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OPEN Recreation of the periodic table
with an unsupervised machine
learning algorithm
1* 2 3 4 1,2,3*
Minoru Kusaba , Chang Liu , Yukinori Koyama , Kiyoyuki Terakura & Ryo Yoshida
In 1869, the first draft of the periodic table was published by Russian chemist Dmitri Mendeleev. In
terms of data science, his achievement can be viewed as a successful example of feature embedding
based on human cognition: chemical properties of all known elements at that time were compressed
onto the two-dimensional grid system for a tabular display. In this study, we seek to answer the
question of whether machine learning can reproduce or recreate the periodic table by using observed
physicochemical properties of the elements. To achieve this goal, we developed a periodic table
generator (PTG). The PTG is an unsupervised machine learning algorithm based on the generative
topographic mapping, which can automate the translation of high-dimensional data into a tabular
form with varying layouts on-demand. The PTG autonomously produced various arrangements of
chemical symbols, which organized a two-dimensional array such as Mendeleev’s periodic table or
three-dimensional spiral table according to the underlying periodicity in the given data. We further
showed what the PTG learned from the element data and how the element features, such as melting
point and electronegativity, are compressed to the lower-dimensional latent spaces.
The periodic table is a tabular arrangement of elements such that the periodic patterns of their physical and
chemical properties are clearly understood. The prototype of the current periodic table was first presented by
Mendeleev in 18691. At that time, about 60 elements and their few chemical properties were known. When
the elements were arranged according to their atomic weight, Mendeleev noticed an apparent periodicity and
an increasing regularity. Inspired by this discovery, he constructed the first periodic table. Despite the subse-
2,3
quent emergence of significant discoveries , including the modern quantum mechanical theory of the atomic
structure, Mendeleev’s achievement is still the de facto standard. Regardless, the design of the periodic table
4,5
continues to evolve, and hundreds of periodic tables have been proposed in the last 150 years . The structures
of these proposed tables have not been limited to the two-dimensional tabular form, but also spiral, loop, or
6–8
three-dimensional pyramid f orms .
The periodic tables proposed so far have been products of human intelligence. However, a recent study has
9
attempted to redesign the periodic table using computer intelligence—machine l earning . From this approach,
building a periodic table can be viewed as an unsupervised learning task. Precisely, the observed physicochemi-
cal properties of elements are mapped onto regular grid points in a two-dimensional latent space such that the
configured chemical symbols adequately capture the underlying periodicity and similarity of the elements. Lemes
and Pino9 10
used Kohonen’s self-organizing map (SOM) to place five-dimensional features of elements (i.e. atomic
weight, radius of connection, atomic radius, melting point, and reaction with oxygen) into two-dimensional rec-
tangular grids. This method successfully placed similarly behaved elements into neighbouring sub-regions in the
lower-dimensional spaces. However, the machine learning algorithms never reached Mendeleev’s achievement
as they missed important features such as between-group and between-family similarities.
In this study, we created various periodic tables using a machine learning algorithm. The dataset that we used
consisted of 39 features (melting points, electronegativity, and so on) of 54 elements with the atomic number
1–54, corresponding to hydrogen to xenon (Fig. S1 for the heatmap display). A wide variety of dimensionality
11
reduction methods has so far been made available, such as principal component analysis (PCA), kernel P CA ,
isometric feature mapping (ISOMAP)12, local linear embedding (LLE)13, and t-distributed stochastic neigh-
14
bour embedding (t-SNE) . However, none of these methods could well visualize underlying periodic laws
1 2
The Graduate University for Advanced Studies, SOKENDAI, Tachikawa, Tokyo 190-8562, Japan. The Institute
of Statistical Mathematics, Research Organization of Information and Systems, Tachikawa, Tokyo 190-8562,
Japan. 3 4
National Institute for Materials Science, Tsukuba, Ibaraki 305-0047, Japan. National Institute of Advanced
*
Industrial Science and Technology, Tsukuba, Ibaraki 305-8560, Japan. email: kusaba@ism.ac.jp; yoshidar@
ism.ac.jp
Scientific Reports | (2021) 11:4780 https://doi.org/10.1038/s41598-021-81850-z
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Figure 1. Workflow of PTG that relies on a three-step coarse-to-fine strategy to reduce the occurrence of
undesirable matching between chemical elements and redundant nodes.
(Supplementary Fig. S3). To begin with, none of these methods offers a tabular representation. The task of build-
ing a periodic table can be regarded as the dimension reduction of the element data to arbitrary given ‘discrete’
points rather than a continuous space. To the best of our knowledge, no existing framework is available for such
table summarization tasks. Therefore, we developed a new unsupervised machine learning algorithm called the
periodic table generator (PTG), which relies on the generative topographic mapping (GTM)15 with latent variable
16
dependent length-scale and variance (GTM-LDLV) . One of the advantages of using the GTM-LDLV arises
from its ability to represent complex response surfaces. Elemental data shows a complex response surface on
the feature space. Controlling the two hyperparameters, the GTM-LDLV can flexibly represent functions whose
smoothness and amplitude vary locally in the feature space. With this model, we automate the process of translat-
ing patterns of high-dimensional feature vectors to an arbitrary given layout of lower dimensional point clouds.
The PTG produced various arrangements of chemical symbols, which organized, for example, a two-dimen-
sional array such as Mendeleev’s table or three-dimensional spiral table according to the underlying periodicity
in the given data. We will show what the machine intelligence learned from the given data and how the element
features were compressed to the reduced dimensionality representations. The periodic tables can also be regarded
as the most primitive descriptor of chemical elements. Hence, we will highlight the representation capability
of such element-level descriptors in the description of materials that were used in machine learning tasks of
materials property prediction.
Materials and methods
Computational workflow. The workflow of the PTG begins by specifying a set of point clouds, called
‘nodes’ hereafter, in a low-dimensional latent space to which chemical elements with observed physicochemical
features are assigned. The nodes can take any positional structure such as equally spaced grid points on a rec-
tangular for an ordinal table, spiral, cuboid, cylinder, cone, and so on. A Gaussian process (GP) m odel17
is used
to map the pre-defined nodes to the higher-dimensional feature space in which the element data are distributed.
A trained GP defines a manifold in the feature space to be fitted with respect to the observed element data. The
smoothness of the manifold is governed by a specified covariance function called the kernel function, which
associates the similarity of nodes in the latent space with that in the feature space. The estimated GP defines a
posterior probability or responsibility of each chemical element belonging to one of the nodes. An element is
assigned to one node with the highest posterior probability.
As indicated by the failure of some existing methods of statistical dimension reduction, such as PCA, t-SNE,
and LLE, the manifold surface of the mapping from chemical elements to their physiochemical properties is
highly complex. Therefore, we adopted the GTM-LDLV as a model of PTG, which is a GTM that can model
locally varying smoothness in the manifold. To ensure non-overlapping assignments such that no multiple ele-
ments shared the same node, we operated the GTM-LDLV with the constraint of one-to-one matching between
nodes and elements. To satisfy this, the number of nodes,
K , has to be larger than the number of elements, .
N
However, a direct learning with suffers from high computational costs and instability of the estimation
K >N
performance. Specifically, the use of redundant nodes leads to many suboptimal solutions corresponding to unde-
sirable matchings to the chemical elements. To alleviate this problem, the PTG was designed to take a three-step
procedure (Fig. 1) that relies on a coarse-to-fine strategy. In the first step, we operated the training of GTM-LDLV
with a small set of nodes such that . In the following step, we generated additional nodes such that ,
K N
and the expanded node-set was transferred to the feature space by performing the interpolative prediction made
by the given GTM-LDLV. Finally, the pre-trained model was fine-tuned subject to the one-to-one matching
between the elements and the K nodes for tabular construction. The procedure for each step is detailed below.
N
Step 1 (GTM-LDLV): the first step of the PTG is the same as the original GTM-LDLV. In the GTM-LDLV,
K nodes, , arbitrarily arranged in the L-dimensional latent space are first prepared. Then we build a
u1,...,uK
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nonlinear function that maps the pre-defined nodes to the D-dimensional feature space. The model
f (uk) f (uk)
defines an L-dimensional manifold in the D-dimensional feature space, which is fitted with respect the data
N
L ≤ 3
points of element features. The dimension of the latent space is set to for visualization.
x n
It is assumed that the D-dimensional feature vector of element is generated independently from a mixture
n
of K Gaussian distributions, where the mixing rates are all equal to , and the mean and the covariance matrix
1/K
of each distribution are y =f u and −1 , respectively ( I denotes the identity matrix). According to the GTM-
k ( k) β I
LDLV, the mean is modelled to be the product of two functions, a D-dimensional vector-valued function
f (uk)
g(u ) ′
and a positive scalar function . Here, we introduce a vector of K latent variables, ,
h(u ) z =(z ,...,z )
k k n 1n Kn
n k
that indicates the assignment of element to one of the given K nodes. The th entry z takes the value of 1 if
kn
x k x ,...,x
is generated by the th component distribution, and 0 otherwise. Here, let X denote a matrix of of
n 1 N
the elements, and Z be a matrix of z1,...,zN . Then, their joint distribution is given by
p X,Z|g,H,β = K−N N K N xn|yk,β−1I zkn, (1)
n=1 k=1
y =f u =g u h u ,
k ( k) ( k) ( k) (2)
where N ·|µ,� denotes the Gaussian density function with mean µ and covariance matrix , g is a vector of
( )
g u k = 1,...,K , and H is a matrix of .
( k)( ) h(uk)(k = 1,...,K)
g(u) c (u ,u ;ξ )
The prior distribution of is given as a truncated GP with mean 0 and covariance function g i j g ,
d
which handles positive-bounded random functions. The prior distribution of the th entry of is given
h (u) h(u)
d
0 c (u ,u ;ξ )
as a GP with mean and covariance function . To be specific, the covariance functions,
c (ui,uj) g i j
and , are given by h g
c (ui,uj)
h
�ui−uj�2 ,
c u ,u ;ξ =ν •exp − (3)
g i j g g 2l
g
L
2l u l u 2 2
c u ,u = ( i) ( j) exp − �ui−uj� . (4)
h i j l2 u +l2 u l2 u +l2 u
( i) ( j) ( i) ( j)
ξ
In Eq. (3), the hyperparameter consists of and , referred to as the variance and the length-scale, that
ν l
g g g
g(u)
control the magnitude of variances and smoothness of a positive-valued function generated from the GP. In
u l u = exp r u
Eq. (4), the length-scale parameter is a function of and parameterized as with the func-
l(u) ( ) ( ( ))
tion following the GP with mean 0 and covariance function . Finally, a gamma prior is placed
r(u) c (u ,u ;ξ )
on the precision parameter in Eq. (1). r i j r
β
The covariance function in Eq. (4) is the key in the GTM-LDLV. In general, a covariance function in a GP
u
governs a degree of preservation between the similarity of any inputs, e.g. and , and the similarity of their
i uj
outputs. The heterogeneous variance over the latent space in Eq. (3) can bring locally varying smoothness in
resulting manifolds in the feature space. In addition, the variance function is statistically estimated with the
hierarchically specified GP prior based on the covariance function .
c (u ,u ;ξ )
r i j r
The unknown parameter to be estimated is . In the GTM-LDLV, the posterior distribution
θ = Z,β,g,H,r
is approximately evaluated using a Markov Chain Monte Carlo (MCMC) method. Iteratively sampling
p(θ|X)
from the full conditional posterior distribution for each , we obtained a set of ensembles that fol-
{Z,β,g,H,r}
low the posterior distribution approximately. By taking the ensemble average over the samples from ,
p(θ|X)
the parameters of the GTM-LDLV are estimated. A detailed description of the GTM-LDLV is given in the Sup-
plementary Information section.
Step 2 (Node expansion): to avoid the occurrence of improper assignments of the N elements to a redundant
set of nodes, we adopt a coarse-to-fine strategy. Starting from an initially trained GP model of at step 1,
K
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