]Corresponding Author: Johan H. Mentink<EMAIL_ADDRESS>
# Benchmarking energy consumption and latency for neuromorphic computing in
condensed matter and particle physics

Dominique J. Kösters  University of Twente, Faculty of Science and Technology,
P.O. Box 217 7500 AE, Enschede, The Netherlands  Radboud University, Institute
for Molecules and Materials, Heyendaalseweg 135, 6525 AJ, Nijmegen, The
Netherlands  IBM Research Europe - Zürich, 8803 Rüschlikon, Säumerstrasse 4,
Switzerland  Bryan A. Kortman  University of Twente, Faculty of Science and
Technology, P.O. Box 217 7500 AE, Enschede, The Netherlands  Nikhef, P.O.Box
41882 1098 XG Amsterdam, The Netherlands  Irem Boybat  IBM Research Europe -
Zürich, 8803 Rüschlikon, Säumerstrasse 4, Switzerland  Elena Ferro  IBM
Research Europe - Zürich, 8803 Rüschlikon, Säumerstrasse 4, Switzerland
Eidgenössische Technische Hochschule Zürich, Department of Information
Technology and Electrical Engineering, Gloriastrasse 35, 8092 Zürich,
Switzerland  Sagar Dolas  SURF Cooperation, Innovation Team, Moreelespark 48,
3511 EP, Utrecht, The Netherlands  Roberto Ruiz de Austri  University of
Valencia-CSIC, Instituto de Física Corpuscular, Parc Científic UV, c/
Catedrático José Beltrán 2, E-46980 Paterna, Spain  Johan Kwisthout  Radboud
University, Donders Institute for Brain, Cognition and Behaviour, P.O. Box
9104 6500 HE, Nijmegen, The Netherlands  Hans Hilgenkamp  University of
Twente, Faculty of Science and Technology, P.O. Box 217 7500 AE, Enschede, The
Netherlands  University of Twente, MESA+ Institute for Nanotechnology, P.O.
Box 217 7500 AE Enschede, The Netherlands  Theo Rasing  Radboud University,
Institute for Molecules and Materials, 6525 AJ Nijmegen, Heyendaalseweg 135,
The Netherlands  Heike Riel  IBM Research Europe - Zürich, 8803 Rüschlikon,
Säumerstrasse 4, Switzerland  Abu Sebastian  IBM Research Europe - Zürich,
8803 Rüschlikon, Säumerstrasse 4, Switzerland  Sascha Caron  Nikhef, P.O.Box
41882 1098 XG Amsterdam, The Netherlands  Radboud University, Institute for
Mathematics, Astrophysics and Particle Physics, Heyendaalseweg 135, 6525 AJ,
Nijmegen, The Netherlands  Johan H. Mentink  [  Radboud University, Institute
for Molecules and Materials, 6525 AJ Nijmegen, Heyendaalseweg 135, The
Netherlands

###### Abstract

The massive use of artificial neural networks (ANNs), increasingly popular in
many areas of scientific computing, rapidly increases the energy consumption
of modern high-performance computing systems. An appealing and possibly more
sustainable alternative is provided by novel neuromorphic paradigms, which
directly implement ANNs in hardware. However, little is known about the actual
benefits of running ANNs on neuromorphic hardware for use cases in scientific
computing. Here we present a methodology for measuring the energy cost and
compute time for inference tasks with ANNs on conventional hardware. In
addition, we have designed an architecture for these tasks and estimate the
same metrics based on a state-of-the-art analog in-memory computing (AIMC)
platform, one of the key paradigms in neuromorphic computing. Both
methodologies are compared for a use case in quantum many-body physics in two
dimensional condensed matter systems and for anomaly detection at 40 MHz rates
at the Large Hadron Collider in particle physics. We find that AIMC can
achieve up to one order of magnitude shorter computation times than
conventional hardware, at an energy cost that is up to three orders of
magnitude smaller. This suggests great potential for faster and more
sustainable scientific computing with neuromorphic hardware.

††preprint: AIP/123-QED

##  I Introduction

The energy demand of modern high-performance computing systems is currently
rapidly rising owing to the massive use of artificial neural networks
(ANNs)Mehonic and Kenyon (2022). Moreover, for several of the most challenging
compute tasks in scientific computing, applications of ANNs offer competitive
advantages over standard algorithms. This includes various examples in
condensed matter physicsButler  _et al._ (2018); Carleo  _et al._ (2019);
Bedolla, Padierna, and Castaneda-Priego (2020) and in particle physicsRadovic
_et al._ (2018); Feickert and Nachman (2021); Karagiorgi  _et al._ (2022),
which features data rates as high as 1 Pb/second in bursts separated by just
25 nanosecondsGovorkova  _et al._ (2022). Such workloads are very challenging
for conventional hardware since the corresponding energy consumption is simply
too high, even for inference tasks with pre-trained networks.

Neuromorphic hardware offers great potential as accelerator for such highly
demanding compute tasks. For example, spiking neural networks are considered
advantageous for optimization problemsDavies  _et al._ (2021); Schuman  _et
al._ (2022); Aimone, Parekh, and Severa (2017); Fonseca Guerra and Furber
(2017) and have recently also been applied to entangled quantum statesCzischek
_et al._ (2022). Another efficient realization of ANNs involves physically
instantiating the synaptic weights in memory devices and exploiting the
physical attributes of these memory devices to implement the ANN in hardware
Sebastian  _et al._ (2020); Yao  _et al._ (2020); Fick  _et al._ (2022). This
approach, typically referred to as analog in-memory computing (AIMC), would
obviate the need to shuttle millions of synaptic weights between the memory
and processing units and could lead to significant gains in energy efficiency
and latency. However, rather little is known about the actual benefits of
running ANNs on AIMC for concrete physics use cases. Moreover, physics users
often do not even consider the energy cost of complete ANN workloads as a
relevant figure of merit.

In this Letter, we aim to assess the potential of AIMC accelerators for ANN-
based use cases in condensed matter and particle physics by benchmarking them
against conventional hardware implementations of the ANNs. To this end we
develop a generic methodology for measuring the energy cost and compute time
for inference tasks with ANNs on CPU and GPU hardware. In addition, we have
designed an architecture to estimate these metrics for inference tasks on a
Mixed-Precision Analog In-Memory Computing (MP-AIMC) platform. The scientific
use cases chosen feature computation of quantum many body states in two
dimensions Carleo and Troyer (2017); Fabiani and Mentink (2019) as well as the
40 MHz challenge for anomaly detection at the Large Hadron Collider (LHC) in
particle physicsAarrestad  _et al._ (2022); Caron, Hendriks, and Verheyen
(2022). By comparing the measurements on conventional hardware against the
estimations for MP-AIMC, we find that the latter can reach up to an order of
magnitude shorter compute time and down to three orders of magnitude lower
energy cost.

##  II Scientific Use cases

###  II.1 Condensed Matter Physics

Understanding the effect of correlations on the properties of quantum many-
body systems is one of the most fascinating research fields in condensed
matter physics. Recently, a new method for the simulation of quantum many body
systems has been pioneered inspired by machine learning Carleo and Troyer
(2017). In this approach, the many-body wave function is approximated by an
ANN and the quantum states generated this way are termed neural-network
quantum states (NQS). Already the simplest network, the Restricted Boltzmann
Machine (RBM), was found to give competitive advantages over conventional
methods, in particular in two dimensions (2D) for which quantum correlations
are strongestCarleo and Troyer (2017); Fabiani and Mentink (2019); Nomura and
Imada (2021); Fabiani, Bouman, and Mentink (2021). Despite this potential, the
method suffers from long training times when applied to large systems.
Moreover, extracting observables from an already trained network takes up a
large part of the computational effort.

For this use case we consider NQS for the 2D antiferromagnetic Heisenberg
model on a square lattice. The Hamiltonian is defined as

$\hat{H}=J\sum_{\langle i,j\rangle}\hat{S}_{i}\cdot\hat{S}_{j},$ (1)

where $J$ is the exchange constant, $\hat{S}_{i}$ the quantum spin operators
for $S=1/2$ and the sum runs over the nearest neighbours. In the context of
NQS, an inference task is defined by evaluation of the RBM wave function,
which for translation invariant systems can be written as

$\psi(s)=\prod_{i=1}^{\alpha N}2\cosh{\left(\left[Ws+b\right]_{i}\right)},$
(2)

where $\alpha$ is the ratio between the number of spins and hidden layer
nodes, $N$ the number of spins in the lattice, $W$ a matrix of dimensions
$\alpha N\times N$ with the weights, $b$ a vector of dimension $\alpha N$ with
the biases and $s$ represents the input spin configuration, which is encoded
in a binary tuple (column vector) $s=(s_{1},\ldots,s_{N})^{T}$, with
$s_{j}=\pm 1$. A schematic representation of the RBM is shown in Figure 1. For
large systems, $\psi(s)$ is evaluated only for a subset of the $2^{N}$
possible states by Monte Carlo sampling. For our assessment, it is sufficient
to consider a $4\times 4$ system for which the input data set is defined by
all possible states with zero magnetization.

In order to obtain ground state properties, the network is trained by
minimizing the energy
$E={\langle\psi|\hat{H}|\psi\rangle}/{\langle\psi|\psi\rangle}$ using the
stochastic reconfiguration methodSorella (1998), while a closely related
training procedure is obtained for quantum dynamics based on the time-
dependent variational principleIdo, Ohgoe, and Imada (2015); Carleo and Troyer
(2017). In either case, the training itself relies on repeated evaluation of
$\psi(s)$ at fixed RBM parameters, before updates are computed. Moreover,
evaluation of observables exclusively relies on inference. Therefore, we focus
on the inference procedure alone and take already trained weights for the RBM
ground stateFabiani and Mentink (2019). In this case, both the weights and
biases can be chosen to be real valued.

Figure 1: Schematic representation of the RBM. the hidden layer size is
$\alpha N$, with $\alpha$ a positive integer (in this schematic representation
$N=4$ and $\alpha=2$). Each hidden neuron features a hyperbolic cosine
activation. Their outputs are multiplied to produce the network output.

###  II.2 Particle Physics

One of the most important challenges in High Energy Physics (HEP) today is to
find rare new physics signals among an abundance of Standard Model (SM)
proton-proton collisions. This use caseCaron, Hendriks, and Verheyen (2022)
uses Deep Learning (DL) techniques in order to find anomalous signals at the
LHC among many SM events, also known as anomaly detection. The event rate from
which events must be selected at the ATLAS detector is 40 MHz, brought down to
the final collection rate of 300Hz using a three staged trigger system. The
network in this use case is designed to run on the Level-1 trigger system
responsible for reducing the rate to less than 75kHz. Therefore, very low
latency is required to ensure the network can keep up with the proton-proton
collision rate.

The network in this use case is designed to detect events that are forbidden
in the SM. This is achieved by a One-Class Deep Support Vector Data
Description (Deep SVDD) approach, trained to map every input of the Deep SVDD
onto a predefined multidimensional point. The distance to this point is then
regarded as the final anomaly score. A schematic representation of the SVDD is
shown in Figure 2. The models are trained under the assumption that all SM
data falls inside the predetermined manifold. During the testing, the beyond
the Standard Model (BSM) data will fall outside this manifold. Several Deep
SVDD networks are combined into an ensemble to maximise the efficiency. The
networks are fully connected networks which output a constant vector for every
input. The loss is defined as

$S(x)=\left[O^{z}_{n}-\mathrm{model}(x)\right]^{2}.$ (3)

The model maps the input x to the same tensor shape as the vector $O$, with
$z$ and $n$ referring to the number of elements and the scalar value,
respectively. The activation function of the hidden layers is an exponential
linear unit (ELU). In this use case, we employ an ensemble of in total 63
networks. They contain the same network structure but different combination of
values [5,8,13,21,34,55,89,144,233] for $z$ and [0,1,2,3,4,10,25] of $n$.

The data used to test and train this use case consists of 5 data
recordsGovorkova  _et al._ (2022). One record containing a mixture of SM
processes and four separate records of BSM processes. The data is generated
using Pythia 8.240Bierlich  _et al._ (2022) using a collision energy of
$\sqrt{s}=13$ TeV. The BSM processes were $A\rightarrow{}4l$,
$LQ\rightarrow{}b\tau$, $H\rightarrow{}\tau\tau$ and the
$H^{\pm}\rightarrow{}\tau\nu$ process. The detector response is modelled with
DELPHES 3.3.2.Ovyn, Rouby, and Lemaitre (2009) Input variables available for
each event are the $p_{T}$, $\eta$ and $\phi$ values of the missing energy, 4
electrons, 4 muons and 10 jets.

Figure 2: A schematic representation of the Deep SVDD model. The input is the
event data of proton-proton collisions and the output is a z-dimensional
vector denoted model$(x)$. An anomaly score can be given to the event by
comparing the model output with a target ($O^{z}_{n}$). In this case the model
output lays far from the target and the event can therefore be considered an
anomaly.

##  III Methods

###  III.1  Energy-measurement methodology CPU and GPU

The energy measurements were performed on the ESC cluster within Innovation
Labs at SURF. The inference of the networks is performed on a dual socket
system with 2 Intel Xeon Gold CPU’s and a NVIDIA V100 GPU. The measurements
are performed using the Energy Aware Runtime (EAR) software packageCorbalan
(2017) and provides an energy management framework for experimental computing.
The inference jobs are monitored by measuring the DC node power using a
Baseboard Management Controller (BMC), which is a specialized service
processor that monitors the physical state of the hardware device using
sensors and communicating with the system administrator through an independent
connection. The BMC is part of the Intelligent Platform Management Interface
(IPMI), which is a standardized message-based hardware management interface.

EAR is currently implemented to give the metrics of an entire submitted job on
a specific node. The energy metric used employs the DC node energy
$E^{\text{node}}_{DC}$. The metrics presented below are the average energy per
inferred sample defined as

$E_{\text{sample}}=\frac{E^{\text{node}}_{DC}}{N},$ (4)

where $N$ is the number of inferred samples, and the throughput $T$ defined as

$T=\frac{N}{\Delta t},$ (5)

where $\Delta t$ is the elapsed time. An effective latency could be derived
from the throughput by taking the inverse, i.e. $L=T^{-1}$.

The samples where supplied in batches to the ANNs. The batch size was chosen
to minimize $E_{\text{sample}}$ by doing a sweep search. The most energy
efficient batch size was in these cases also the batch size with the highest
throughput. To ensure that initialization processes are insignificant in terms
of energy and time, the total computation time of inference is forced to be
larger than $99\%$ by increasing the number of inference steps. The scripts
developed for energy measurements with EAR, applicable for any neural network,
are available open access at githubKosters  _et al._ (2022).

###  III.2 Energy estimation for mixed-precision analog in-memory computing
hardware

In this section we present a dedicated MP-AIMC hardware design, which can
support both physics use cases described earlier. Although originating from
vastly different fields of physics, the operations necessary to compute the
wave function $\psi(S)$ and the anomaly score are similar and therefore the
two use cases are easily combined in one design.

AIMC hardware is based on memory crossbar arrays with stationary weights,
implementing matrix-vector multiplication (MVM) directly in hardware. The
proposed MP-AIMC architecture, shown in Figure 3, is composed of four analog
tiles and a Digital Processing Unit (DPU). The analog tile is assumed to be
similar to a phase-change memory-based AIMC tile designed and fabricated in 14
nm CMOS technology nodeKhaddam-Aljameh  _et al._ (2022); Narayanan  _et al._
(2021). The DPU components are also estimated based on circuit designs at the
same technology node.

The four tiles are identical and consist of a local controller, 512 Digital-
to-Analog Converters (DACs), $512\times 512$ memory crossbar with Phase Change
Memory (PCM) devices, 512 Analog-to-Digital Converters (ADCs) and a Local
Digital Processing Unit (LDPU).

The DAC converts an 8-bit signed integer to a voltage pulse with fixed height
(negative and positive numbers have a different fixed height) where the width
of the pulse corresponds to the value of the integer. The DACs are connected
to the wordlines of the memory crossbar. The crossbar evaluates the vector
matrix multiplication by accumulating current along the bitlines according to
Ohm’s and Kirchoff’s law. The ADCs are variants of the current-controlled
oscillator-based ADCs as described in Ref. Khaddam-Aljameh  _et al._ (2022).
The instantaneous current flowing through the bitlines is digitized and
accumulated in a digital counter (which is also part of the ADC) during the
application of the input pulse. The ADC outputs two 10-bit unsigned integers,
one integer corresponding to positive values and one corresponding to negative
values. Due to the compact design of the ADCs, it is possible to operate all
512 ADCs in parallel. The combined latency of the DACs, ADCs and the memory
crossbar is estimated to be 40 ns.

The LDPU implements the affine scaling, addition of biases and optionally the
ReLU activation function. Due to practical considerations, the Deep SVDD
implementation on the MP-AIMC architecture employs a ReLU instead of an ELU
activation function. Both activation functions result in a similar accuracy.
The LDPU and all the components used inside the DPU are synthesized to verify
that both timing and area constraints are met and custom designed for
relatively low-precision operations.

The additional DPU takes the network output and computes $\psi(s)$ for NQS and
for the SVDD it evaluates the euclidean distance between the target,
$O_{n}^{z}$, and the network output, which can be viewed as the anomaly score.
This evaluation can be done with the use of look-up tables for activation
functions and an adder tree for both use cases. For NQS, only one analog tile
and the DPU is utilized (shown in Figure 3 c), while for the SVDD application
all 4 tiles and DPU are utilized (Figure 3 b).

Figure 3: Proposed MP-AIMC architecture. (a) Block diagram of the proposed
AIMC-based architecture showing four identical tiles and the DPU. The sub-
figure include a zoom-in of one tile and a zoom-in of the DPU where the
corresponding data types are mentioned per block. Included in blocks A and B
are a hard-wired look-up table, floating point multiplier and floating point
adder. Sub-figure (b) shows the utilization of the proposed AIMC-based
architecture for the condensed matter physics use case, where only tile 4 and
the DPU are utilized. Sub-figure (c) shows the utilization for the particle
physics use case, where all 4 tiles and the DPU are utilized.

Due to the stationary nature of the weights in the proposed MP-AIMC
architecture, this design is well suited for a data-flow in pipeline fashion.
This means that the data-flow can be divided in several sequential pipeline
stages. The throughput is then limited by the slowest pipeline stage, which in
this design is estimated to be 50 ns, and is independent of network size as
long as the individual layers fit on the $512\times 512$ crossbar arrays. When
running the workloads of the two use cases with maximum load (entire 512 x 512
crossbar is utilized) the power consumption of the crossbar (the memory array
as well as the peripheral circuitry including the data converters) is
estimated to be 0.13W, where the peripheral circuitry contributes contributes
about 90% of the power consumption. The average power consumption of the LDPU
and DPU after synthesis are estimated at 0.33 W and 0.18 W respectively. In
this case the average power consumption for the computational and particle
physics use cases are 0.63 W and 2.03 W respectively. Note that, in the former
case just one tile and the DPU are used whereas in the later case, all four
tiles and the DPU are employed. Energy consumption of communication between
tiles and between the final tile and the DPU is included in the energy
estimation.

Figure 4: Results benchmarks on CPU (Intel Xeon Gold dual socket) and GPU
(NVIDIA V100) and estimations for the proposed AIMC architecture. (a) The
average energy per inference (or average energy per state) in Joule and (b)
the throughput inferences per second (or states per second) for the condensed
matter physics use case. (c) The average energy per inference (or average
energy per event) in Joule and (d) the throughput inferences per second (or
events per second) for the particle physics use case. The measurements on the
CPU and GPU are repeated 10 times and the standard deviation is included in
the plots. The batch size used for the condensed matter physics use case is
equal to all states with zero magnetization ($12870$) and the batch size used
for the particle physics use case is $10^{6}$. Horizontal axes indicate a
measure defining the network complexity/size; this is the hidden layer density
$\alpha$ for the condensed matter physics use case and the output dimension
$z$ for the particle physics use case.

##  IV Results

###  IV.1 Condensed Matter Physics

Figure 4a shows the average energy of inference and 4b the throughput, both as
function of the hidden layer density $\alpha$. For the NVIDIA GPU the energy
consumption (throughput) increases (decreases) with hidden layer size and by
increasing the data precision from floating point 32 to floating point 64, two
commonly used precision settings in the field. For both metrics the GPU
outperforms the CPU up to a factor 10. The dual socket Intel Xeon Gold CPU
features a non-monotonic dependence on network size and floating point
precision. This is attributed to irregular utilization of the cores (40 in
total). Forcing execution on a single core recovers the expected linear
scaling with hidden layer density as we have confirmed by independent
measurements (data not shown).

The energy and throughput estimates from the proposed MP-AIMC architecture are
also shown in Figure 4a and 4b (dashed lines). The estimated energy use of
this architecture is up to a factor $10^{3}$ lower than the CPU and GPU. We
observe that the MP-AIMC design yields a throughput comparable to CPU.
However, GPU outperforms the MP-AIMC throughput up to a factor 7 for
$\alpha\leq 4$. For large networks the GPU throughput reduces whereas MP-AIMC
remains flat by design, given that the hidden layer fits into a single MP-AIMC
tile. This suggests that for even larger networks MP-AIMC will have an
advantage also for the throughput. In addition, we note that unlike both the
CPU and the GPU, the proposed MP-AIMC architecture features no
parallelisation. The throughput of the MP-AIMC architecture naturally
increases if multiple networks are run in parallel using more tiles and DPU
resources at the expense of an increase in area of the chip.

###  IV.2 Particle Physics

Figure 4c shows the average energy consumption of a batch of inferred events
and Figure 4d the throughput as function of the individual network design of a
single SVDD in the SVDD ensemble, labeled with $z$, the dimension of the
manifold. Distinct from the condensed matter physics use case, GPU and CPU
score very similar results for both metrics, in particular for single
precision. Interestingly, in this case, the CPU outperforms the GPU for the
energy cost at the largest network sizes ($z>144$). Also for the throughput,
the performance difference between CPU and GPU is small. GPU is found to be
superior to the CPU for double precision, while for single precision again CPU
performs better, now for all network sizes. This nontrivial behavior is
attributed to the computational cost stemming from the distance calculation,
Eq. (3), which in turn determines the anomaly score, which we confirmed by
leaving out the distance calculation in the code.

Figure 4c and 4d also includes the energy estimation of the proposed MP-AIMC
architecture, which is observed to be up to a factor of $10^{3}$ more
efficient than CPU and GPU. By the pipelined dataflow, the MP-AIMC throughput
is the same as for condensed matter physics use case. However, on GPU and CPU
the inference with the more complex and deeper structure of the SVDD requires
more computational efforts than evaluation of the shallow RBM. As a result,
MP-AIMC can yield over a factor $20$ faster throughput than CPU and GPU at the
largest SVDD output dimensions.

##  V Conclusion

We have presented a methodology to measure energy cost and compute time on CPU
and GPU for inference tasks based on ANNs. By applying this methodology to NQS
for quantum many-body systems in condensed matter physics and SVDD networks
for anomaly detection in particle physics, we found that benefits of GPU as
compared to CPU for energy efficiency and throughput strongly depend on the
ANN architecture and on non-MVM operations. In particular, CPU can outperform
GPU even for the largest networks considered. Therefore, energy benchmarks are
always important, especially when working with unorthodox experimental ANN-
based algorithms.

Furthermore, we have proposed a dedicated MP-AIMC architecture capable of
implementing both physics use cases, based on which the energy consumption and
throughput can be estimated. By comparing the measured energy on CPU and GPU
with the energy estimations for MP-AIMC, it is found that the latter improves
the energy efficiency up to a factor $10^{3}$ for both the condensed matter
physics and particle physics use case. The throughput is flat as a function of
the network size in the proposed MP-AIMC architecture, as long as the network
fits in the $512\times 512$ crossbar array. For the relatively small networks
used for NQS tested for the condensed matter physics use case, this yields a
MP-AIMC throughput comparable with that of the CPU, whereas GPU throughput is
up to factor 7 higher. Importantly, for the larger SVDD networks in the
particle physics use case, the MP-AIMC throughput is always higher, over a
factor 20 for the largest network tested.

The benchmarks performed suggest great potential for neuromorphic accelerators
based on MP-AIMC. A key challenge associated with AIMC is computational
imprecisionLe Gallo  _et al._ (2018). Yet, solutions on deviceGiannopoulos
_et al._ (2018), unit cell and circuitLe Gallo  _et al._ (2022), as well as
algorithmic levelJoshi  _et al._ (2020) have been shown to be effective in
compensating for the low-precision analog computing. It is foreseen that
similar approaches can be effective for the networks presented in this work
and consider studying the impact of precision as a very interesting topic for
future work. Moreover, future work may focus on next-generation cross-bar
structures and parallel architectures which can further improve the
throughput. Combined with the fundamentally flat scaling of the compute time
with the network size, this suggest great potential for scientific workloads
with exceptionally high inference demands, potentially enabling so far
uncomputable tasks in scientific computing.

## Acknowledgements

We thank Giammarco Fabiani for stimulating discussions and support on the
condensed matter physics use case, as well as for providing the trained
networks, Julita Corbalan for support on operating the EAR software package,
We thank our colleagues at IBM Research, in particular Y. Kohda, S. Munetoh
and G. Karunaratne for discussions on the MP-AIMC power/performance analysis
and acknowledge the support from the IBM Research AI Hardware Center. This
research received funding from the Interdisciplinary Research Platform (IRP)
for Green Information Technology at the Faculty of Science of Radboud
University that aims to promote pioneering research ideas in emerging
interdisciplinary themes, the Nederlandse Organisatie voor Wetenschappelijk
Onderzoek (NWO) and is part of the Shell NWO/FOM inititative “Computational
sciences for energy research” of Shell and Chemical Sciences, Earth and Life
Sciences, Physical Sciences, Stichting voor Fundamenteel Onderzoek der Materie
(FOM) and Stichting voor de Technische Wetenschappen (STW). It also received
funding from the European Research Council under ERC Grant Agreement No.
856538 (3D-MAGiC), European Union’s Horizon Europe research and innovation
programme under Grant Agreement No. 101046878 (HYBRAIN), and we gratefully
acknowledge the computer resources at Artemisa, funded by the European Union
ERDF and Comunitat Valenciana as well as the technical support provided by the
Instituto de Fisica Corpuscular, IFIC (CSIC-UV). R. RdA also acknowledges the
Ministerio de Ciencia e Innovación (PID2020-113644GB-I00).

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