from simulations during a uniaxial pulling of the top edge up to = 0.1 (second column). Experimental realization of the same cellular solids under pulling confirms negative Poisson's ratios for both designs. The experimental measurements also in strong numerical agreement with simulation results (third and fourth columns). Scale bars: 5cm.
design criterion, we customized the loss function to enforce points on Ω and Ω2 edges to have mean positive and negative displacements, respectively. We were able to optimize p6 and pg symmetric cellular solids shown in Figure 7 with Vef = 0.15 and Vef -0- 0.2 under compression, respectively, that they also shrink.
Finally, we examine the ability of our framework in designing cellular meta-materials with more complicated mechanical response: achieving zero effective Poisson's ratios under both uniaxial compression and tension via having zero displacements resulting successful structure that leverages p symmetry group, indicating zero displacement at both left and right edges for either pushing or pulling of 10% of the cellular solid height.
Ω2 and Ω, edges. Figure 8 shows the
RELATED WORK
Existing cellular solids design are mainly focused on structures constructed from translational symmetry of an optimized pore unit cell. For example, Wang et al. (2014) utilized level set method
2
1.28.
%
0.88
0.658p
0.86.r
strain
0.1
Figure 6: Undeformed configurations (i.c., 0) of three cellular solids designs with 50% volume fraction and pore shapes respecting pl symmetry group, and their stress-strain responses with the corresponding target curves during uniaxial compression of the top edge up to 0.1.

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...p6. design with -0.15 Poisson's ratio ............... undeformed
PS design with -0.2 Poisson's ratio undeformed
pushed
pushed
Figure 7: Undeformed and deformed configurations of p6 and pa symmetric cellular solids designs that have negative Poisson's ratios of -0.15 and -0.2, respectively, under uniaxial compression.
pulled
undeffrmed
pushed
D.6
loss
iteration #
200
Undeformed and deformed configurations of cellular solid design with 50% volume fraction and pore shapes respecting p6 symmetry group indicating zero effective Poisson's ratios under both 10% uniaxial pushing and pulling, with loss values during optimization.
Figure 8:
design cellular meta-materials via unit cells topology optimization. Xue Mao (2022) proposed an FEM-based mapped shape optimization technique to design periodic meta-materials. Medina et al. (2023) employed a higher order moving-mesh method to optimize the pore shapes of cellular solids. Du al. (2024) optimized pore structures through describing them by moving morphable voids method. Machine learning methods are becoming an emerging tool for both accelerating the design process and discovering new meta-materian materials beyond intuitions. For example, Xue et al. (2020b) introduced data-driven neural network homogenization approach for cellular meta-materials design. Ma et al. (2020) applied machine learning tools for characterization of non-uniform arrangement of cells in meta-materials. Ha et al. (2023) leveraged generative machine learning techniques for inverse design of meta-materials with target stress-strain responses. In this study, we propose equivariant neural network flow to learn volume- volume-preserviee preserving cellular solids via performing shape optimization of the pores while exploring all 2D crystallographic symmetry groups to find the optimal arrangement of the pores.
7 LIMITATIONS AND FUTURE WORK
We introduce a machine learning framework that enables an efficient approach for inverse design of two-dimensional cellular solids with various nontrivial mechanical behaviors beyond the reach of existing methods. Scaling up three-dimensional cellular solids design while considering crystallo- graphic symmetric arrangements could potentially unlock uncommon mechanical functionalities not observed in two dimensional meta-materian materials. But this requires efficient and faster solvers that our framework suffers from at large number of degrees of freedom in the mechanics modeling. A future work could focus on speeding up the simulations leveraging learned surrogate models (Sun et al., 2021), neural network-based order reduction techniques (Beatson et al., 2020), and also amortized optimization methods (Xue et al., 2020a). Since all crystallographic symmetry groups include trans- lational symmetry, further computational cost reduction also be achieved by simulating single unit cell with periodic boundary conditions instcad modeling the entire cellular solid (Mizzi et al., 2021).
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ACKNOWLEDGMENTS
We would like thank Deniz Oktay for his computational mechanies insights. This work was partially supported by NSF grants IIS-2007278 and OAC-2118201, NSF NSF under grant number 2127309 to the Computing Research Association for the CIFellows 2021 Project. PO is supported by the Gatsby Charitable Foundation.
NSF under grant number 21 18201, the
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Eder Medina, Chris Rycroft, and Katia Bertoldi. Nonlinear shape optimization of flexible mechanical metamaterials, Extreme Mechanics Letters, 61:102015, 2023.
Luke Mizzi, Daphne Attard, Ruben Gatt, Krzysztof K. Dudek, Brian Ellul, and Joseph N. Grima. Implementation of periodic boundary conditions for loading of mechanical metamaterials and other complex geometric microstructures using finite element analysis. Engineering with Computers, 37 (3):1765-1779, 2021.
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Tianju Xue, Alex Beatson, Maurizio Chiaramonte, Geoffrey Roeder, Jordan T. Ash, Yigit Mengue, Sigrid Adriaenssens, Ryan P. Adams, and Sheng Mao. A data-driven computational scheme for the nonlinear mechanical properties of cellular mechanical metamaterials under large deformation. Soft Matter, 16(32):7244-533, 2020b.
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A APPENDIX
A.1 THE 17 WALLPAPER GROUPS
The following figure illustrates the 17 distinct (up to isomorphy) space groups on R22 also known as the wallpaper groups. In each figure, the region marked red corresponds to the polytope IT in (1). The letter F inscribed to mark orientation. Figures from Adams & Orbanz (2023).
F
F
F
1
T
F
7
P
7
P
P
P
F
F
d
L
J
F
F
H
F
F
F
F
F
T
F
1
F
P
F
F
F
E
d
'd
J
R.
F
E
F
F.
F
F
J
1
7
P
H
F
7
H
R
A.2
PROOFS
A.2.1 PROOF OF THEOREM
This proof draws one of the fundamental properties of space groups, namely that, if G is space group and TG its subgroup of pure translations, the quotient group G/Tę is finite. See for example (Vinberg & Shvartsman, 1993).
Step I. We first show G is finite. Consider the relation = G defined as
for some E Te
Since Tg is an equivalence relation, and by definition of quotients, the equivalence classes of correspond to the elements of the quotient group G/Tg. Since G is a space groups, this implies has finite number of equivalence classes. Recall that each element of G is of the form
((x) AgT + by Agæ + Eien c(())b
where c(Ø) = (cr (ø)..... ..... Cn (Ø)) is vector in R.. It follows that there are unique vectors é(σ) E [0, 1) and c()) E Z" such that c(0) = a()) + z()). The transformations
)(z)
Agr + E (0)b, and (x(x)
+ Eien a(0)b;
are hence indeed elements and T E To that satisfy = + T and are the only such elements. follows that each equivalence class of contains exactly one element of G which shows the set C is indeed finite.
Step 2. To verify the properties of I, we must first establish two simple properties G. namely G G and
(ń(x) (x((x) b, for some To
Here, Gy is the set obtained by composing each element of G with zi from the right, and GD denotes application of the definition of G Gt. Since G is group and one of its elements, have
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G G, property (i) holds. To verify (ii), observe that the "hat operation" & preserves the equivalence class of Since e = a + To and T E Tg, have e = Φ. If t/ is further group element, we also have dx((x) = ((vx)) - br and therefore oi = Φs.. In summary, we have
क O Φ
which shows (ii)
Step 3. Now consider a Te- ee-invariant function f. For any db E G, we have
Egeâ A f(xxxx)
substitute π == dub Enedy - + b,) by (9i), for some r E To
Encro A-3 (x(x)
Enero -
since h is To-invariant since A-1 zv
A Enedy
A (
A A A
Av Exe@ Az f(fx) by (9ii).
follows that
(Tf))a)) Ejet A ((11(x A Eδeü A5x1(xx) A+(Tf())
for any E G.
A.2.2 PROOF OF THEBREM 2
The existence and uniqueness of solutions of (4) is given by the Picard-Lindedes theorem (e.g. Deufhaard Bornemann, 2002). Here a statement that suffices for our purposes:
Fact 1 (Version of the Picard-Lindelûf theorem). Fix n, k E N and a bounded interval : [0, tmar]. Let H R² x I + Rd be a Srnschitz,functton that is k 2 times continuously differentiable in its second argument. Then
F(zo,t) == =0 + ('' ((F(a,, 8),
defines a function 1R x 1 + Rd that continuous in its first argument, + times continuously differentiable in the second, and is the unique solution of (4).
FR
We break the main ingredients of the proof of Theorem 2 up into three lemmas. First, we show that function that satisfies (5) yields flows with the symmetry properties we require.
Let be a group of isometries IR", and fix some max > O. Let R" XI function is Lipschitz, and k times continuously differentiable in its second argument, for k 2 1. Then there is unique function Th. x [0, mmax) - IR." that satisfies (4) for all (x. 1) in its domain, and this function satisfies
Lemma 1.
F(ox, t) = (2)
for all G. all R and St S tmax
(10)
if and only if H satisfies (5). The solution continuous in its first argument, and (k + 1) times continuously differentiable in the second.
Proof. We first construct the solution F R" x - R2 the larger interval / = [-1,tt (This to ensure that 0 is in the interior of 7, which we need for step 3 below.) The restriction of F to R [0, fmax] is then solution in the statement of the result.
Step 1. Existence, uniqueness, and smoothness of F _ / then follow immediately from Fact where we set Emin 1 and t = 0. Fact I also shows that F solves
F(x,1) áu
((F(x,,),() with F(x,0) 0)
(11)
What we have show is that F is equivariant if and only if H satisfies (5).
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Step 2. Suppose H satisfies (5). Write F (z. for all Φ.
o 6=FF(drr t). Thus, F is equivariant iff F F
(x,
d (Ag- 1) + b0-1)
since Φ o Ag-1 + b/-1
di
d F(der, 1)
A
A and b do not depend by (11)
dt
A A 1),1)
= F(oxt t),t)
by (5)
= (z,
definition of F
Substituting the initial condition into the definition of F shows
1 (zz 0) o oI 0)) o o
Thus, both F and 1 solve the differential equation (11) for the initial condition Since the solution unique by Fact 1, it follows that F(x, 1) = (x,,)) for all summary, (5) implies equivariance of F.
Step 3. Conversely, assume F is equivariant, F F. Since H is continuously differentiable, Fact shows + Fxx, 1) is twice continuously differentiable. Its Taylor expansion around any in the interior -1, hence
d F(x,(+e) = (xx,)) + Fxx(() ol6?)
di F(x,() H(F(x, (), 1) + o(E?)
(12)
for each R" and 0.
If set f and use the initial condition F(x, 0) I. obtain
oF oF e) As(x + (x, o(²2) + bæ oz + (2H dl?) where we note that Ago(E2) - =o(²) since Ag is an isometry. Substituting dx into (12) shows
F(øx, c) = or H(áx, 0) +
Since F = F, we hence have
④x + (dx, 0) E E ole2) ox Ag o(e2)
and therefore
H(dr, 0) = AgH(I, 0) + o(c)
Since that true for all c, it follows that H ( -) (x, 0). In summary, equivariance of F
implies that (5) holds t 0. Since F is flow, the flow F* s) == F(x, s) satisfies (4) for the function (r, s) = H f + s) and the initial value =.f- := F (a. t). If F is equivariant, so is Ft. We can hence apply the same argument at
which shows H H(x t) (xr,
Li
Given differentiable function f R" - Rm we denote by Df its differential. If m n, this is the Jacobian matrix, and if m = 1, the transpose (Df) is the gradient of J.
Lemma 2. For any differentiable function g R2n - The and any m E N, the differential D(g p) is Tsginnariant. In particular, Dp is T iivariiaa
Proof. Since the Jacobian matrix of p has entries
2 2xp2)-1(x)
if j 2i
(Dp(x))s
(x)
if 2i otherwise
it can be written as a function Dp = M o of p. where M 2 + 12x22 That shows
D)gop)(x) (DD)(p(x)) (D))(xx) (Dg (((=))
and hence D(g o = (Dg M) o pod o= (Dg M) db
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Our final auxiliary result shows that the symmetrization operator I does not increase divergence. For Theorem 2, this means that if a function ff is divergence free,, then so is ff.
Lemma 3 ( does not increase divergence). If G a spuce group on IR", and f : I2" - R is continuously differentiable function, then
(i luup.
I
divf Ilaup
Proof. Since the divergence is the trace of the differential, and both trace and differential are linear,
(x) tr D 1 Each summand has differential
G Eost A f((x))))
|G| Eset tr D(A
D f (x) A (Df (D)
Since the trace is linear and orthogonally invariant, it follows that tr D(A (=)) [r((Df))zz) (dx)(f)()))) s [|div(f)llaa
The result then follows via the triangle incquality,
(di)(T)) = |3] Eeea |ldiv/llaup
1
||div =|div/llaup
which is all we had to show.
Proof of Theorem 2. Since p is Tn-invariant and infinitely often differentiable, the function h o is T -s-invariant and (k + 1) times continuously differentiable. The function A(h p). by definition of A, a function of the gradient V (h o p). Since this gradient is Te-invariant by Lemma 2, A(h p) is To- invariant. By Theorem 1, I'Ahop)) satisfies (5). is also divergence-free, since A (hcpp p) divergence- free and I' does not increase divergence, by Lemma 3. We can therefore choose H := I'Ahh p) in Lemma and the lemma shows that there a unique solution F that is an equivariant flow. Since H is also divergence-fie free, suitable version of Liouville's 's theorem (e.g. Arnold (1998), Theorem 1.3.1) shows F hluue--pesserving
A.3 MECHANICAL MODEL
We nearly incompressible neo-Hookean hyperelastic model, which effectively describes the mechanical behavior of the elastomeric materials commonly employed in manufacturing of flexible meta-materials. The mechanical behavior of such hyperelastic material can be described by introduc- ing a strain energy density function Y that is independent of the path of deformation and is function of the deformation gradient tensor F Vu+II I. Here, the displacement field, which is function u f - IR? A representation of this function is computed by the differentiable mechanics solver described in Section 4. Since the mesh representation of the shape 89 uses N mesh points, and the vector field has two degrees of freedom at each point, the representation of u is an element of 2R With these ingredients, the function is given by
1 tr (r(FFF) - 2 2 2 ledde((F) ) (Indet(F)) (
V (F)
(13)
where μ = E/2(1+12) and == >>==E/(1+)(( 2v) are Lamé parameters of material with Young's modulus F., and is the Poisson's ratio. The displacement field of the structure at the equilibrium u² is computed from the space of all admissible displacement fields H that satisfy Dirichlet boundary conditions (that is, u takes a suitable prescribed value on the boundary), by minimizing the total stored potential energy from material deformation under mechanical loading
u = arg min (F))..
(14)
uEH nn
The minimizer is computed using a second order Newton method for sparse Hessians.
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A.4 MESH INDEPENDENCE ANALYSIS
We performed mesh independence analysis for all the cases to ensure simulation results from mechanics solver these analyses with associated time for each optimization step is shown in Table 2 for designing under uniaxial tension of pl group. Mesh independence studied in initial reference model.
affected by insufficient mesh resolution. The details of example of
Table 2:
Mesh independence analysis based on the deformation of the middle right edge cellular solid with p1 group symmetry under uniaxial tension.
number of DoF
9,000
14,000
20,000
25,000
29,000
middle deformation (mm)
3.05
3.31
3.49
3.65
3.65
optimization time (8)
15
34
49
61
95
A.5 METTR-ATERIIAL DESIGNS WITH 21 SYMMETRY UNDER UNIAXIAL TENSION
Three cellular solids designs while pores respecting p1 symmetry group under uniaxial tension when aiming for linear responses with & equals to 0.1, 0.5, and 1.5. The design 4 is with p1 symmetry group when targeting linear response with 1.5.
3
1.58
stress
taaret
0.584
0.184
strain
A.6 META-MATERIAL DESIGNS WITH NEGATIVE POISSON'S S RATIOS
All eight cellular solids designs with best performances in achieving Vef -0.5 under uniaxial tension while pore shapes respecting different symmetry groups.
pl
p2
p2mg
p2gg
p3
D6
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A.7 FABRICATION AND EXPERIMENTAL DETAILS
140mm-by-140mm samples with pull tabs were laser cut from butyl-based laser-engravable rub- ber sheets of 2.38 thickness. Laser cutting was performed using Universal Laser Systems PLS6.110D) machine with two 75-W CO2 lasers emitting at 10.6 μum Cut patterns were programmed using the Universal Laser Systems software suite with the following settings: 3 % travel speed, 20 % power. Cutting was repeated three times.
Experimental tensile tests were performed using Instron 5969 UTM outfitted with Instron 2530-series 50N static load cell. Samples were mounted in the UTM's clamps by their pull tabs; controlled displacement of the top clamp between -5mm and 20mm was applied over the course of three cycles. Video footage of the sample was used to extract the vertical and lateral strain of each sample midway between the clamps, allowing calculation the average experimental Poisson's 's ratio Ver. at 0.1 vertical strain.
A.8 META-MATERIAL DESIGNS UNDER UNIAXIAL COMPRESSION
Examples of cellular solids designs with target force-displacement responses under uniaxial com- pression respecting symmetry groups pm, p2mg, and p4 that not match target values. These simulations ran with several different initialization of the neural network parameters.
target
target
target
p2mg
P4
18


Reviews:
[{"review_id": "tpUy94r1i6", "reviewer": "Reviewer_rTA7", "summary": "The paper is unclear, but to the best I can understand, the authors investigate an approach to perform inverse design of mechanical metamaterials by utilizing crystallographic symmetry groups.", "questions": "1) Could the authors explain more clearly what general problem they are trying to solve?  Presumably there is some data-driven learning problem so that this work is suitable for this research community: could the authors explain more clearly what it is?  \n\n2) Why don't the authors discuss, or compare with, any other existing approaches for metamaterial design? There are a large number of data-driven inverse design approaches.  Perhaps there is a good reason, but given the problem I cited in (1), it is unclear to me. \n\nSee the following reference for some examples, but there are many:  Ardizzone, Lynton, et al. \"Analyzing inverse problems with invertible neural networks.\" arXiv preprint arXiv:1808.04730 (2018).", "limitations": "", "rating": 5, "confidence": 3, "soundness": 2, "presentation": 1, "contribution": 2, "strengths": "1) Methods to solve inverse problems in the sciences is an important open problem.  \n2) Building models/methods that leverage symmetry is potentially interesting approach.", "weaknesses": "1) Lack of clarity.  Based upon the introduction, I have only a very vague idea of what the authors are proposing, or why they are proposing it.  Specifically, the problem setting is not clear, there is little/no discussion of existing methods to solve the problem, and the limitations of existing methods are also unclear.  These problems make it very difficult to review the work.    \na) Line 39: \"Existing approaches to cellular solids..\".  What is an \"approach\" to cellular solids referring to?...an approach to what? \nb) Line 45: What is \"shape optimization\"?  We're optimizing the shape for what purpose? This is machine learning conference, and this topic might not be clear to your audience (e.g., me). \nc) Line 48: The authors begin talking about their solution, but it is unclear at this point (i) what problem they're trying to solve, (ii) what approaches exist to solve this problem, and (iii) how using \"flow\" is going to mitigate these problems.  \nd) Line 48:  What exactly is \"flow\" in this context?   I see there is a \"Related Work\" section at the end of the paper, and it cites three papers that the authors claim have studied \"Neural network flows\" in recent years, however, I looked at the most recent of these cited references, Cranmer et al. 2020, and the word \"flow\" is never mentioned.  It remains unclear to me what exactly the authors are discussing. \n\n2) The authors present some results with their approach, but there is no mention or comparison with any other existing approaches.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "6xognLOdQ7", "reviewer": "Reviewer_Z228", "summary": "This paper explores using symmetry preserving flows and a mechanical simulator in order to design new meta-materials with desirable properties. The authors demonstrate how to construct neural networks equivariant to space groups and show successful designs that exhibit good force-displacement responses and effective Poisson ratios. They also manufacture real world designs and validate the simulation results on them.", "questions": "- Line 319: Is there any specific reason why such small networks were used?", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "- Novel application of equivariant/invariant networks to meta-materials design\n- The paper is clearly written and explains various meta-materials related terms for the unfamiliar reader.\n- The authors describe, in great detail, the space groups and the construction of the symmetry preserving flows\n- The experiments seem to show good results and the authors further experiment in the real world", "weaknesses": "I imagine that there could be multiple designs that satisfy the desired properties. However the proposed method doesn't seem to allow for generating multiple different valid designs as it is not a generative model. The space group and the desired property must first be fixed, and as the network trains, it converges to some design that optimizes the loss. There doesn't seem to be an easy way to generate multiple valid designs without training every time.\n\nOne other concern is that I wonder if this paper is suitable for this venue. As a reviewer who doesn't know much about meta-materials design, it is difficult for me to judge the impact of the proposed method and the results. The best I can tell about the results is that the manufactured design seems to match the simulation measurements.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "48dcSWROnH", "reviewer": "Reviewer_DtCf", "summary": "The paper presents a method to design metamaterials by learning equivariant flows. The method is capable of designing materials for all 17 space groups in two dimensions.", "questions": "The following typos were found:\n\n-\t125: grammar of the sentence “In the case…”\n-\t354: “preserved”\n-\t531: grammar of the sentence\n-\t849: grammar: “vector field has to degrees”\n\nPlease compress the pdf; the current version has 27.5MB. It can be compressed without noticeable loss of quality to ~3MB.\n\nPlease 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\n\nPlease 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 $\\beta=1.5$)", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The paper is well written and the presented approach is mathematically grounded. The problem of designing mechanical metamaterials is an active area of research. The work presented here involves metamaterials where the unit cell is a general porous structure where each point in the domain can be air $(\\rho=0)$ or material $(\\rho=1)$. This is a good focus at times when much of the work in the field is on strut-based lattice metamaterials.\n\nThe idea of starting out with an initial model with well-behaved connected topology and morphing it into a flowed model is important – for a well-behaved flow, one will not end up with defective topology and disconnected patches of material (as opposed to what one might obtain by using fully generative models such as diffusion).\n\nPart of having a well-behaved flow is ensuring it is volume-preserving. The authors incorporated an idea from the literature in which a divergence-free flow is obtained as vector whose components are appropriately arranged partial derivatives of a scalar function. This enables to set an original relative density $\\bar{\\rho}$ which is then preserved by the flow.", "weaknesses": "1. The presented approach is in 2d which has limited relevance as we often want materials with 3d structure (not extrusions of 2d cross-sections)\n\n2. The emphasis on validation with real-world prototypes is only tangentially relevant to the presented work. \nE.g. in abstract: “We validate these new designs in simulation and by fabricating real-world prototypes.” There are two statements to unpack: \n    1.\tvalidate these new designs in simulation - this is not really validation since the training loop already involves the differentiable simulator\n    2. validate these new designs by fabricating real-world prototypes – this is not really the validation of the presented equivariant flows, but rather a validation that the differentiable mechanics simulator captures real-world behaviour of the structures\n\n    Overall please change the way in which such claims are formulated:\n        (i)\tany claimed “validation of simulations with respect to target behaviour” is only a test that the target behaviour is within the expressive power of the framework. It would be interesting to push this to the limit – for instance, what would happen if you tried to design for tension while setting as the target the same stress-strain curve which you use in compression? \n        (ii)\t“validation by manufacturing real-world specimens” is only a check of the ability of the differentiable mechanics simulator to capture real-world behaviour of the structures.\n\n3. One limitation identified in section 7 for scaling up to 3d structures is the lack of efficient/fast solvers and the high number of degrees of freedom. In section 5 it is mentioned that 25000 degrees of freedom were needed for force-displacement experiments. Please include the convergence study with respect to mesh resolution (accuracy/stability vs time per iteration vs number of DoFs)", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "pLCV3MonZG", "reviewer": "Reviewer_4yCV", "summary": "The authors present a machine learning framework that efficiently tackles the inverse design of two-dimensional cellular solids. By training a neural network to learn a symmetry-preserving flow, they could expand the design space of a class of mechanical metamaterials, known as cellular solids, into more generalized isometries. Their approach ensures that the solution avoids disconnected regions and the associated numerical challenges. Moreover, they demonstrate that such neural flows are divergence-free, enabling the preservation of the total volume while focusing solely on optimizing the geometry. The effectiveness of their approach is validated  through simulations and the fabrication of real-world prototypes.", "questions": "- It appears that the neural network is not always Lipschitz-continuous or ($k+1$)-times continuously differentiable. Is there any proof supporting this, or did you relax the constraint based on certain assumptions?\n- It is highly suggested to compare the proposed approach to other machine learning alternatives. These models can also satisfy the conditions outlined in Theorem 2 and offer greater interpretability compared to MLP. \n- How did you determine the configuration of the neural network? Does the configuration—such as the number of hidden neurons, layers, or the choice of activation function—affect the simulation accuracy? Additionally, which components of the neural network play the most critical role in the simulation performance? Whether the learned function exactly meets the constraint stated in Theorem 2?", "limitations": "", "rating": 5, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "- This paper demonstrates how to expand the design space of cellular solids beyond translational symmetry using symmetry-preserving flow. It also provides detailed derivations and proofs.\n- This paper leverages a neural network to construct the symmetry-preserving flow and optimizes materials to achieve desirable mechanical properties. The effectiveness of the approach has been validated through simulations and real-world fabrication.", "weaknesses": "- The contributions and related work are not clearly presented. Does this optimization formulation for extending the design space already exist, but cannot be effectively solved by classical optimization methods, or is it first introduced in this paper? Are there any other formulations or approaches that can also extend the design space of cellular solids? If so, why is this formulation considered superior to others?\n- My main concern is the motivation of using a neural network for such design problem. Many other machine learning methods could also satisfy the conditions outlined in Theorem 2 and the loss function. Why is the neural network preferred? There is no justification, proof, or experimental comparison to support this choice of neural networks as a proxy model.  This issue downsides this paper quality significantly. I strongly suggest the authors provide strong motivation from this aspect.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)