How to See Hidden Patterns in Metamaterials with Interpretable
Machine Learning:
Supplementary Information
A Simulation Settings
In-house/Infinite-medium Simulator
To determine whether a unit-cell design has a bandgap, we evaluate the dispersion relation along the
wavevector contour (Irreducible Brillouin Zone contour) shown in Supplementary Figure 1. To evaluate the
dispersion relation, we solve a series of eigenvalue problems given by the harmonic elastic wave equation with
Bloch-Floquet periodic boundary conditions.
Dispersion relation analysis is performed only on the domain of a single unit-cell, and assumes that
the metamaterial has a unit cell that is infinitely repeating in each direction that Bloch-Floquet periodic
boundary conditions are applied.
The harmonic elastic wave equation is:
ρ
(
x
)
ω
2
u
=
∂
∂x
j
C
ijkl
(
x
)
1
2
∂u
k
∂x
l
+
∂u
l
∂x
k
i,j,k,l
= 1
,
2
.
(1)
where
C
is the material stiffness tensor,
ρ
is the material density,
u
is the displacement field (eigenfunction),
ω
is the eigenfrequency, and
x
represents positions in the 2-D space. Displacements (and therefore wave
modes) are confined to the same 2-D plane that contains the metamaterial. Bloch-Floquet theory says
that the displacement field solutions,
u
, should be spatially periodic, so we introduce the wavevector
γ
to
characterize this spatial periodicity. To enforce Bloch-Floquet periodicity, we apply Bloch-Floquet periodic
boundary conditions:
u
(
x
+
a
n
) =
u
(
x
)
e
−
i
γ
·
a
n
∀
x
∈{
x
∈
Ω
|
x
+
a
n
∈
Ω
}
, n
= 1
,
2
.
(2)
where
i
is the imaginary unit,
a
n
are the lattice vectors representing the periodicity of the material, and
Ω
⊂
R
2
represents the domain of the unit-cell.
To discretize and solve this eigenvalue problem, we apply the Finite Element method using bilinear
quadrilateral elements. The resulting discretized eigenvalue problem is
[
K
(
γ
)
−
ω
2
M
(
γ
)]
u
= 0
,
(3)
where
K
and
M
are the stiffness and mass matrices, respectively. The boundary conditions are baked
into the wavevector-dependent stiffness and mass matrices. This code is implemented in
MATLAB
.
Figure 1: Left: Wavevector contour (also known as Irreducible Brillouin Zone contour). Right: Example of a
dispersion relation evaluated along the wavevector contour.
Commercial/Finite-medium Simulator
Dispersion relation analysis is performed only on the domain of a single unit-cell, and assumes that the
metamaterial has a unit cell that is infinitely repeating in each direction that Bloch-Floquet periodic boundary
conditions are applied. However, real periodic metamaterials do not extend infinitely in any directions. While
dispersion analysis is a critically useful method for converging on a unit-cell design with desirable properties,
it is important to check that these properties are still present when the metamaterial is only finitely periodic.
To perform this validation, we perform frequency domain analysis on finitely periodic geometries using
the commercial Finite Element software
COMSOL Multiphysics
[
12
] with the Structural Mechanics Module.
1
Material Properties
The material is a 2-D metamaterial made by tiling a
10
×
10
pixelated unit-cell. Each unit-cell is made of two
constituent materials: one is soft and light, with elastic modulus and density of
E
= 2[
GPa
]
,
ρ
= 1
,
000[
kg/m
3
]
respectively, and the other is stiff and heavy with
E
= 200[
GPa
]
, and
ρ
= 8
,
000[
kg/m
3
]
.
B
Relationship of Proposed Methods to Other Machine Learning
Methods
Comparison of shape-frequency features to CNN filters
: One might think that the sliding windows of the
shape-frequency features are similar to convolution filters in CNNs, but they are not. Shape-frequency
features count only exact matches between each window and the soft constituent material, while a traditional
convolutional layer would output a real number for the degree of match. In order to attempt to replicate the
calculation of shape-frequency with a CNN, one would concatenate a convolutional layer, a Rectified Linear
Unit (ReLU) layer and a linear layer in a CNN [
7
]: the convolutional layer calculates the matching score, the
ReLU layer sets a threshold to round the matching score to 0 or 1, and the linear layer averages the rounded
score over all locations. However, if we train such a CNN, the weights cannot be integers and thereby the
model cannot be as sparse and interpretable as our method using the shape-frequency features.
Comparison of unit-cell template sets to prototype learning and decision sets:
The unit-cell template set
technique is closely related to prototype-based machine learning models (e.g., [
1
,
6
]). The prototype-based
models learn a set of prototypical cases (which can be whole observations or parts of observations) from the
training data. Given a test observation, the model makes a decision by comparing the test observation to the
prototypical cases. For example, in classification, the model would output the label of the the prototypical
case that most closely resembles the test observation. A unit-cell template can be viewed as a part-based
prototype because it is part of a training observation, and the decision of the model is made by matching the
templates and the test observation. A difference between classical prototype-based learning and our approach
is that classical objectives are classification losses while our method optimizes support and precision.
A unit-cell template set can also be viewed as an instance of a decision set. A decision set (e.g.,
[
2
,
3
,
10
,
8
,
11
]), is a logical model comprised of an unordered collection of rules, where each rule is a
conjunction of conditions. In other words, the entire model is a disjunctive normal form (DNF), or “OR of
ANDs.” A positive prediction is made if at least one of the rules is satisfied. One unit-cell template in the set
is essentially a conjunction of conditions that compares whether the pixel in the template is the same as the
pixel in the test observation. That is, there is an “OR” over unit-cell templates and an “AND” over pixels in
the template. As far as we know, decision sets have not previously been utilized for material discovery.
C
Visual Illustration of Sampling in Finer Resolution Design Space
Figure 2(a) shows an example of how to sample a
20
×
20
design with GOSDT+SFF trained on
10
×
10
data.
This is essentially a rejection sampling process, where a random
20
×
20
design is selected, and accepted if
the algorithm predicts existence of a band gap, and rejected if not. However, as the design space has changed
from
10
×
10
to
20
×
20
, the calculation of SFF has also changed, with details discussed in Section 4.1.3 in
the main text.
Figure 2(b) shows an example of how to sample a
20
×
20
design with a
10
×
10
unit-cell template. While
still following the templates learned from coarse resolution (
10
×
10
) data, most of these finer resolution
designs can never be found in the coarse resolution (
10
×
10
) design space.
D More Experimental Results
D.1 Structure-to-property - Prediction Accuracy
We now compare the testing balanced accuracy (bacc) of various machine learning methods, including linear
models like support vector machines with linear kernels (SVMs) and logistic regression (LR); tree-based
models like CART, random forest (RF), and boosted trees (LightGBM [
5
]); as well as neural networks
including the multi-layer perceptron (MLP) and convolutional neural networks (CNNs). LightGBM [5] was
used with 300 trees, and the MLP had 2 hidden layers, each containing 100 neurons. The CNN architecture
is ResNet18 [
4
] adapted to one-channel input, and default parameters were used for other methods. All other
models were trained on shape-frequency features (SFF) and raw features respectively, while the CNN was
trained on the entire 2-D design. We trained each model 5 times to average out randomness in the training
process. Figure 3 shows the balanced accuracy results. Models trained on shape-frequency features generally
perform much better than models trained on raw features, with the exception of MLP. Also, LightGBM
trained on SFF outperforms CNN.
D.2 Property-to-structure Sampling - Precision and Support
Raw+LightGBM
SFF+LightGBM
SFF+GOSDT (ours)
Unit-Cell Template Sets (ours)
5
.
1
×
10
−
6
s
3
.
0
×
10
−
4
s
1
.
8
×
10
−
4
s
1
.
9
×
10
−
7
s
Table 1: Average sampling time of a new
10
×
10
design.
2
(a) Sampling finer resolution design with SFF+GOSDT trained on coarse resolution data. Sampling consists of
rejection sampling with GOSDT, and adapting the calculation of SFF from coarse to fine resolution (see Section 4.1.3).
(b) Sampling finer resolution design with unit-cell template trained on coarse resolution data.
Figure 2: Visual illustration of how to sampling finer resolution design with proposed methods.
3