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Machine learning the electronic structure of matter across temperatures

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Abstract

We introduce machine learning (ML) models that predict the electronic structure of materials across a wide temperature range. Our models employ neural networks and are trained on density functional theory (DFT) data. Unlike other ML models that use DFT data, our models directly predict the local density of states (LDOS) of the electronic structure. This provides several advantages, including access to multiple observables such as the electronic density and electronic total free energy. Moreover, our models account for both the electronic and ionic temperatures independently, making them ideal for applications like laser-heating of matter. We validate the efficacy of our LDOS-based models on a metallic test system. They accurately capture energetic effects induced by variations in ionic and electronic temperatures over a broad temperature range, even when trained on a subset of these temperatures. These findings open up exciting opportunities for investigating the electronic structure of materials under both ambient and extreme conditions.
Machine learning the electronic structure of matter across temperatures
Lenz Fiedler,1, 2, Normand A. Modine,3Kyle D. Miller,3and Attila Cangi1, 2,
1Center for Advanced Systems Understanding (CASUS), D-02826 Görlitz, Germany
2Helmholtz-Zentrum Dresden-Rossendorf, D-01328 Dresden, Germany
3Sandia National Laboratories, Albuquerque, NM 87185, USA
(Dated: June 12, 2023)
We introduce machine learning (ML) models that predict the electronic structure of materials
across a wide temperature range. Our models employ neural networks and are trained on density
functional theory (DFT) data. Unlike other ML models that use DFT data, our models directly
predict the local density of states (LDOS) of the electronic structure. This provides several advan-
tages, including access to multiple observables such as the electronic density and electronic total
free energy. Moreover, our models account for both the electronic and ionic temperatures indepen-
dently, making them ideal for applications like laser-heating of matter. We validate the efficacy
of our LDOS-based models on a metallic test system. They accurately capture energetic effects
induced by variations in ionic and electronic temperatures over a broad temperature range, even
when trained on a subset of these temperatures. These findings open up exciting opportunities for
investigating the electronic structure of materials under both ambient and extreme conditions.
I. INTRODUCTION
Predicting the electronic structure of matter is essen-
tial for advancing scientific progress across various ap-
plications. Electronic structure calculations, which typi-
cally employ density functional theory (DFT) [1,2], have
become a routine tool in materials science and chemistry
due to their accuracy and computational efficiency [3,4].
However, as the demand for high-fidelity simulation
data in emerging research areas increases, conventional
DFT simulations face significant limitations. DFT cal-
culations exhibit unfavorable scaling with both system
size and temperature [5], limiting their applicability for
current scientific challenges, particularly in studying ma-
terials under extreme conditions and within the warm
dense matter regime [68]. Progress in this area not only
contributes to the fundamental sciences by advancing our
understanding of astrophysical objects [912], but also
propels technological developments by enabling the mod-
eling of inertial confinement fusion capsule heating pro-
cesses [13], radiation damage processes in reactor walls
[1417], and advanced manufacturing [18,19]. Addi-
tionally, it supports diagnostics of scattering experiments
conducted at free-electron laser facilities [20,21] and pro-
motes the emerging field of hot-electron chemistry for ac-
celerating chemical reactions [22,23]. A particularly rel-
evant phenomenon in these applications involves rapidly
driven electrons leading to transient non-equilibrium con-
ditions resulting in hot electrons and cool nuclei which
have also been observed in semiconducting and dielectric
materials [24,25].
To address these computational limitations, the elec-
tronic structure community has increasingly turned to
machine learning (ML) techniques [26]. ML algorithms
can accurately predict complicated relationships using
tractable data samples. The application of ML to DFT
has led to numerous approaches, with most focusing on
predicting specific observables of interest [27] or replac-
ing DFT entirely with ML interatomic potentials (ML-
IAPs), which capture the electronic total energy or total
free energy landscape of a system and enable extended
simulations of ionic dynamics [28,29]. While existing
ML-based approaches show promise in accurately pre-
dicting observables or capturing the energy landscape of
quantum systems, most of them do not provide direct
access to the electronic structure of matter. Knowledge
of the electronic structure, however, offers several advan-
tages, including the exploration of multiple observables
beyond those targeted by a specific ML model and accu-
rately modeling of systems at higher temperatures, where
electronic temperature effects play a crucial role. In re-
cent work, the electronic structure has become a focus
of ML models [3032], but these models did not consider
the electronic temperature.
Only a few studies have made efforts to incorporate
electronic temperature into ML models trained to predict
DFT data. Ref. [33] develops an ML-IAP that directly
addresses the electronic temperature, unlike typical ML-
IAPs. Refs. [34,35] use an ML-approximated density
of states (DOS) to capture energetic effects associated
with the electronic temperature and provide corrections
to calculated observables. Similarly, Ref. [36] proposes
machine-learning a correction to the ground state energy
rather than the electronic total free energy.
The models discussed in this work differ from the ap-
proaches mentioned above. They are based on a recently
developed formalism that uses neural networks (NN) to
predict the electronic structure, represented by the lo-
cal density of states (LDOS) [37]. These models can
reproduce multiple observables, such as the DOS, elec-
tronic densities, and electronic total free energies from
the LDOS predictions. Unlike Ref. [33], they predict
the electronic total free energy alongside other electronic
structure properties and directly incorporate the elec-
tronic temperature. In our previous work [38], we demon-
strated that these models can replace electronic structure
calculations for systems larger than those accessible via
conventional DFT. However, to provide ab-initio accu-
arXiv:2306.06032v1 [cond-mat.mtrl-sci] 9 Jun 2023
2
racy for advanced scientific applications, these models
must also be able to make predictions across temper-
ature ranges. Since conventional DFT scales cubically
with the temperature [5], retraining ML models for each
temperature range of interest is impractical. If ML mod-
els can predict the LDOS throughout such a temperature
range one can replace DFT simulations and access elec-
tronic structure data at ab-initio accuracy without any
additional temperature scaling.
Moreover, our models uniquely account for electronic
and ionic temperature separately. The ionic tempera-
ture enters the model through the ionic configurations
sampled during dynamics and used as input to ML in-
ference, while the electronic temperature is an explicit
parameter in the expressions used to evaluate observ-
ables such as the electronic density and total free energy
from the LDOS. As a result, our models can effectively
handle situations where the electronic temperature sur-
passes the ionic temperature, such as in laser heating of
matter. This characteristic of our ML model responds to
a growing need for approaches that extend beyond con-
ventional molecular dynamics to tackle non-equilibrium
conditions between electrons and nuclei, as recently de-
veloped within the framework of two-temperature molec-
ular dynamics [39,40].
In this work, we present a case study of aluminum
as evidence that LDOS-based ML-DFT models can ac-
curately predict the electronic structure of matter over a
range of temperatures. By directly predicting the LDOS,
our models inherently incorporate thermal excitation ef-
fects. In contrast to conventional DFT simulations, our
models mitigate the temperature scaling issue, requir-
ing training data only at a few select temperatures to
facilitate modeling of materials over a broad tempera-
ture range. This ability to interpolate across temperature
regimes suggest that LDOS-based ML-DFT models are
particularly well-suited to investigating materials under
extreme conditions.
II. METHODS
A. Finite-Temperature DFT
In electronic structure theory, a system of Lelectrons
and Nions is typically described using collective elec-
tronic coordinates r={r1, ..., rL}and ionic coordi-
nates R={R1, ..., RN}, where rjR3and RαR3.
Within the framework of quantum statistical mechanics
[41], thermodynamic properties of such a system can be
obtained from the grand potential
Ω[ˆ
Γ] = Tr(ˆ
Γˆ
Ω) ,(1)
with the grand canonical operator
ˆ
= ˆ
Hµˆ
NkBτeˆ
S , (2)
where ˆ
Ndenotes the particle number operator, ˆ
Sthe en-
tropy operator, µthe chemical potential, and τethe (elec-
tronic) temperature. In the following, we will distinguish
between the ionic temperature τiand the electronic tem-
perature τe. In Eq. (2), ˆ
Hdenotes the Hamiltonian that
represents all interactions between electrons and ions. In
electronic structure theory, the Born-Oppenheimer ap-
proximation [42] is typically employed, resulting in the
Born-Oppenheimer Hamiltonian
ˆ
H=ˆ
T+ˆ
Vee +ˆ
Vei +ˆ
Vii ,(3)
where ˆ
T=PL
j−∇2
j/2denotes the kinetic energy oper-
ator of the electrons, ˆ
Vee =PL
jPL
l=j1/(2|rjrl|)the
electron-electron interaction, ˆ
Vei =PL
jPN
αZα/|rj
Rα|the electron-ion interaction (with the α-th ion hav-
ing the charge Zα), and ˆ
Vii =PN
αPN
β=αZαZβ/(2|Rα
Rβ|)the ion-ion interaction. The Born-Oppenheimer ap-
proximation assumes that since the mass of the ions far
exceeds that of the electrons, the electrons can be as-
sumed to reach thermal equilibrium with each other on a
comparatively small time scale. Thus, the kinetic energy
of the ions and the ion-ion interaction can be treated
classically and contribute simple additive terms to the
energy. The resulting equations thus only depend on the
ionic positions parametrically, and we will omit the R
dependence in the following.
The statistical density operator ˆ
Γin Eq. (1) is used to
compute statistical averages. It is defined as a sum over
all L-particle eigenstates ΨL,j of the Born-Oppenheimer
Hamiltonian as
ˆ
Γ = X
L,j
wL,j |ΨL,j ΨL,j |,(4)
where wL,j denote statistical weights that satisfy the nor-
malization condition PL,j wL,j = 1. In the grand canon-
ical ensemble, thermal equilibrium is defined as the sta-
tistical density operator that minimizes the grand poten-
tial.
The framework outlined here does not provide a prac-
tical approach for performing large-scale electronic struc-
ture calculations due to the electron-electron interaction
in Eq. (3) and the dimensionality of ΨL,j. Instead, DFT
based on the Hohenberg-Kohn theorems [1], their gener-
alization to finite temperatures [43], and the Kohn-Sham
scheme [2] is used to describe the grand potential as a
functional of the electronic density n. Specifically, the
grand potential can be expressed as
Ω[n] = E[n]kBτeSS[n]µL , (5)
with the electronic total energy
E[n] = TS[n] + EH[n] + EXC[n] + Eei[n] + Eii ,(6)
where the electronic density is restricted to densities cor-
responding to those stemming from many-body wave-
functions ΨL,j . Here, TS[n]denotes the kinetic energy
of the auxiliary system of non-interacting fermions, SS[n]
the non-interacting electronic entropy, EH[n]the Hartree
3
energy, which is the classical electrostatic interaction en-
ergy, Eei[n]the electron-ion interaction, which reduces
to the interaction of the electronic density with an ex-
ternal potential, while Eii refers to the constant shift
in energy due to ion-ion interaction. EXC[n]refers to
the exchange-correlation energy, which incorporates any
electronic contributions not captured by the aforemen-
tioned energy terms, and which needs to be approxi-
mated in practice. Consequently, the accuracy of DFT
calculations chiefly depends on appropriate approxima-
tions for this term. A plethora of practical approxi-
mations exist, such as the local density approximations
(LDA, e.g. PW91 [2,44]) or generalized gradient approx-
imations (GGA, e.g. PBE [4547]). Furthermore, the
XC functional should depend explicitly on temperature
[48]. However, this explicit temperature-dependence is
often unclear and usually omitted in standard calcula-
tions. Instead, the temperature dependence is crudely
included through the density in thermal equilibrium. Re-
cent advances [4961] have reignited the development
of temperature-dependent XC approximations. Stud-
ies of the electron liquid [62] and uniform electron gas
[6366] have aided in the construction of local [6770]
and generalized gradient approximations [71,72] to the
temperature-dependent XC contribution.
The subscript Sin Eq. (5) indicates that the kinetic
energy and the electronic entropy are usually calculated
via the Kohn-Sham ansatz [2]. This ansatz employs an
auxiliary system of non-interacting single-particle wave-
functions ϕj, which are constrained to reproduce the in-
teracting electronic density through
n(r,R) = X
j
fτe(ϵj)|ϕj|2,(7)
where fτedenotes the Fermi-Dirac distribution, while
ϕjand ϵjare the eigenfunctions and eigenvalues of the
Kohn-Sham equations
1
22+vS(r)ϕj(r) = ϵjϕj(r),(8)
where ϕj(r)are referred to as Kohn-Sham wave func-
tions or orbitals. The number of Kohn-Sham eigenstates
and eigenvalues to be considered is Lfor calculations at
τe= 0K, i.e., the sum in Eq. (7) runs from j= 1 to j=L.
However, in the finite temperature picture, one has to
account for thermal excitations of the electrons by cal-
culating ηadditional eigenstates, where ηis chosen such
that fτe(ϵj)is negligible for j > L +η. As one moves to
higher temperatures, ηhas to be increased, which causes
the unfavorable scaling of DFT with temperature.
The Kohn-Sham potential vSis determined self-
consistently to reproduce the interacting density via
Eq. (7). Note that several terms within the Kohn-Sham
framework implicitly depend on τe, such as the eigen-
states and eigenfunctions of Eq. (8), which are not ex-
plicitly denoted here for readibility.
With the finite-temperature DFT framework [51],
practical calculations become feasible. The electronic to-
tal free energy can be calculated using Eq. (5) as
A[n] = Ω[n] + µL =E[n]kBτeSS[n],(9)
which enables calculating the force acting on the α-th ion
Fα=∂A/∂Rα. These forces can be used in dynamical
studies, such as molecular dynamics simulations, which
yield thermodynamic observables. Here, the ionic tem-
perature τiis once again relevant and controlled via ther-
mostats, such as the the Nosé-Hoover thermostat [73,74].
These thermostats ensure that thermodynamic sampling
is performed in the correct thermodynamic ensemble,
such as a canonical ensemble of ions, where N,τiand
the volume of the simulation cell Vis held constant.
B. DFT machine learning model
While finite-temperature DFT enables practical calcu-
lations of many systems, it still has inherent computa-
tional scaling limitations. The standard DFT approach
scales with N3, making it very difficult to simulate sys-
tems involving more than a few thousand atoms. While
there are techniques that reduce this scaling behavior
to N, such as orbital-free DFT [75] or linear-scaling
DFT [76,77] through appropriate approximations, nei-
ther route enables a general replacement of KS-DFT sim-
ulations.
As discussed above, more Kohn-Sham wavefunctions
need to be included in Eq. (8) at higher temperatures,
leading to an additional computational overhead that
scales unfavorably with temperature [5]. These limita-
tions pose a challenge for generating first-principles data
for practical applications, particularly for increasingly
relevant investigations into matter under extreme condi-
tions. As a result, ML is emerging as a promising route
for overcoming these limitations.
ML is based on algorithms that can improve their per-
formance through observed data, i.e., they can learn [78].
By training models on a representative set of data, it is
possible to make predictions in a fraction of the time it
would take to perform original data collection with con-
ventional algorithms. In the context of DFT, ML models
can be trained on a number of potentially costly simu-
lations and thereafter replace the need for DFT simula-
tions, leading to drastically reduced simulation times.
There are various ML approaches for DFT simulations
that differ widely in their goals and application pur-
poses. In Ref. [26], we have identified common avenues
of research, including ML models that directly predict
quantities of interest for a subset of components (e.g.,
Ref. [28,79,80]) and models that predict total (free) en-
ergies and atomic forces, known as machine-learned in-
teratomic potentials (ML-IAPs, e.g., Ref. [27,81,82]).
The latter can easily be integrated into MD frameworks,
replacing DFT simulations as the primary engine for
4
extended dynamical simulations. IAPs built on (semi-
)empirical approximations have been used in this capac-
ity before the advent of ML. However, ML-IAPs generally
provide an even better reproduction of the electronic to-
tal free energy surface of a system of interest, allowing for
highly accurate thermodynamic sampling of observables
[83].
Current ML approaches for DFT calculations are lim-
ited in the quantities they predict and do not provide the
full electronic structure of simulated systems. We have
recently developed an ML-DFT framework [37,84] that
overcomes this issue. It provides the entire electronic
structure of systems of arbitrary size by predicting the
LDOS. The LDOS is defined as
d(ϵ, r) = X
j
|ϕj(r)|2δ(ϵϵj)(10)
and, from a computational point of view, is a vector in
the energy domain ϵfor each point in real-space r. Corre-
sponding to the Kohn-Sham system as discussed above,
the upper boundary of the summation in Eq. (10) de-
pends on the number of Kohn-Sham eigenstates sampled,
i.e., L+η. In order for models to be transferable in the τe
domain, we choose ηsuch that unoccupied Kohn-Sham
orbitals are included in the LDOS as well; thus, the same
LDOS can be used at higher electronic temperatures to
accurately compute energies.
Through the LDOS, the electronic density and the den-
sity of states Dare obtained as
n(r) = Zdϵfτe(ϵ)d(ϵ, r),(11)
D(ϵ) = X
j
δ(ϵϵj) = Zdrd(ϵ, r),(12)
which are both of direct interest for electronic structure
theory. Moreover, the electronic total energy in Eq. (6)
and the electronic total free energy in Eq. (9) can be
expressed solely in terms of das
E[d] =Eb[D[d]] + EXC[n[d]] EH[n[d]] ,(13)
ZdrvXC(r)n[d](r),
A[d] =E[d]kBτeSS[D[d]] ,(14)
where Eb=Rdϵfτe(ϵ)ϵD(ϵ)denotes the band energy and
vXC the exchange-correlation potential. For a complete
derivation of this framework refer to Ref. [37]. Conse-
quently, if the LDOS of a system can be predicted ac-
curately, it becomes possible to determine both the elec-
tronic structure and the energetics of the system, and au-
tomatic differentation can be used to obtain the atomic
forces.
The LDOS can be predicted by a model M[λ]in the
form of
˜
d(ϵ, r) = M(B(J, r)[R])[λ],(15)
where λrepresents hyper-parameters that describe both
the model (i.e., type of ML algorithm, characteristic fea-
tures) and the fitting techniques (i.e, training parameters
of the model, data employed), and for which we have de-
veloped techniques for rapid optimization [84]. In our
ML models, we use neural networks (NN) for the actual
ML task. NNs can learn complicated relationships be-
tween sets of data through nested linear transformations
and non-linear activation functions, based on individual
units called artificial neurons or perceptrons [85]. The
output of the l+ 1-th layer of an NN is calculated from
the outputs of the l-th layer as
yl+1 =σ(Wlyl+bl),(16)
where Wland blare the weights and biases of the l-th
layer, which are the tuneable parameters, and σis a non-
linear activation function. The process of determining
the optimal Wand bis known as training [86].
The first layer of the NN receives input data, which
in the case of our ML models are descriptors with di-
mensionality Jmax with J= 1, ..., Jmax, denoted as Bin
Eq. (15). These descriptors capture information about
the ionic structure locally around each point in space r.
The locality of this mapping is essential for scalability;
since Mlearns to predict the electronic structure at each
point rpindependently of distant points rq, the model
can be applied to diverse or large-scale systems as long
as the observed descriptors Bare close to those in the
training set.
We employ a grid-point generalization of bispectrum
descriptors [87] to encode the local ionic structure, as
described in Refs. [37,84]. The bispectrum descriptors
at point rrepresent the atomic density around rin terms
of a basis set expansion.
This framework is transferable to different system sizes
[38] and is applicable to a wide range of systems, as
long as the LDOS can be accurately calculated via DFT
for model training. We have developed an open-source
software package called Materials Learning Algorithms
(MALA) [88] which implements this LDOS based frame-
work and interfaces with popular open-source libraries
such as Quantum ESPRESSO, LAMMPS, and PyTorch.
Fig. 1, adapted from Ref. [38], illustrates the framework.
Additionally, the transferability of models extends to dif-
ferent temperature ranges, which will be discussed in the
following.
C. Computational details
We investigate the temperature transferability of ML-
DFT models using an aluminum data set covering ionic
temperatures from 100K to 933K. The 933K data cor-
responds to the solid phase and has been previously
used in Ref. [37]. For the other temperatures, we gener-
ated uncorrelated ionic configurations by sampling every
147th step from DFT-MD trajectories generated using
5
Figure 1. Overview of the MALA framework and the open
source libraries used for constructing the full ML pipeline.
The pictograms below depict the individual steps of the
framework, which involves calculating local descriptors at an
arbitrary grid point (green) from atoms (red) within a cer-
tain cutoff radius (orange), a neural network, and the elec-
tronic structure, specifically the electronic density for a cell
of aluminum atoms (red). The pictograms are adapted from
Ref. [38].
the VASP code [8991], employing a 2×2×2Monkhorst-
Pack [92]k-grid, a plane-wave basis set with a cutoff
energy of 500 eV and a PAW pseudopotential [93,94].
The calculations were carried out using the PBE [4547]
exchange-correlation functional.
The LDOS was then calculated for these ionic con-
figurations using the Quantum ESPRESSO code [9597]
with computational parameters consistent with those in
Ref. [37]. For a subset of the configurations, we also per-
formed calculations at electronic temperatures that differ
from ionic temperatures. This was done to investigate ef-
fects of the electronic temperature, as shown in Sec. III.
For actual model training, which is described in Sec. IV,
the same electronic and ionic temperatures were consis-
tently used. The full dataset can be found in Ref. [98].
MALA version 1.2.0 [88] was used to build and train
the ML models, and the model parameters were kept
consistent with the 933K model described in Ref. [98],
with the exception being the second set of models cal-
culated for Fig. 8, where the layer width was increased
from 4000 neurons to 6000 neurons. Models and training
scripts can be obtained via Ref. [99]. Accessing individ-
ual data points in a randomized order is essential when
training neural networks, and it is a standard practice in
this field. However, for the volumetric data used in our
training scheme, loading all the relevant data files into
memory at the same time, so as to perform randomiza-
tion in memory was not feasible due to their size. There-
fore, we used "lazy loading", i.e., loading volumetric data
corresponding to one ionic configuration at a time. This
initially prevented the training code from properly ran-
domizing the data during training. In order to avoid is-
sues that arise when training with non-randomized data,
we used separate scripts to read data and mix relevant
volumetric data files into new sets of randomized files to
ensure randomized access to data points for each training
experiment. The relevant scripts for this can be found in
Ref.[99].
III. LEARNING THE ELECTRONIC
TEMPERATURE
ML frameworks have been established for learning the
energy of materials. These models, known as ML-IAPs,
take a set of ionic positions, R, and predict the energy
and often forces which are the derivative of the energy
with respect to the positions. Usually, these models learn
the relative energy with respect to some reference config-
uration (e.g., isolated atoms) rather than total energies or
total free energies. The ionic temperature τi, which can
be regulated though thermostats in dynamic simulations,
determines the distributions of ionic velocities and posi-
tions. Even though ML-IAPs are not directly dependent
on the ionic temperature, they can learn to correctly pre-
dict dynamics for a range of temperatures once trained
on ionic configurations representative of those tempera-
tures.
Revisiting Eq. (5) reveals that the Kohn-Sham elec-
tronic total free energy depends on the electronic temper-
ature τe, both directly through the electronic kinetic en-
ergy and entropy terms as well as implicitly through the
electronic density, which is calculated via Eq. (7) with the
Fermi-Dirac distribution. Thus, for a given set of ionic
positions, the Kohn-Sham electronic total free energy can
have a whole range of values depending on the electronic
temperature. In contrast, conventional ML-IAPs assume
that the energy depends only on ionic positions, and thus
it is impossible for them to capture the effects of different
electronic temperatures.
The LDOS models discussed in Sec. II B address this
shortcoming of regular ML models for finite-temperature
DFT by learning the LDOS instead of directly learning
the energy. Specifically, the electronic total free energy
is obtained from the LDOS using Eq. (14), where the
direct dependence of the electronic kinetic energy, en-
tropy, and density on τeis taken into account. Note that
the LDOS d(ϵ, r)also depends on τeimplicitly through
the Kohn-Sham eigenvalues and eigenfunctions, which in
turn depend on the self-consistently determined Kohn-
Sham potential. In principle, τecould be added to the
descriptors used to calculate d(ϵ, r)and a suitable set of
τe-dependent training data could be used to capture the
dependence of the LDOS on τe. Instead, we will investi-
gate the simple approximation in which the training data
for the LDOS ML model is generated with τe=τi, and
the ML model does not include any explicit dependence
on τe. We will find that this approximation captures the
great majority of the energetic effects of changing τe.
This brings up the question of the orders of magnitude
for energy effects arising from both τiand τe. Changes
in either temperature are expected to have noticeable
6
impact on the calculated total free energies, and it is
important to investigate their magnitudes and relative
behaviors, since the ML models discussed here aim to
recover these effects. Furthermore, it is necessary to in-
vestigate whether LDOS models can capture electronic
temperature effects from a theoretical point of view. To
this end, we have performed DFT calculations in which τi
or τewas kept constant while the other temperature was
varied. These calculations were conducted for 256 alu-
minum atoms at 100K, 500K, and 933K (melting point,
solid phase) for the ionic temperature experiments; for
the electronic temperature experiments, temperatures up
to 6000K were investigated. The results of this investi-
gation are shown in Fig. 2and Fig. 3.
Figure 2. Changes in DFT energy when keeping electronic
temperature constant while varying ionic temperature. For
each point depicted, the electronic total free energy of ionic
configurations sampled at the given ionic temperature has
been calculated and averaged.
Fig. 2shows the impact of the ionic temperature. Here
the electronic total free energy has been calculated for ten
ionic configurations sampled at τi= [100K,500K,933K]
with DFT and a constant τe= 100K. Afterwards, the
energies have been averaged per temperature. As ex-
pected, sampling ionic configurations at different ionic
temperatures results in a change in energy, even when
the electronic temperature is kept constant. Fig. 2gives
an insight into how large this change in energy is to be
expected for changes in the ionic temperature. Here,
we observe an energy change of slightly more than 100
meV/atom for the range of 100K to 933K. This result
agrees well with the classical Harmonic heat capacity,
which would give a 3
2kB(T1T0) = 108 meV/atom in-
crease in potential energy for a temperature change from
T0= 100Kto T1= 900K.
In contrast, Fig. 3shows the behavior of the electronic
total free energy and electronic total energy when elec-
tronic temperature is varied (both in black), while ionic
temperature is kept constant. As mentioned above, one
advantage of the models discussed here is their ability
to treat electronic and ionic temperature independently,
which is an inherent capability of DFT simulations not
easily reproduced in ML-DFT frameworks. This prop-
erty becomes relevant, e.g., when electrons are heated to
very high temperatures by a laser source and only sub-
sequently heat the cold ions. In the case of aluminium,
one can estimate the electronic temperature at which hot
electrons are able to melt solid aluminium at low temper-
atures at roughly 5800K (see App. A). We thus investi-
gate electronic temperatures up to 6000K in Fig. 3and
further show ML inference results for this temperature
range in Sec. IV.
The black curve in Fig. 3was calculated using ionic
configurations sampled at ionic temperature 100K and
DFT simulations with increasing electronic tempera-
ture. It can clearly be seen that overall, one observes a
quadratic dependence for both energies w.r.t. electronic
temperature. While electronic total energy increases, the
electronic total free energy decreases, due to an increase
of the electronic entropy term, which is included in the
electronic total free energy according to Eq. (9) and is
negative in sign. It should be noted that for the temper-
ature range of 100K to 933K, on which the models dis-
cussed in Sec. IV were originally trained, the associated
energy change is of the magnitude of a few meV/atom,
and thus smaller than for changes in ionic temperature
within the same range.
Additionally, Fig. 3shows the hypothetical behavior
of ML models for changes in electronic temperatures.
There, the blue curve corresponds to the (mimicked) be-
havior of a regular ML-IAP. Since ML-IAPs generally
only operate on ionic configuration data, any such model
has no concept of electronic temperature, and thus its en-
ergy predictions would be entirely unaffected by changes
in electronic temperature, yielding a constant prediction.
Naturally, for an actually trained ML-IAP, this line may
be shifted upwards or downwards, based on the train-
ing routines; the relative behavior would be the same,
however.
Conversely, the red curve in Fig. 3shows the effects of
using a τeindependent LDOS to predict electronic total
energies and total free energies over a range of electronic
temperatures. Here, we take the LDOS calculated at τe=
100K for each ionic configuration and predict electronic
total (free) energies by substituting τein Eq. (13) and
(14). Ideally, this represents the results that would be
obtained from a LDOS based ML model without any
explicit dependence on τein the LDOS prediction. It
can be seen that one almost perfectly recovers the relative
behavior of the electronic total (free) energies with this
approximation. There is a small energy offset of roughly 5
meV/atom separating the DFT and LDOS based energies
in terms of absolute value, but as has been discussed in
Ref. [37] and [38], this is expected when moving from a
wave function based estimation of the energy to an LDOS
based one. This offset is further unproblematic, since
physical properties are related to energy differences, not
absolute energy values and as such, it is important that
7
Figure 3. Changes in electronic total and total free energy as defined by Eqs. (6), (9), (13) and (14) when keeping ionic tem-
perature constant while varying electronic temperature. Ionic temperature was kept constant by sampling ionic configurations
at 100K and not changing the ionic structure thereafter. Data was collected either by averaging energies over 10 ionic config-
urations (circles) or selecting one ionic configuration (for the temperature range above 933K, shown as triangles). The dotted
lines are quadratic fits to the entire data set. In (b) and (d), a magnification of the results given in (a) and (c) are shown,
to give more detail for the training range of the ML models and energetic differences between LDOS and wave function based
energy equations. For the grey curves, regular DFT calculations with varied electronic temperatures were performed. The blue
curves show the expected behavior of a regular ML-IAP, which reproduces the same energies for all electronic temperatures,
since the same ionic configurations are used. The red curve show the best case scenario for an LDOS based model - here, the
LDOS sampled at 100K has been evaluated at higher electronic temperatures using Eq. (13) and (14).
8
energy predictions reproduce correct relative behaviors,
which the LDOS based models achieve.
The results shown in Fig. 2and 3lead to three impor-
tant observations. First, it is evident that both electronic
and ionic temperature influence the calculation of the
electronic total free energy, i.e., energies change consid-
erably with increasing temperatures. Secondly, while for
lower temperatures changes in ionic temperature lead to
larger changes in the total free energy, electronic temper-
ature effects become relevant at higher temperatures as
well. In general, electronic effects may become dominat-
ing at high temperatures for specific observables, such as
the thermal conductivity [100]. And finally, LDOS based
models, are in principle capable of recovering most ener-
getic effects related to the electronic temperature, which
is an important property when going to high temperature
regimes.
In order to comprehend why the LDOS accurately en-
capsulates these effects, it is crucial to recognize that the
LDOS and hence the DOS exhibit only minimal varia-
tion with electronic temperature, as depicted in Fig. 4.
This can be discerned from the definitions of the LDOS
and DOS in Eqs. (10) and (12). Both are dependent on
the electronic temperature through the temperature de-
pendence of the Kohn-Sham potential, which in turn de-
pends on the electronic temperature via the correspond-
ing Kohn-Sham orbitals and eigenvalues. Two primary
energy contributions influenced by the electronic tem-
perature are the kinetic energy, which can be accessed
through the band energy Eb, and the electronic entropy
SS, both expressed using the DOS as
Eb=Z fτe(ϵ)ϵD(ϵ),(17)
SS=Z fτe(ϵ) ln [fτe(ϵ)]
+ [1 fτe(ϵ)] ln [1 fτe(ϵ)]D(ϵ).(18)
Thus, provided that the (L)DOS has been sampled to
include enough (unoccupied) energy states at lower elec-
tronic temperatures, the LDOS can be used to accurately
evaluate the energy at higher electronic temperatures.
Therefore, the LDOS is a suitable target quantity
for ML models across temperature ranges and for high-
temperature regimes. The task at hand is to correctly
predict the LDOS for ionic configurations sampled at
varying τi. Any model that performs well at this task
will, by default, capture most of the relevant effects re-
lated to τe.
IV. TEMPERATURE TRANSFERABLE
MODELS
Even with LDOS-based models correctly treating ef-
fects related to electronic temperature, it remains to be
determined whether ML models can correctly learn the
electronic structure of ionic configurations across a tem-
perature range. In the case of the models employed here,
Figure 4. Density of states (DOS) for the same aluminum
configuration but varying electronic temperatures, including
absolute differences D(ϵ)shown in black and red.
the ionic structure is encoded in bispectrum descriptors,
and naturally, these descriptors change with the ionic
temperature. Furthermore, we need to assert that mod-
els can recover the LDOS accurately enough to repro-
duce energy effects related to the electronic temperature,
which they in theory should be able to, as shown in Fig. 3.
To investigate this, we construct models using different
aluminum training data sets and evaluate them across a
temperature range of 100K to 933K, the melting point
of aluminum, in increments of 100K. In all DFT calcu-
lations used to generate the training data, the ionic and
electronic temperatures were set to the same values. This
is why τ=τe=τiis used to represent both the ionic
and electronic temperatures in Fig. 5, Fig. 6, Fig. 7, and
Fig. 8. The first step is to establish a proper baseline for
this experiment. It is expected that models trained only
on singular temperatures would struggle to accurately
capture the electronic structure of both higher and lower
temperatures. We quantified this by training models on
data for either 100K, 500K, or 933K representing the be-
ginning, middle, and end of the temperature range.
The results are shown in Fig. 5, which displays the
electronic total free energy errors for models trained on
only one temperature. We trained five models per tem-
perature to assess the robustness of the models. Both the
average and standard deviation across the initializations
are given, along with the model that performs best. It
is evident that for all three temperature, the models be-
come increasingly inaccurate as one moves away from the
training temperature. Specifically, we seek electronic to-
tal free energy errors of below 10 meV/atom. In the best
case, i.e., the models shown in Fig. 5(b), this thresh-
old is barely met for temperatures directly adjacent to
the training temperature, and quickly exceeded as one
9
moves to lower or higher temperatures.
Figure 5. Performance of MALA models when trained on sin-
gle temperature data. Five models with different initializa-
tions were trained for each temperature, where (a) shows the
average and standard deviation, while (b) shows the perfor-
mance of the best model. Training was performed using data
from four ionic configurations, while validation was performed
with data from one ionic configuration per temperature.
As mentioned above, such a behavior is expected, since
NN based models usually perform bad at extrapolation
tasks, and training a model on merely one ionic temper-
ature inevitably constitutes an extrapolation in the ionic
temperature domain. In order to build transferable ML
models for the electronic structure one has to take multi-
ple temperatures into account, and the principal question
is how many and which temperatures need to be incor-
porated into such a model in order to produce accurate
results across the selected temperature range.
There are two conceivable ways to construct such a
model. One potential route is to take training data from
the entire temperature range into account. However,
since ML models based on NNs should perform well when
used in an interpolative fashion, a different strategy is to
investigate the number and proper selection of temper-
atures needed to train models that perform well across
the temperature range.
To this end, we have trained models using training sets
with different combinations of temperatures. In all cases,
the amount of training data per temperature has been
kept constant, and five models per approach have been
trained with different initializations, to quantify the ro-
bustness of the method. The results are shown in Fig. 6.
The computational cost for training models increases as
the number of data points and the temperature range
expands. We provide a brief discussion on this trade-off
between accuracy and cost in Appendix B.
The first type of model, shown in blue in Fig. 6, uses
training data from the beginning and end of the tempera-
ture range. Although technically the inference across the
temperature range is an interpolation task, the reported
accuracies are unsatisfactory in the middle of the temper-
ature range. As previously observed in Fig. 5, accuracies
are only sufficient for 200K around the training tempera-
ture, i.e., close to temperatures observed in training, and
thus the resulting models cannot make accurate predic-
Figure 6. Performance of MALA models when trained on mul-
tiple temperatures. Two types of models were explored: one
incorporating data from the beginning and end of the tem-
perature range, and one additionally incorporating data from
the middle of the temperature range, i.e., 500K. Each model
was trained five times with different initializations. Panel (a)
shows the average and standard deviation of the performances
across these models, while panel (b) shows the performance
of the best model. For each included temperature, data from
four ionic configurations was used for training, while valida-
tion was performed with data from one ionic configuration.
tions from around 300K to 600K.
A obvious solution to address this problem is to include
of training from the middle of the temperature range.
A first attempt at this is shown in Fig. 6in orange,
where training data at 500K was added to the model.
As expected, this substantially reduces the inference er-
rors across the middle of the temperature range. In fact,
the resulting model achieves competitive accuracy almost
throughout the entire temperature range.
Further analysis of errors reveals two main sources of
error. Firstly, there seems to be a noticeable decrease in
accuracy between 100K and 500K compared to temper-
atures above 500K. The reason for this lies in the fact
that while diversity within the data set increases as one
moves to higher temperatures, differences between tem-
peratures become more subtle, and conversely, differences
in ionic structure are more pronounced at lower temper-
atures. This can be further verified by looking at the
radial distribution function (RDF) [101103], which is
calculated by averaging the ion density contained in a
shell of radius [r, r +dr]for a cell of volume V(r)and
isotropic system density ρ=N/V , i.e.,
g(r) = 1
ρ N V (r)
N
X
i=1
N
X
j=1
j=i
δ(r |rirj|).(19)
Differences in the RDF can best be analyzed through
the cosine similarity, which is a standard metric in data
science analysis [104] and is defined as
sC(x,y) = x·y
x∥∥y(20)
for vectors xand y. By interpreting the RDFs of in-
dividual temperatures as vectors of dimensionality Nr,
10
where Nris the number of radii sampled for the calcula-
tion of the RDFs, one can calculate their similarities. A
similarity sC= 1 indicates that the two RDFs are iden-
tical, while sC= 0 implies fully orthogonal RDFs. We
have carried out such an analysis for the temperatures
included in the model shown in Fig. 6. The results are
presented in Fig. 7.
Figure 7. Cosine similarities between an arbitrary ionic con-
figuration at reference temperatures 100K, 500K and 933K
and a different configuration at all investigated ionic temper-
atures. The maximum cosine similarity per model is shown for
the first two presented multi-temperature models, providing a
metric of model performance. The pictrogram below summa-
rizes the main result found in this analysis, i.e., configuration
diversity across temperatures decreases with increasing tem-
perature, while configuration diversity within temperature in-
creases with increasing temperature.
It is evident that a drop in model accuracy corresponds
to areas for which none of the included configurations at
the training temperatures exhibit high cosine similari-
ties to the inference configurations. For instance, a clear
lack of cosine similarity can be observed in the region
between 200K and 600K when only considering 100K
and 933K reference data; this corresponds to the lack
of accuracy observed for the first model shown in Fig. 6.
Similarly, even with the inclusion of 500K training data,
such a gap persists for temperatures of around 200K and
298K, which corresponds to the sudden drop in accu-
racy observed for the second model in Fig. 6. Visualizing
the combined maximum cosine similarity provides a good
metric of expected model performance.
Therefore, to construct a final temperature transfer-
able model, we incorporate 298K data into the model
training process. The resulting model is shown in Fig. 8,
depicted in blue. We incorporate the same amount of
data for each training temperature, and train the model
with five different initializations to quantify the robust-
ness of the method.
The resulting models exhibit excellent performance
throughout the temperature range. For the best of the
five initializations, errors consistently remain below 10
meV/atom throughout the entire test set, and on aver-
age, this threshold is only slightly exceeded. While there
is a small linear increase in error when moving to higher
temperatures, in general however, the models are gener-
ally capable of delivering consistent accuracy across the
entire temperature range.
However, as mentioned above, a second issue remains
with the models both shown in Fig. 6and Fig. 8(blue).
When considering the entire ensemble of trained mod-
els, it is apparent that the models become less robust as
temperature increases. This can be attributed to the fact
that while pronounced differences between temperatures
play an important role for lower temperatures, a larger
diversity of data points per temperature is observed at
higher temperatures, necessitating consideration of more
diverse ionic and electronic environments.
One way to mitigate these issues is to include more
data at the problematic temperatures, such as 933K.
However, this approach may exceed the information cap-
turing capacity of the employed model for the given data.
To address this, we trained models with two additional
ionic configurations at 933K with the results also given
in Fig. 8, depicted there in red. Slightly larger NNs were
used to successfully train these models (see Sec. II C for
detailed information). The inclusion of additional data
at 933K leads to an increase in the systematic model
robustness across the temperature range, i.e., while the
average accuracy did not change, the standard deviation
decreased. There is a slight decrease in accuracy for lower
temperatures, yet errors are in the same range and the
10 meV/atom threshold is only slightly exceeded. Over-
all, models trained with additional data at 933K demon-
strate excellent accuracy and robustness across the entire
temperature range.
The models shown in Fig. 8show good transferabil-
ity across a range of ionic temperatures. As introduced
in Sec. III, LDOS based models should further be able
to accurately predict the electronic total free energy at
electronic temperatures different, esp. higher than they
were originally trained on, since the (L)DOS is roughly
constant with changing electronic temperature. To this
end, one can calculate the total (free) energies for the
same configurations as used in Fig. 3. In Fig. 9this has
been done for two MALA models discussed here, namely
the model trained solely on 100K data shown in Fig. 5
and the final multi-temperature model shown in Fig. 8.
In each case, the model initialization with the best per-
formance as shown in the respective figures have been
used.
In Fig. 9it can clearly be seen that overall, both MALA
models reproduce the quadratic dependence of the energy
with respect to the electronic temperature quite well. It
should be noted that this result represents an interpola-
tion from a pure ML perspective, since the LDOS was
11
Figure 8. Performance of MALA models trained on four temperatures, i.e., 100K, 298K, 500K, and 933K. The models depicted
in blue were trained with four ionic configuration and the same model architecture used for Fig. 5and Fig. 6. The models
depicted in red were trained with two additional ionic configurations for 933K and a slightly larger architecture (see Sec. II C
for more information). All models were trained five times with different initializations, and per temperature included, data
from four (resp. six in the 933K case for the models depicted in red) ionic configurations was used for training, while validation
was performed with data from one ionic configuration.
12
predicted from ionic configurations similar to those ob-
served in training, while from a physics perspective, these
results constitute an extrapolation. Due to the afore-
mentioned weak dependence of the (L)DOS on electronic
temperature, this becomes feasible. Results for two dif-
ferent models are shown to illustrate the subtle point that
the accuracy of this temperature extrapolation depends
on a model’s capability of reproducing the (L)DOS for an
ionic temperature of 100K. Since this ionic temperature
was fixed in the results shown in Fig. 9, it is evident from
the results of an idealized LDOS model shown in Fig. 3
that this must be the case. The model trained solely
on 100K data thus performs slightly better at this task,
almost exactly recovering the DFT total energy in the
low temperature case and only slightly deviating from
it in the electronic total free energy case. Conversely,
the multi-temperature model, which shows generally ex-
cellent performance across the ionic temperature range,
performs slightly worse in these conditions, especially in
the total free energy prediction shown in Fig. 9. The dif-
ference in accuracy between electronic total free and elec-
tronic total energy may be explained by the fact that for
the former, the electronic entropy is factored in; for this,
a DOS integration has to be performed with a numeri-
cally determined Fermi energy as explained in Ref. [37],
which can lead to slight inaccuracies.
Generally however, both models recover the rela-
tive energy behavior with increasing electronic temper-
ature quite well. Especially with respect to the multi-
temperature model this is an important result, since it
shows that a singular ML model can be used to predict
the electronic structure of a material at varying ionic
and electronic temperatures. Both temperatures may
be varied independently of one another, within reason-
able boundaries, yet the same model can be employed.
Therefore, MALA models can replace DFT fully in this
regard, since both parameters are independent inputs for
DFT simulations as well.
V. DISCUSSION AND OUTLOOK
In summary, our work has demonstrated the potential
of ML models in replacing DFT calculations for electronic
structure predictions across a range of temperatures. By
targeting the LDOS, we have shown that ML models can
accurately reproduce the electronic structure of ionic con-
figurations at temperatures unseen during the training
process, and are further capable of extrapolation in the
electronic temperature domain. This is due to the fact
that the LDOS by default recovers effects at higher elec-
tronic temperatures than it was originally calculated at.
Thus, any model that is capable of predicting the LDOS
at arbitrary ionic configurations will be less susceptible
to errors due to changes in electronic temperature. This
only holds true if the model actually reproduces an ac-
curate LDOS for an arbitrary ionic configuration and if
the LDOS has been sampled for large enough energies in
the energy domain. The resulting models are capable of
reproducing DFT fully within the temperature ranges for
which they apply, and one may use them to model any
combination of ionic and electronic temperatures.
We have also investigated the number and selection of
temperatures needed to train transferable ML models.
We found that a careful selection of temperatures, aided
by analyzing the RDFs of the system, can lead to highly
accurate and robust models.
While our work focusses on the simple system of alu-
minum in the solid phase up to the melting point, our
findings provide a foundation for future investigations
into more complicated systems and conditions, such as
warm dense matter. In these cases, more elaborate de-
scriptors and powerful model representation may be nec-
essary, and we look forward to further exploration in
these areas. Overall, our work highlights the potential of
ML models in accurately predicting the electronic struc-
ture across a range of temperatures, with broad implica-
tions for materials science and beyond.
ACKNOWLEDGMENTS
The authors acknowledge helpful feedback from Chen
Liu, SambaNova Systems, with regards to data set prepa-
ration. The authors thank Kieron Burke for motivat-
ing the scientific question addressed in this paper. San-
dia National Laboratories is a multi-mission laboratory
managed and operated by National Technology and En-
gineering Solutions of Sandia, LLC, a wholly owned sub-
sidiary of Honeywell International, Inc., for DOE’s Na-
tional Nuclear Security Administration under contract
DE-NA0003525. This paper describes objective techni-
cal results and analysis. Any subjective views or opinions
that might be expressed in the paper do not necessar-
ily represent the views of the U.S. Department of En-
ergy or the United States Government. This work was
supported by the Center for Advanced Systems Under-
standing (CASUS) which is financed by Germany’s Fed-
eral Ministry of Education and Research (BMBF) and by
the Saxon state government out of the State budget ap-
proved by the Saxon State Parliament. The authors ac-
knowledge the Center for Information Services and High
Performance Computing [Zentrum für Informationsdien-
ste und Hochleistungsrechnen (ZIH)] at TU Dresden for
providing its facilities for high-performance calculations.
Appendix A: Estimation of relevant electronic
temperatures for aluminium
As discussed in Sec. III and IV, we have investigated
the behavior of the electronic total (free) energy when
keeping ionic temperature fixed at relatively small values,
while increasing electronic temperature. The overarching
motivation for this kind of experiment are conditions in
which electrons are heated to temperatures far exceeding
13
Figure 9. Comparison of MALA model inference with varying electronic temperature as compared to DFT, idealized LDOS
models and ML-IAPs. The same methodology and configurations as in Fig. 3have been used, i.e., the ionic temperature was
kept constant at 100K, ten configurations were sampled for the points denoted with circles and one for the points denoted with
a triangle, while dotted lines denote a quadratic fit. The model shown in red corresponds to the 100K model shown in Fig. 5,
while the model shown in green corresponds to the model trained on data from four temperatures (i.e., the model shown in red
in Fig. 8).
14
ionic temperatures, e.g., by means of laser heating. As
our models extend only to solid ionic configurations, we
have selected the upper limit for electronic temperature
as the temperature at which the electrons would be just
hot enough to melt the material. Naturally, this is only
a rough estimate based on classical formulas intended to
guide the temperature range investigated.
One can make such an estimation by first calculating
the electronic thermal energy as
Ee=1
2γτ 2
e.(A1)
This derivation is based on the free electron model [105],
in which the electronic volumetric heat capacity is given
as cv=γτe, with the material constant γ. Conversely,
the ionic thermal energy above the Debye temperature
(which is 433K in the low temperature limit for alu-
minium [106]) can be expressed as
Ei(τi) = 3
2NAkBτi,(A2)
according to the Dulong-Petit law [107], to which, e.g.,
the Debye model [108] reduces at high temperatures.
These two simple approximations enable estimating the
heat energy necessary to heat the electronic and ionic sys-
tems; in order quantify the heat energy needed to melt a
material, one further needs to factor in the heat of fusion
Efusion, i.e., the energy associated with the phase change
from solid to liquid phase.
With these considerations, the energy necessary to
melt aluminium purely through electronic heating can
be defined as
Emelt
e=Ee(τmelt
e),
=Ei(933 K) + Ee(933 K) + Efusion .(A3)
Thus, Eq. (A3) states that the electronic thermal energy
necessary to melt initially solid aluminium at τi=τe=
0 K is equal to the sum of both ionic and electronic ther-
mal energies at the melting point as well as the heat of
fusion itself. Conceptually speaking, the external heat-
ing of the electrons would have to provide sufficient heat
energy to heat both ionic and electronic systems to 933K
and provide the heat of fusion.
With the experimentally determined values γ= 1.35 ·
103J/(mol K) [109] and Efusion = 10431.1 J/mol [110],
one can determine Emelt
eas Emelt
e= 22654.8 J/mol. This
corresponds to a temperature τmelt
e= 5793.3 K.
Therefore, we show both DFT and ML results for elec-
tronic temperatures up to 6000 K and, in doing so, assert
that our models are capable of reproducing the correct
physics up to physically relevant temperatures.
Appendix B: Model training times and data volume
The models presented in Section IV can be evaluated
in terms of their accuracy-cost trade-off, which is typi-
cal for ML models. More accurate models, such as those
Table I. Training times and data volumes for models trained
in Section IV. All models were trained on a single GPU, so
the reported timings in hours (h) also correspond to the com-
putational cost in GPU-hours. Timings are averaged over all
five model initializations per experiment. Data volume in-
cludes the total size of both the training and validation data
sets used in the model training.
textbfModel Training Data
time [h] volume
[GB]
Model 1 3.24 107.50
100 K
(Fig. 5)
Model 2 2.19 107.50
500 K
(Fig. 5)
Model 3 2.05 107.50
933 K
(Fig. 5)
Model 4 7.51 215.00
100 K, 933 K
(Fig. 6)
Model 5 15.19 322.51
100 K, 500 K, 933 K
(Fig. 6)
Model 6 14.19 430.01
100 K, 298 K, 500 K, 933 K
(Fig. 8)
Model 7 24.21 473.01
100 K, 298 K, 500 K, 933 K
(more 933 K data, Fig. 8)
depicted in Fig. 8, generally require a larger computa-
tional effort for training. Hence, we provide the asso-
ciated training times and data volume in Table I. The
reported training times are averaged over all five model
initializations per model type, while the data volume rep-
resents the cumulative size of the training and validation
configuration data used during training.
The relationship between the computational cost for
training a model and the number of training data points
associated with a given temperature is nearly linear, with
a few notable exceptions. For instance, the initial four-
temperature model (Model 6) shown in Figure 8can be
trained in less time compared to the three-temperature
model (Model 5) illustrated in Figure 6. This discrepancy
arises from the use of early stopping during the training
process, where training is terminated once the accuracy
on the validation data set no longer improves. This in-
dicates that a model has effectively captured all the nec-
essary information from the training data set. A more
balanced data set in Model 6 allows it to reach this point
earlier than Model 5, which includes fewer temperature
15
data points. Moreover, the four-temperature model in-
corporating the additional 933 K data (Model 7) requires
relatively longer training times due to the utilization of
a larger neural network compared to the other models
listed in Table I. Nevertheless, the increase in compu-
tational effort needed to tune temperature transferable
models is moderate when considering the enhanced ac-
curacy demonstrated in Fig. 5, Fig. 6, and Fig. 8.
l.fiedler@hzdr.de
a.cangi@hzdr.de
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