# Neural networks for a quick access to a digital twin of scanning physical
properties measurements

Kensei Terashima<EMAIL_ADDRESS>National Institute for Materials
Science, 1-2-1 Sengen, Tsukuba, Japan  Pedro Baptista de Castro  National
Institute for Materials Science, 1-2-1 Sengen, Tsukuba, Japan  University of
Tsukuba, 1-1-1 Tennoidai, Tsukuba, Japan  Miren Garbiñe Esparza Echevarria
National Institute for Materials Science, 1-2-1 Sengen, Tsukuba, Japan
University of Tsukuba, 1-1-1 Tennoidai, Tsukuba, Japan  Ryo Matsumoto
National Institute for Materials Science, 1-2-1 Sengen, Tsukuba, Japan
Takafumi D Yamamoto  National Institute for Materials Science, 1-2-1 Sengen,
Tsukuba, Japan  Akiko T Saito  National Institute for Materials Science, 1-2-1
Sengen, Tsukuba, Japan  Hiroyuki Takeya  National Institute for Materials
Science, 1-2-1 Sengen, Tsukuba, Japan  Yoshihiko Takano  National Institute
for Materials Science, 1-2-1 Sengen, Tsukuba, Japan  University of Tsukuba,
1-1-1 Tennoidai, Tsukuba, Japan

###### Abstract

For performing successful measurements within limited experimental time,
efficient use of preliminary data plays a crucial role. This work shows that a
simple feedforward type neural networks approach for learning preliminary
experimental data can provide quick access to simulate the experiment within
the learned range. The approach is especially beneficial for physical
properties measurements with scanning on multiple axes, where derivative or
integration of data are required to obtain the objective quantity. Due to its
simplicity, the learning process is fast enough for the users to perform
learning and simulation on-the-fly by using a combination of open-source
optimization techniques and deep-learning libraries. Here such a tool for
augmenting the experimental data is proposed, aiming to help researchers to
decide the most suitable experimental conditions before performing costly
experiments in real. Furthermore, this tool can also be used from the
perspective of taking advantage of reutilizing and repurposing previously
published data, accelerating data-driven exploration of functional materials.

Neural networks, Physical property measurements, Repository data augmentation

##  I Introduction

The recent global rise of databases on materials as well as accessible
repositories and advanced text mining techniques has led to the evolution of
materials informatics as an effective tool for exploration and design of
functional materials [1]. Along with the increase of the available crystal
structural input data, the emergence of high-throughput first-principles
calculation systems has enhanced a variety of materials estimated property
data useful for statistical analysis [2, 3, 4]. In addition, the application
of machine-learning methods has proved to be quite efficient for dealing with
a large amount of materials data [1, 5], allowing the extraction of useful
information that has been hidden or too complicated for an ordinal human to
percept. As a result of the recent rapid growth of such tools for handling
materials data, the process for obtaining candidate materials in cyberspace
has been significantly accelerated, while the bottleneck for the discovery of
functional materials remains on the actual synthesis and experimental
evaluation in real space [6]. This is because the cost for obtaining
experimental data is usually high, especially when researchers have to use a
state-of-the-art shared facility with fixed machine time. Furthermore, not
only the experiment itself but also its preparation can be time-consuming as
sometimes one has to start from optimization of the sample synthesis process,
a typical aspect for physics and materials science measurements.

Such a costly-in-real situation is even more serious in industrial fields,
where a large amount of money has to be spent for the production of ”real”
items. To overcome this issue, there an idea called “digital twin” was
produced, where one builds a simulation model that takes account of the data
in real space and tries to optimize the parameters in the cyber space before
real synthesis [7]. For being beneficial, the model has to correspond enough
to the objective, and the cost for the simulation both in money and time has
to be lower than the actual production. Here again, the application of
machine-learning methods has greatly reduced the duration required for the
execution of simulation as compared to that for performing conventional
simulation algorithms. Recently, the use cases of such a digital-twin approach
aided by machine learning are becoming popular also in academic research. For
instance, machine-learning was used for approximating time-consuming
simulation results for fluid dynamics, allowing researchers to perform a quick
optimization of the growing condition of objective crystals [8].

In this context, such a digital-twin approach is expected to be effective and
highly useful also for physics and materials science measurements, where the
objective properties are sometimes deduced from experimental data by applying
mathematical operations such as derivatives or integration. Especially in
physics, the development of new approaches or ideas to data analysis, creates
an opportunity to revisit or reanalyze previously published data, allowing
researchers to draw new insights from this previous knowledge. However, it is
quite often that the data are non-evenly spaced along multiple axes due to
experimental confinements, and it may contain non-linear responses that cannot
be interpolated easily. In such a case, it is quite difficult to estimate the
target quantity from preliminary data. On the other hand, artificial neural
networks are known to have high flexibility and adaptivity in principle [9,
10]. It has been also discussed [11] that neural networks learning can be as
efficient as other approximation methods from a viewpoint of minimizing loss
using the number of parameters as low as possible. They have proved to be even
useful when the objective has inhomogeneity in smoothness [12], or when the
objective has several discontinuities [13]. Thus, approximation using this
method is expected to be suitable for materials science and physics
experimental data that tend to contain discrete change due to first-order
transition and/or multiple peak structures.

In this article, we show that learning preliminary data of materials science
by neural networks provides us a way to quickly build a model that can
describe the target property dependencies on scanning axes (treated as
features). In order to ensure the accuracy of the model that can be built on-
the-fly during the experiment, several open python libraries have been used in
combination, as well as several tools for efficient learning. Once the model
is built, the simulation within the learned feature range can be done
instantly, which enables researchers to evaluate the experimental plan as well
as its cost (Figure 1(a)). We propose that such an approach of data
augmentation helps researchers to use preliminary- or deposited data
effectively for performing the data-driven search of functional materials.

Figure 1: (a) Graphical concept for application of digital-twin approach to
materials property measurements for performing efficient experiments using
preliminary data as train data. (b) Combination of open libraries used in this
study to quickly build a fully-connected feedforward neural networks model for
simulating experimental data.

##  II Construction of neural networks model

In our approach, the constructed code relies mainly on 4 open libraries
available to the public in addition to the Anaconda package [14], shown in
Figure 1(b). As a machine-learning body, we built fully-connected feedforward
neural networks, using Keras with Tensorflow backend (v.2.x) [15, 16]. Keras
has several optimizers by default, and among them, we tried stochastic
gradient descent, Nesterov accelerated gradient [17], root mean square
propagation [18], and adaptive moment estimation (Adam) [19] for several times
to our data shown in Section 3. For our data, Adam tended to be the most
stable and fast for minimizing loss of mean squared error between predicted
values and training data (not shown). We set the number of layers, nodes,
learning rate, and batch size as hyperparameters to be optimized. Once those
hyperparameters are suggested, random weights and biases are given as initial
values in Keras and they are tuned by the optimizer as the epoch proceeds in a
single training run. Thus the learned result in a single run varies by the
initial random values, even though there could be a general trend for
obtaining a moderate set of hyperparameters for example the case in learning
rate and batch size [20, 21, 22]. To be quick, we confined the search space of
the hyperparameters in the following range: $2\leq$ number of layers $\leq
10$, $50\leq$ number of nodes $\leq 200$, $1e^{-5}\leq$ learning rate $\leq
1e^{-4}$, and $16\leq$ batch size $\leq 1024$ varying on the data size. To
obtain the best possible learned result in a fixed time, we used the Bayesian
optimization package Optuna [23]. By combining a relational database provided
by MySQL [24], Optuna receives each training result with a set of
hyperparameters running in parallel, so that it can determine a set of
hyperparameters for the next run based on Tree Parzen Estimator [25].
Typically, the best set of hyperparameters is found by at most $\sim$30 runs.
After many trials, we found that overfitting is welcome for our purpose to
learn the detail of the real experimental data (including noise level and
characteristics of experimental machines), and that cross-validation simply
increases the required time for training. Thus we used the whole data for
training. For visualization of data and controlling multiple Optuna runs in
parallel, we used the Streamlit library [26] to construct a graphical user
interface (GUI) that works on the web browser.

To obtain the best model during the hyperparameter search, we used three
functions equipped in the above libraries. First, is early stopping
implemented in Keras, with the option of saving the best model in the run,
where the maximum epoch number is 500. The second is the learning rate
scheduler implemented in Keras, where we change the learning rate to be 10
times smaller value from its initial value after 100 epochs. The third is
pruning with the Asynchronous Successive Halving algorithm [27] implemented in
Optuna, where it examines if each run shall be pruned based on the comparison
of current score with those of running parallel by 25n epochs where $n$ is an
integer greater than or equal to zero.

The approach used here might be primitive as compared to recent state-of-the-
art combinations of neural networks architectures, optimizers, and other
techniques, but we tried to keep the system simple, fast, and easy to use as
much as possible, considering that the target users are not specialist of data
science but materials scientists like us. As a result, we could quickly finish
learning small datasets typical for physical property measurements (number of
data $\sim$ a few thousand) shown in the next section, even without using
graphics processing units (GPUs). More detail on case studies for the duration
and used computers are in Supporting Information. A tutorial for how to get
ready and perform the neural networks learning and simulation shown in this
paper will be available at https://www.github.com/kensei-te/mat$\\_$interp.

##  III Results and Discussion

In this section, we show 4 typical use cases as an example. The first is the
magnetization curve of a well-known magnetocaloric material ErCo2, where
$1^{st}$ order transition occurs that accompanies a steep change in the
magnetization as a function of temperature and the transition temperature
migrates by application of an external magnetic field. The second example is
the magnetization curve and estimation of magnetocaloric effect in another
magnet Fe3Ga4 that shows $1^{st}$ order transition but its transition
temperature changes more drastically by applied field than in the first
example of ErCo2. The third example is the resistance data of the PdH
superconductor which is relatively noisy as compared to above examples. The
fourth example is a mapping data of angle-resolved photoelectron spectroscopy
(ARPES) intensity on Fermi energy of La(O,F)BiS2 superconductor taken above
superconducting temperature ($T_{c}$) as a demonstration of the application of
the method for two-dimensional intensity map, consisting of a number of peak
structures.

###  III.1 $M(T,H)$ of ErCo2 magnet and estimation of its magnetocaloric
effect

First, we show a case for physical property data that have 1st order
transition, namely a steep change in the observed value. ErCo2 is known to be
a ferrimagnet with its ordering temperature $T_{Curie}$ of $\sim$ 35 K [28].
It exhibits a $1^{st}$ order magnetic transition accompanied by
magnetostriction, and the material is one of the most popular materials for
magnetic refrigeration [29, 30, 31]. To evaluate the magnitude of magnetic
entropy change $|\Delta S_{M}|$, one of the most common ways [30] is to
measure the magnetization of the sample as a function of temperature and
applied magnetic field change $\Delta H$(from 0 to $H$) either isothermal or
isofield, and deduce $|\Delta S_{M}|$ by applying following Maxwell’s equation
for the observed magnetization data:

$\Delta S_{M}(T,\Delta H)=\mu_{0}\int_{0}^{H}\frac{\partial
M(T,H^{{}^{\prime}})}{\partial T}dH^{{}^{\prime}}$

The equation includes an integration along the applied field ($H$).
Practically, with such discrete experimental data, trapezoidal integration can
be applied for a numerical integration, which corresponds to a linear
interpolation along the $H$-axis. However, if we estimate $|\Delta S_{M}|$ in
this manner from the experimental data shown in Figure 2(a) taken by a coarse
magnetic field step, it ends up with extrinsic oscillation of estimated
$|\Delta S_{M}|$ values as shown as a gray solid line in Figure 2(e). This is
simply because of the failure of linear interpolation for data having $1^{st}$
order transition (see Supporting Information for technical details). To avoid
this extrinsic effect for estimation of the target property, one has to use a
finer magnetic field step, so that linear interpolation starts to work well
since $\partial M/\partial T|{{}_{H}}$ between two steps has a finite overlap
each other. This means one has to know the proper measurement step before the
measurement itself. In such a case, neural networks simulation based on
preliminary data works efficiently as it can perform a non-linear
interpolation. Figure 2(b) shows the predicted magnetization values by a
constructed model for the conditions in the train data of Figure 2(a), which
corresponds well to the train data, indicating that a plausible model has been
made. Once such a model is constructed, the model can perform quick
predictions for any desired feature values within the learned range. After a
couple of trials, we found in Figure 2(f) that extrinsic oscillation in the
simulation is expected to be significantly suppressed when the magnetic field
step of the measurement is smaller than 0.2 T. We measured the magnetization
in the proposed step as in Figure 2(c), and found in Figure 2(e) that the
experimentally evaluated $|\Delta S_{M}|$ corresponds very well to what has
been predicted by the simulated model.

Figure 2: (a) Preliminary magnetization data of ErCo2, which is used as the
train data for constuction of model. (b) Simulated magnetization by the
trained model for the conditions in the train data. Gray solid lines show the
train data. (c) Magnetization of ErCo2 with magnetic field step proposed by
the model. (d) Simulated magnetization by the trained model. (e) Estimated
$|\Delta S_{M}|$ of ErCo2 for a field change of 0-5 T. (f) Simulated $|\Delta
S_{M}|$ for different experimental steps of magnetic fields. Gray solid line
shows the estimated $|\Delta S_{M}|$ from (a) by applying trapezoidal
integration along magnetic field direction. Blue arrows between figures
indicate an order of workflow of the measurements and simulations.

###  III.2 $M(T,H)$ of Fe3Ga4 magnet and estimation of its magnetocaloric
effect

Here we show the case of another magnet Fe3Ga4 and evaluation of its
magnetocaloric property, to show the possibility of applying neural networks
study for analyzing literature data. The magnetic property and magnetization
as a function of temperature and magnetic field have been reported [32], while
$|\Delta S_{M}|$ of this material has not been reported yet. Figure 3(a) shows
our preliminary experimental data that corresponds well to the literature data
[32], but both of them do not have fine enough experimental steps to evaluate
$|\Delta S_{M}|$ of this material. As shown in Figure 3(d) and 3(f), we have
performed a simulation of expected $|\Delta S_{M}|$ taken on each field step
case and found that extrinsic effect for $|\Delta S_{M}|$ would not be
observed when we measure magnetization by a magnetic field step less than 0.1
T for this material. Considering the required experimental time, we chose a
0.075 T step, and the simulated data showed an excellent correspondence with
the real experimental data as shown in Figure 3(c)-(f). The simulation also
tells us that, the estimated $|\Delta S_{M}|$ is at most 0.25 J kg-1 K-1 by a
field change of 0-1 T (which is small as compared to those of other
magnetocaloric materials [30]) despite this measurement covers wide
temperature range of 10-200 K with fine magnetic field step, hence it is
expected to take approximately 1 week. Therefore if one is simply looking for
a material with high $|\Delta S_{M}|$, the simulation tells us that this
material may not be suitable to be synthesized and measured by spending a fair
amount of time. In other words, such a simulation is not only useful for
analyzing the past preliminary or repository data from a different point of
view, but also it can help researchers to evaluate the cost of performing the
experiment by leveraging the confined knowledge available from preliminary- or
literature data.

Figure 3: (a) Preliminary magnetization data of Fe3Ga4, which is used as the
train data for constuction of model. (b) Simulated magnetization by the
trained model for the conditions in the train data. Gray solid lines show the
train data. (c) Magnetization of Fe3Ga4 with magnetic field step proposed by
the model. (d) Simulated magnetization by the trained model. (e) Estimated
$|\Delta S_{M}|$ of Fe3Ga4 for a field change of 0-1 T. (f) Simulated $|\Delta
S_{M}|$ for different experimental steps of magnetic fields. Gray solid line
shows the estimated $|\Delta S_{M}|$ from (a) by applying trapezoidal
integration along magnetic field direction. Blue arrows between figures
indicate an order of workflow of the measurements and simulations.

###  III.3 $R(T,H)$ of PdH superconductor

Next, we show the case for data with a lower signal-to-noise ratio. For this
purpose, we used a part of our resistance data of PdH [33] as train data
(Figure 4(a)). As can be seen in Figure 4(b), the model does not necessarily
capture the ultrafine structure coming from noise, however it grabs not only
the overall tendency of the lineshape but also several fine structures that
might be influenced by noise. The inability of tracking the full noise is
because of the fact that the number of layers and nodes in the model is
limited and lower than the number of total train data points. Indeed, neural
networks are sometimes used for denoising purposes [34, 35]. On the other
hand, our intention here is to simulate the experiment as it is, including the
extrinsic effect such as noise and unique characteristics of the measurement
apparatus, based on the non-evenly spaced preliminary data. Comparing the
experimental data in Figure 4(c) with simulated data in Figure 4(d), it can be
said that the model in the current approach can tell us how the expected
lineshapes can be affected by the noise level appeared in the train data, with
a finite inevitable denoising effect.

Figure 4: (a) Train data of resistivity measurements for PdH, taken from a
part of the whole experimental data in (c). (b) Simulated resistance by the
trained model for the conditions in the train data. Gray solid lines show the
train data. (c) Experimental data of resistivity measurements for PdH
superconductor [33]. (d) Simulated resistance by the trained model for the
same conditions in (c). Gray open circles show the conditions seen by the
model during training.

###  III.4 Angle-resolved photoelectron spectroscopy intensity map of
La(O,F)BiS2

As a last example, we show that our quick and simple approach is also
applicable to two-dimensional intensity map, namely, ARPES intensity map at
Fermi energy in La(O,F)BiS2 superconductor on normal state (above $T_{c}$).
ARPES intensity at fixed energy tends to be high in angular area where
corresponding electronic state exists, thus the data consists of a number of
peak structures (see Supporting Information for lineshape of each cut). In
Figure 5, the ARPES intensity is shown as a function of two-dimentional angles
with respect to the normal angle to the sample surface, that corresponds to
two-dimentional wave vectors in reciprocal space [36]. For constructing such a
mapping data, it is quite often that one measurement is performed with fixing
one of the axis (for instance $\theta_{2}$ in the figure) and keep
concatenating data with different angle of $\theta_{2}$. Therefore a
preliminary measurement with coarse angle step of $\theta_{2}$ (Figure 5(a))
can be performed before a serious measurement with fine angular steps and
better statistics (Figure 5(c)), so that one can determine the objective
measurement area of angles. With help of neural networks learning, we can
simulate how data would become with specific measurement area and steps as in
Figure 5(d), thus the simulation helps researchers to make a decision for
planning the experiment. Here we note that the simulated pattern captures
characteristics of the measurement apparatus, namely high sensitivity at edge
and detection limit in certain angle, so it is suitable for estimating the
effect of such instrumental conditions as well. However, the simulated pattern
is affected by the presence of sizable noise, hence the pre-processing of
train data with other noise reduction method might help further though such
process may take longer time. Alternatively, it is also possible to take an
avarage of predictions by several models found during learning process. We
also note that although our simple approach is applicable on-the-fly to such a
mapping data including non-evenly spaced ones, other sophisticated methods
such as super resolution convolutional neural networks-based approach [37, 38,
39] would perform certainly better especially for evenly spaced data such as
photographs if one can spare enough time for training and has an access to
powerful GPUs.

Figure 5: (a) Roughly-scanned ARPES intensity distribution as a function of
angles with respect to sample normal, used as train data. (b) Simulated ARPES
intensity map by the trained model for the conditions in the train data. Green
dotted square shows the simulated angular area in (d). (c) Same as (a) but
with proposed angular steps by simulation in (d). (d) Simulated ARPES
intensity map by the trained model for estimation of measurement conditions in
(c).

###  III.5 Scope and limitation of current approach

Here we note the applicability/inability of our approach. The code is
available at
https://www.github.com/kensei-te/mat$\\_$interp including how to get ready to
use GUI, which runs in Linux and Mac OS (see Supporting Information for
details). It can learn quickly the data so that one can perform the simulation
during the experiment, especially when the total number of data points kept up
to a few tens of thousands. For this reason, it is preferred to keep the
dimension of data (i.e. number of feature columns) low, even though it can
deal with multidimensional data. The approach would be useful mainly for two
cases: (i) Simulation of a costly experiment from preliminary experimental
data: based on the roughly-scanned data, one can simulate how the data and
resultant estimated target property would be, when taken in fixed experimental
steps and ranges. Thus it helps researchers to plan and perform efficient
experiments. This is beneficial especially when the target properties are
deduced from mathematical operations of experimental data and hence are
depending on experimental conditions, as shown in Section 3.1 and 3.2. (ii)
Aligning and augmentation of repository data: there is an increasing number of
accessible materials data sets, that have been published in the past. It is
quite often that one encounters such problems that despite the material being
of interest, there is a mismatch or lack in the experimental conditions as
compared to what is required in order to perform additional analysis. As the
current method can adapt readily to non-evenly spaced data with a non-linear
response, it will help researchers to judge whether the material is worthy to
be examined further by reusing this data for the simulation method proposed
here, to spare a certain amount of experimental cost as shown in Section 3.2.

On the other hand, the method is ultimately a mere interpolation, thus the
signal that is skipped in the train data with too coarse step will never be
predicted in the simulation. For the same reason, though the simulation helps
researchers to plan the experiment, the simulated results and estimated values
of physical properties themselves should be clearly distinguished from
experimental data and be taken with care since it is not certain what will
come out in reality until verified by experiment. Also, the prediction by
learned models includes how the data will be affected by unique
characteristics of experimental apparatus (sensitivity, detection limit, and
so on) and noise, as has been shown in Section 3.3 and 3.4. The effect of
noise could be examined by comparing several simulated results in different
runs (see Supporting Information for details). If the researcher aims to
suppress the effect of noise rather than simulating it, other noise reduction
methods may work better.

##  IV Conclusion

In this work, we present a simple and quick method to simulate experimental
data by learning preliminary data using fully-connected feedforward neural
networks. The approach is shown to be suitable for materials science
experimental data with typical examples. Such a tool would help researchers to
deal with preliminary- or past repository data efficiently, supporting
decision-making during the experimental research. A tutorial on how to get
ready and perform the neural networks learning and simulation shown in this
paper will be available at https://www.github.com/kensei-te/mat$\\_$interp.

##  V Experimental Section

###  V.1 Sample Synthesis and Preparation

ErCo2 sample was obtained by arc-melting of Er and Co rocks in a vacuum
chamber of Ar atomosphere, followed by a 21 hour annealing on 850 ∘C and
quenching by water. Fe3Ga4 sample was obtained by arc-melting of Fe and Ga
chunks. The synthesis procedure of PdH sample is provided in [33], while the
synthesis procedure of La(O,F)BiS2 sample is in [40].

###  V.2 Experimental Conditions

Magnetization of samples has been measured using Magnetic Properties
Measurement System (Quantum Design), in field cooling process. Resistance of
sample under 2.6 GPa has been measured by four probe method using diamond
anvil cell with boron-doped diamond electrodes [33] and Physical Properties
Measurement System (Quantum Design). ARPES measurements have been performed at
BL28A [41] in Photon Factory KEK, Japan (proposal number 2021G686), with DA30
analyzer (Scienta Omicron) at 20 K (estimated $T_{c}$ of the sample is
approximately 2.2 K [42]), $h\nu$ = 70 eV.

##  VI Acknowledgements

The ARPES measurements were performed at BL28A of Photon Factory, KEK
(proposal No. 2021G686). The authors acknowledge Wei-Sheng Wang and Taku Tou
for testing the installation of the code. This work was supported by the JST-
Mirai Program (Grant No. JPMJMI18A3), JSPS Bilateral Program
(JPJSBP120214602), and JSPS KAKENHI (Grant Nos. 20K05070, 19H02177). P.B.C.
and M.G.E.E. acknowledges the scholarship support from the Ministry of
Education, Culture, Sports, Science and Technology (MEXT), Japan.

## References

  * Stanev  _et al._ [2021] V. Stanev, K. Choudhary, A. Kusne, J. Paglione, and I. Takeuchi, Artificial intelligence for search and discovery of quantum materials, Communications Materials 2, 105 (2021).
  * Jain  _et al._ [2013] A. Jain, S. P. Ong, G. Hautier, W. Chen, W. D. Richards, S. Dacek, S. Cholia, D. Gunter, D. Skinner, G. Ceder, and K. A. Persson, The Materials Project: A materials genome approach to accelerating materials innovation, APL Materials 1, 011002 (2013).
  * Kirklin  _et al._ [2015] S. Kirklin, J. E. Saal, B. Meredig, A. Thompson, J. W. Doak, M. Aykol, S. Rühl, and C. Wolverton, The open quantum materials database (oqmd): assessing the accuracy of dft formation energies, npj Computational Materials 1, 1 (2015).
  * Curtarolo  _et al._ [2012] S. Curtarolo, W. Setyawan, G. L. Hart, M. Jahnatek, R. V. Chepulskii, R. H. Taylor, S. Wang, J. Xue, K. Yang, O. Levy, M. J. Mehl, H. T. Stokes, D. O. Demchenko, and D. Morgan, Aflow: An automatic framework for high-throughput materials discovery, Computational Materials Science 58, 218 (2012).
  * de Castro  _et al._ [2020] P. B. de Castro, K. Terashima, T. D. Yamamoto, Z. Hou, S. Iwasaki, R. Matsumoto, S. Adachi, Y. Saito, P. Song, H. Takeya, and Y. Takano, Machine-learning-guided discovery of the gigantic magnetocaloric effect in hob sub 2 /sub near the hydrogen liquefaction temperature, NPG Asia Materials 12, 35 (2020).
  * Xiong  _et al._ [2021] Y. Xiong, Q. T. Campbell, J. Fanghanel, C. K. Badding, H. Wang, N. E. Kirchner-Hall, M. J. Theibault, I. Timrov, J. S. Mondschein, K. Seth, R. Katz, A. M. Villarino, B. Pamuk, M. E. Penrod, M. M. Khan, T. Rivera, N. C. Smith, X. Quintana, P. Orbe, C. J. Fennie, S. Asem-Hiablie, J. L. Young, T. G. Deutsch, M. Cococcioni, V. Gopalan, H. D. Abruña, R. E. Schaak, and I. Dabo, Optimizing accuracy and efficacy in data-driven materials discovery for the solar production of hydrogen, Energy & Environmental Science 14, 2335 (2021).
  * Tao  _et al._ [2019] F. Tao, H. Zhang, A. Liu, and A. Y. C. Nee, Digital twin in industry: State-of-the-art, IEEE Transactions on Industrial informatics 15, 2405 (2019).
  * Dang  _et al._ [2021] Y. Dang, C. Zhu, M. Ikumi, M. Takaishi, W. Yu, W. Huang, X. Liu, K. Kutsukake, S. Harada, M. Tagawa, and T. Ujihara, Adaptive process control for crystal growth using machine learning for high-speed prediction: application to sic solution growth, CrystEngComm 23, 1982 (2021).
  * Cybenko [1989] G. Cybenko, Approximation by superpositions of a sigmoidal function, Mathematics of Control, Signals, and Systems 2, 303 (1989).
  * Mhaskar and Micchelli [1992] N. Mhaskar, H. and A. Micchelli, C., Approximation by superposition of sigmoidal and radial basis functions, Advances in Applied Mathematics 13, 350 (1992).
  * Mhaskar [1996] N. Mhaskar, H, Optimal approximation of continuous functions by very deep relu networks, Neural Computation 8, 164 (1996).
  * Suzuki [2019] T. Suzuki, Adaptivity of deep reLU network for learning in besov and mixed smooth besov spaces: optimal rate and curse of dimensionality, in  _International Conference on Learning Representations_ (2019).
  * Imaizumi and Fukumizu [2019] M. Imaizumi and K. Fukumizu, Deep neural networks learn non-smooth functions effectively, in  _Proceedings of the Twenty-Second International Conference on Artificial Intelligence and Statistics_, Proceedings of Machine Learning Research, Vol. 89, edited by K. Chaudhuri and M. Sugiyama (PMLR, 2019) pp. 869–878.
  * ana [2020] Anaconda software distribution (2020).
  * Chollet  _et al._ [2015] F. Chollet _et al._ , Keras (2015).
  * Abadi  _et al._ [2015] M. Abadi, A. Agarwal, P. Barham, E. Brevdo, Z. Chen, C. Citro, G. S. Corrado, A. Davis, J. Dean, M. Devin, S. Ghemawat, I. Goodfellow, A. Harp, G. Irving, M. Isard, Y. Jia, R. Jozefowicz, L. Kaiser, M. Kudlur, J. Levenberg, D. Mané, R. Monga, S. Moore, D. Murray, C. Olah, M. Schuster, J. Shlens, B. Steiner, I. Sutskever, K. Talwar, P. Tucker, V. Vanhoucke, V. Vasudevan, F. Viégas, O. Vinyals, P. Warden, M. Wattenberg, M. Wicke, Y. Yu, and X. Zheng, TensorFlow: Large-scale machine learning on heterogeneous systems (2015), software available from tensorflow.org.
  * [17] Y. Nesterov, A method of solving a convex programming problem with convergence rate mathcal $\\{$O$\\}$(1/kˆ $\\{$2$\\}$), in  _Sov. Math. Dokl_ , Vol. 27.
  * Tieleman  _et al._ [2012] T. Tieleman, G. Hinton, _et al._ , Lecture 6.5-rmsprop: Divide the gradient by a running average of its recent magnitude, COURSERA: Neural networks for machine learning 4, 26 (2012).
  * Kingma and Ba [2014] D. P. Kingma and J. Ba, Adam: A method for stochastic optimization, arXiv preprint arXiv:1412.6980 (2014).
  * Krizhevsky [2014] A. Krizhevsky, One weird trick for parallelizing convolutional neural networks, arXiv preprint arXiv:1404.5997 2014 (2014).
  * Goyal  _et al._ [2017] P. Goyal, P. Dollár, R. Girshick, P. Noordhuis, L. Wesolowski, A. Kyrola, A. Tulloch, Y. Jia, and K. He, Accurate, large minibatch sgd: Training imagenet in 1 hour (2017).
  * Jastrzebski  _et al._ [2018] S. Jastrzebski, Z. Kenton, D. Arpit, N. Ballas, A. Fischer, Y. Bengio, and A. J. Storkey, Width of minima reached by stochastic gradient descent is influenced by learning rate to batch size ratio, in  _ICANN (3)_ (2018) pp. 392–402.
  * Akiba  _et al._ [2019] T. Akiba, S. Sano, T. Yanase, T. Ohta, and M. Koyama, Optuna: A next-generation hyperparameter optimization framework, in  _Proceedings of the 25rd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining_ (2019).
  * Widenius  _et al._ [2002] M. Widenius, D. Axmark, and K. Arno,  _MySQL reference manual: documentation from the source_ (” O’Reilly Media, Inc.”, 2002).
  * Bergstra  _et al._ [2015] J. Bergstra, B. Komer, C. Eliasmith, D. Yamins, and D. D. Cox, Hyperopt: a python library for model selection and hyperparameter optimization, Computational Science Discovery 8, 014008 (2015).
  * [26] Streamlit, software available from streamlit.io.
  * Li  _et al._ [2018] L. Li, K. Jamieson, A. Rostamizadeh, E. Gonina, M. Hardt, B. Recht, and A. Talwalkar, A system for massively parallel hyperparameter tuning, arXiv preprint arXiv:1810.05934 10.48550/ARXIV.1810.05934 (2018).
  * Ross and Crangle [1964] J. W. Ross and J. Crangle, Magnetization of cubic laves phase compounds of rare earths with cobalt, Phys. Rev. 133, A509 (1964).
  * Wada  _et al._ [1999] H. Wada, S. Tomekawa, and M. Shiga, Magnetocaloric properties of a first-order magnetic transition system erco2, Cryogenics 39, 915 (1999).
  * GschneidnerJr  _et al._ [2005] K. A. GschneidnerJr, V. K. Pecharsky, and A. O. Tsokol, Recent developments in magnetocaloric materials, Reports on Progress in Physics 68, 1479 (2005).
  * Tang  _et al._ [2022] X. Tang, H. Sepehri-Amin, N. Terada, A. Martin-Cid, I. Kurniawan, S. Kobayashi, Y. Kotani, H. Takeya, J. Lai, Y. Matsushita, T. Ohkubo, Y. Miura, T. Nakamura, and K. Hono, Magnetic refrigeration material operating at a full temperature range required for hydrogen liquefaction, Nature Communications 13, 1817 (2022).
  * Kawamiya and Adachi [1986] N. Kawamiya and K. Adachi, Magnetic and mössbauer studies of metamagnetic fe3ga4, Journal of the Physical Society of Japan 55, 634 (1986), https://doi.org/10.1143/JPSJ.55.634 .
  * Matsumoto  _et al._ [2021] R. Matsumoto, S. Nakano, S. Yamamoto, and Y. Takano, Synthesis and electrical transport measurement of superconducting hydrides using diamond anvil cell with boron-doped diamond electrodes, Japanese Journal of Applied Physics 60, 090902 (2021).
  * Kim  _et al._ [2021] Y. Kim, D. Oh, S. Huh, D. Song, S. Jeong, J. Kwon, M. Kim, D. Kim, H. Ryu, J. Jung, W. Kyung, B. Sohn, S. Lee, J. Hyun, Y. Lee, Y. Kim, and C. Kim, Deep learning-based statistical noise reduction for multidimensional spectral data, Review of Scientific Instruments 92, 073901 (2021).
  * Restrepo  _et al._ [2022] F. Restrepo, J. Zhao, and U. Chatterjee, Denoising and feature extraction in photoemission spectra with variational auto-encoder neural networks (2022).
  * Damascelli  _et al._ [2003] A. Damascelli, Z. Hussain, and Z.-X. Shen, Angle-resolved photoemission studies of the cuprate superconductors, Reviews of Modern Physics 75, 473 (2003).
  * Kim  _et al._ [2016] J. Kim, J. K. Lee, and K. M. Lee, Accurate image super-resolution using very deep convolutional networks, in  _2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR)_ (2016) pp. 1646–1654.
  * Chai  _et al._ [2020] X. Chai, H. Gu, F. Li, H. Duan, X. Hu, and K. Lin, Deep learning for irregularly and regularly missing data reconstruction, Scientific Reports 10, 3302 (2020).
  * Peng  _et al._ [2020] H. Peng, X. Gao, Y. He, Y. Li, Y. Ji, C. Liu, S. A. Ekahana, D. Pei, Z. Liu, Z. Shen, and Y. Chen, Super resolution convolutional neural network for feature extraction in spectroscopic data, Review of Scientific Instruments 91, 033905 (2020), https://doi.org/10.1063/1.5132586 .
  * Nagao  _et al._ [2014] M. Nagao, A. Miura, S. Demura, K. Deguchi, S. Watauchi, T. Takei, Y. Takano, N. Kumada, and I. Tanaka, Growth and superconducting properties of f-substituted robis2 (r=la, ce, nd) single crystals, Solid State Communications 178, 33 (2014).
  * Kitamura  _et al._ [2022] M. Kitamura, S. Souma, A. Honma, D. Wakabayashi, H. Tanaka, A. Toyoshima, K. Amemiya, T. Kawakami, K. Sugawara, K. Nakayama, K. Yoshimatsu, H. Kumigashira, T. Sato, and K. Horiba, Development of a versatile micro-focused angle-resolved photoemission spectroscopy system with kirkpatrick–baez mirror optics, Review of Scientific Instruments 93, 033906 (2022).
  * Yamamoto  _et al._ [2021] S. Yamamoto, R. Matsumoto, S. Adachi, and Y. Takano, High-pressure effects on la(o,f)BiS2 single crystal using diamond anvil cell with dual-probe diamond electrodes, Applied Physics Express 14, 043001 (2021).