# Why Reinforcement Learning in Energy Systems Needs Explanations

Hallah Shahid Butt<EMAIL_ADDRESS>Karlsruhe Institute of
TechnologyKarlsruheGermany and  Benjamin Schäfer<EMAIL_ADDRESS>Karlsruhe Institute of TechnologyKarlsruheGermany

(2024; Date: December 2023)

###### Abstract.

With economic development, the complexity of infrastructure has increased
drastically. Similarly, with the shift from fossil fuels to renewable sources
of energy, there is a dire need for such systems that not only predict and
forecast with accuracy but also help in understanding the process of
predictions. Artificial intelligence and machine learning techniques have
helped in finding out well-performing solutions to different problems in the
energy sector. However, the usage of state-of-the-art techniques like
reinforcement learning is not surprisingly convincing. This paper discusses
the application of reinforcement techniques in energy systems and how
explanations of these models can be helpful.

explainable artificial intelligence (XAI), reinforcement learning, machine
learning, interpretations, energy systems

††conference: 46th International Conference on Software Engineering; April
2024; Lisbon, Portugal††journalyear: 2024††copyright:
rightsretained††conference: 2024 Workshop on Explainability Engineering; April
20, 2024; Lisbon, Portugal††booktitle: 2024 Workshop on Explainability
Engineering (ExEn ’24 ), April 20, 2024, Lisbon, Portugal††doi:
10.1145/3648505.3648510††isbn: 979-8-4007-0596-0/24/04

##  1\. Introduction

Infrastructure, such as hospitals, power plants, and universities, play an
important role in our daily lives. Economic and global development has
increased infrastructure complexity. With the help of advancements in
technology and automation of labour-intensive tasks, these complex systems
require electrical power to be functional and operative. Therefore, to meet
the increasing demand for electrical power, policymakers are encouraged to
think about other unconventional sources of energy that not only fulfill the
demand but also are more environmentally friendly. Conventional sources of
energy, such as oil, gas, and coal energy, are not good for the environment as
they emit immense amounts of CO2. Moreover, the reserves of conventional
resources are depleting with time, and this scarcity of resources can lead to
severe problems for future generations. Consequently, sustainable distributed
energy systems (DERs) are integrated into the energy systems
(shengren2023optimal, ). While these systems emit less CO2, their inclusion
has significantly increased the complexity of the power grids since these DERs
are distributed across the power system (potentially far away from consumers)
and are weather-dependent. This later property of the DERs adds uncertainty to
their nature; hence, it demands the need for proper management and planning of
the systems to meet the power demand and supply.

In 1956 when the term ”Artificial intelligence (AI)” was coined
(hamet2017artificial, ), it was presented as an idea of simulating learning
and solving every real-world problem on computers (aiml, ) with less or no
human intervention (hamet2017artificial, ). Since computational resources have
become readily available, it has become easy for computers to learn the
problem-solving process and solve complex tasks in diverse areas with the help
of AI and machine learning (ML) algorithms. These techniques have made a foray
into every scientific field whether it is healthcare, banking (aiml, ),
transportation (su14084832, ), and many other areas. For instance, in the
medical field, researchers have applied AI and ML techniques to diagnose
diseases like cancer (huang2020artificial, ), outbreak prediction
(nayak2018epidemic, ), and medical image processing with the help of computer
vision (olveres2021new, ; sathya2020application, ). Similarly, researchers
have found opportunities to apply AI and ML techniques to address issues of
”systems of systems”.

AI has gained significant importance in scientifically addressing
infrastructure issues. A lot of studies have been carried out on
infrastructure systems and researchers have done an extensive review on how AI
can assist us in dealing with problems arising in infrastructure systems like
energy, transportation, water systems, and telecommunications
(mcmillan2022review, ). In energy systems, AI and ML techniques can be used to
tackle numerous challenges of the ongoing phase-out of conventional resource
usage and the increasing usage of renewable energy. This includes predicting
the power generation and load, enhancing monitoring of the system, and finding
optimal solutions to reduce power losses (10.1145/3485128, ). The above-
mentioned problems can be optimally solved with the help of state-of-the-art
methods like reinforcement learning. RL methods have proven their credibility
in other fields but in energy systems, their application is not convincing
which we will discuss in a later section.

Despite the accuracy and precision of the models generated by advanced and
state-of-the-art algorithms, policymakers and companies are unable to trust
the outputs of many of these algorithms because of their black-box nature
(abduljabbar2019applications, ). Even the developers of these black-box models
are often unable to comprehend how the model processed the different input
variables to convert to a certain output function and how the inputs are
related to each other (rudin2019we, ). Interpretation of these models is
crucial in all domains, especially in medicine, energy systems, and other
complex infrastructures, as we need to trust the systems concerned with such a
vital part of our lives.

In this paper, we will discuss the applications of ML and RL in energy
systems, explain why the interpretation of ML models is important, which
techniques are used for explaining these models, and in how existing knowledge
should be advanced by new research.

##  2\. Energy Systems and Reinforcement Learning

As discussed earlier, energy systems have reached a level of complexity, and
with more sensors, an increasing amount of data becomes available so that we
can find solutions to these challenges to the energy system with the help of
ML-based algorithms. Already, with the digitization of the systems, the usage
of machine learning methods has increased drastically. Including DERs made it
crucial to monitor the energy systems so they can operate effectively. To
model these systems, it is necessary to have a good knowledge of the
parameters on which such systems depend, necessitating inference (su14084832,
).

The most commonly used learning types in energy systems are supervised, and
unsupervised learning. These techniques are used to forecast the power
generation, optimally manage the operational and maintenance cost (shin2021ai,
), and schedule the storage and generation operations. Authors in (shin2021ai,
) have reviewed the number of articles published to apply machine learning
methods in the energy domain, and they have found out that after 2012,
researchers are more interested in using ML techniques in energy systems, and
the number of publications made annually reached up to 2000 (PERERA2021110618,
). Authors in (mcmillan2022review, ) support this increased interest as they
reviewed the work of researchers who have used ML-based methods to effectively
forecast the generation, manage demand and supply, and monitor the performance
of the systems. Hence, both reviews stress the importance of how AI and ML
approaches have efficiently solved complex problems in energy systems.

Even though algorithms behave differently in different domains, authors in
(mcmillan2022review, ) have curated a list of well-performing algorithms that
include artificial neural networks (ANN), support vector machines (SVM),
decision trees and random forests, and K-mean clustering. Previously, in
energy systems, the community leaned mostly toward traditional approaches.
However, with the transition and advancements in the methods, researchers in
the energy sector are also increasingly adopting ANN-based techniques. Areas
like demand forecasting, weather predictions, and price forecasting are
popular and are solved by these algorithms (mcmillan2022review, ). For
example, Hafeez et al (HAFEEZ2020114915, ) have proposed a novel electrical
load forecasting hybrid model that outperforms previous ANN and LSTM
algorithms concerning the accuracy, runtime, and convergence rate.
Furthermore, researchers have also reviewed and studied how ANN can be applied
to optimize the performance of solar collectors (GHRITLAHRE201875, ). They
compared the performance with traditional techniques like linear regression
and non-linear models and found that ANNs outperformed the other methods.

In addition to the use of supervised and unsupervised learning methods, RL is
also attracting attention, and researchers are keen to implement RL methods in
different domains. Though we have seen RL-based methods applied in different
sectors, the researchers of the energy domain seem to be reluctant to adopt RL
approaches as can be seen from a low number of RL publications in energy
systems (PERERA2021110618, ). In RL, the learning agent learns a specific
policy based on the feedback given by the environment
(kaelbling1996reinforcement, ). The RL agent receives a state from state
space, and it takes actions either to explore the state space further or to
exploit learned rules. This new action leads the agent to a new state after
receiving a reward or penalty from the environment (li2017deep, ). This data-
driven method helps to learn and create new policies based on trial and error
to attain the maximum reward. RL approaches are more suitable to find optimal
solutions and react to dynamic environments than supervised or unsupervised
learning. Moreover, in real-time decision-making and optimization problems,
the RL-based method can be the best choice (rl, ). In recent times, RL methods
have proven their worth in different fields like gaming, robotics (rl, ;
8787888, ) traffic control systems, advertising, and the stock industry (rl,
).

Since the complexity of energy systems escalated due to the inclusion of DERs,
it is of key importance to optimally control these systems. Fortunately, the
rise of complexity is accompanied by an ever growing amount of operational and
real-time data. Processing and harnessing this bulk of data requires state-of-
the-art ML and specifically RL approaches to control and optimize the systems
(cao2020reinforcement, ). From dispatching generated power optimally over the
network to storing energy for later consumption, highly renewable energy
systems are more complicated in terms of their system management and therefore
offer numerous opportunities for RL applications. The uncertainties of the
system like dependency on weather, energy market variations, and improved
technologies of the energy systems, state-of-the-art techniques are required
to address the problems (PERERA2021110618, ). Regardless of the applications
and domain, reinforcement learning methodologies have attained success in the
scientific field as these methodologies handle uncertainties. Furthermore, for
some complex problems, model-free solutions are suitable rather than model-
based approaches. Thus, RL methods are a powerful and effective technique for
the modern challenges of energy systems (PERERA2021110618, ).

Concluding, more data becomes available for energy systems, while the
integration of DERs increases complexity and requires optimization and control
solutions capabale of processing real-time multi-dimensional data efficiently.
RL-based approaches are uniquely suitable to address this challenge. The
scalability of RL-based approaches has improved with the improvement of state-
of-the-art methods like Deep Q network and Proximal Policy Optimization (PPO)
since they deal with multi-dimensional data (8787888, ). Consequently, we are
also proposing to using Deep Q network and PPO to address our research
question, as discussed in section 4.

##  3\. Energy Systems and Interpretation of Models

Because of the advancements in the field of data science, many data-driven
complex tasks are solved with high-level accuracy with the help of ML
algorithms. However, most of these models and output functions are
uninterpretable, and even the designers of those algorithms are unable to
understand how these inputs are converted into the following output. In ML,
most of the models are black-box, i.e. these models are generated in an
untransparent way that does not reveal how results are obtained, e.g. because
a large amount of intrinsically simple units (neural layers) are combined into
one extensive complex system. This is in contrast to transparent (or white
box) models, such as linear regression, where we clearly see how the model
operates but which unfortunately are often performing worse than black boxes
(rudin2019we, ). AI and ML techniques face backlash from being unreliable and
vulnerable to bias. This behaviour of producing unfair and biased results is
analyzed by studying AI-designed tools where a famous criminal risk assessment
tool was biased in judging African-Americans as criminal, and a beauty pageant
winners tool judging black-skinned toned participants unfairly (8631448, ;
survey, ). Such unreliable results can impact the decision-making process and
it will lead to wrong decisions which can later on affect the process and
management. Hence, it is of key importance to have interpretability and
transparency of the models so that it can help in improving the credibility of
models and elevate acceptance of their decisions.

In this digital age, it becomes inevitable to use AI-based systems for the
decision-making processes. The intelligence of systems is helping decision-
makers and policymakers to think in different ways. It is also evident that
the fairness, transparency, and interpretability of the predictions or
suggestions provided by the AI systems are crucial because these systems have
a great impact on human lives (ferreira2021human, ) . According to
(ferreira2021human, ), the most important domains where explainability of
these systems are medical and law. However, this idea to make the models
explainable is not confined to only these areas. As we discussed earlier,
energy systems are also complex systems; hence, the application of AI and ML
techniques is beneficial in the decision-making process. However, these ML-
based systems are still not fully adopted in reality despite the accuracy and
advancement in ML technologies because the black-box nature of such algorithms
is a key hurdle (towardsxai, ). Furthermore, the unreliability and lack of
understanding of these systems is another barrier to adopting ML-based systems
specifically in the energy domain (WENNINGER2022118300, ). So, to implement
these ML-based systems, it is evident that there should be a system that is
interpreted by humans and adds to model validation and inspection. Thus,
increasing transparency of ML solutions will likely lead to the adaptation of
more such models in the energy field.

The notion of explanations is defined differently, depending on the context.
Sometimes it is presented as an explanation of an intrinsic process which
helps users to understand the process (ferreira2021human, ), and some
researchers categorize the explanations in various sub-categories to achieve
different goals like generating trust and transparency, following some
renowned regulations, ensuring fairness, generating accountable and reliable
models, reduction of misinterpretation in model performance, and validating
models (gerlings2020reviewing, ). In energy systems, we need explanations as
just discussed (e.g. trust, reliability, fairness). Current methods do not yet
provide full explanations but so far focus on deriving interpretations of the
trained models that identify and quantify the importance of different features
or input in generating any model as (ferreira2021human, ) has done work while
modeling energy systems. The necessary explanations also vary depending on the
target group, for instance, experts need explanations in terms of finding out
the potential significant features in predictions as researchers in
(sebastian, ) have also interpreted their models by computing the feature
importance, whereas, for a layperson, the underlying mechanics of the system
are not important. Instead, they are interested in actionable and causal
explanations of the predictions as mentioned in (WENNINGER2022118300, ).

Though we are witnessing the increasing trend of applying explainable
artificial intelligence (XAI) techniques in the energy field, it is still a
new and emerging aspect in the field of energy systems, and a lot of work is
yet to be done as it is shown in Table 1. It is important to note that these
publications have used a variety of explainability approaches to elaborate a
range of energy system problems using again a spectrum of different ML-based
methods, including for example SHAP on regression or attention for forecasts
(MACHLEV2022100169, )

Years | XAI and Energy systems
---|---
2020-2023 | 64
2010-2019 | 4
2000-2009 | 0
Table 1. This table shows the number of publications published within these
years that were related to keywords “XAI and energy systems“

More and more researchers are shifting their attention to explaining the ML
models used in energy systems. There are some areas where the concept of
interpreting the models is widely used, including power grid applications, the
energy sector, and energy management in buildings (MACHLEV2022100169, ).
Authors in (MACHLEV2022100169, ) have listed the commonly used XAI approaches
in the energy domain including Shapley additive explanations (SHAP) (9225146,
; MITRENTSIS2022118473, ; alova2021machine, ; 9448986, ), feature correlation
(konstantakopoulos2019design, ; das2019novel, ), and permutation feature
importance (en14217367, ; WENNINGER2022118300, ).

Similar to the explanations of other ML techniques, reinforcement learning is
also gaining importance in energy systems so that energy experts can build
trust in the systems. Many researchers have implemented deep reinforcement
learning (DRL) to solve complex problems. For example, Huang et al (8787888, )
have addressed the power energy control system with the help of DRL methods.
While in (9347147, ), researchers have worked on implementing DRL to address
power emergency control, and they have used deep explainer to obtain at least
initial interpretations of previously black box models. However, the
explainability of such systems is still mostly missing. Thus, a lot of work
has yet to be done since interpretations, let alone full explanations, of
reinforcement learning algorithms in energy are not very common, as suitable
methods are not readily available. To name two examples of RL applications in
energy systems that require explanations:

  * •

First, an RL controller of an energy storage system needs explanations because
this will help energy experts or laypeople understand and trust the control
laws which are learnt by the system.

  * •

Second, frequency and voltage control of transmission systems could be
realized by an RL agent, especially in emergency situations, where humans are
too slow to react and existing rule-based systems are too inflexible.

However, actions to stabilize a system have to convince operators so that they
take appropriate action and so that potential damage due to failure can be
properly investigated. Both of the above-mentioned RL applications make
explanations for these models inevitable (MACHLEV2022100169, ).

##  4\. Our planned contribution

We wish to focus on explainable reinforcement learning in small-scale energy
systems. Due to the stochastic nature of renewable energy sources, it is
natural to include energy storage in the energy system to ensure a smoother
and more constant supply of power, e.g. to a household. Thermal energy storage
(TES) systems are one candidate for balancing the supply and demand of the
buildings. Wang et al have already developed a deep Q-network RL approach to
optimally control this setup (WANG2023113696, ). Utilizing another technology,
Elseify et al (ELSEIFY2023108986, ) considered a scenario with multiple
batteries to store photovoltaic (PV) generated power and again obtained an
optimal solution using RL methods, specifically an equilibrium optimizer.
However, it is crucial to derive explanations for experts and laypeople alike.
Hence, we require suitable methods and proofs-of-concept for explainable
reinforcement learning (MACHLEV2022100169, ).

In our research, we want to identify the optimal power management of a
building, including balancing renewable generation and consumption via demand
response and storage. Currently, we are considering the use-case of buildings
with roof-mounted PV panels that have to balance PV generation, storage and
purchases from the power grid. We further assume that these buildings have
installed batteries to store excess solar power generated by the roof-mounted
PV so that this power can later be used when insufficient power is generated
by the PV panels, especially at night and on cloudy days. We search for
optimal usage strategies of batteries to minimize the total costs of
purchasing power from the connected utility grid. Rather than using supervised
and rule-based approaches, we aim to solve this optimization problem with the
help of state-of-the-art RL methods like Deep Q-network, and PPO
(schulman2017proximal, ), so that the agent learns the policies as per the
given states of the environment. Technically, we are modeling a single
building in terms of demand $D$, grid electricity prices $p$, stored energy in
the batteries, and the power generated by the PV panel $G_{\text{PV}}$. We
have used the notion of net demand which is the function of actual demand and
power generation of PV panel.

(1) $\text{Net demand}=f(D,G_{\text{PV}})=D-G_{\text{PV}}.$

This net demand indicates the extra power that the building needs from the
grid. The agent takes specific actions, for instance, to charge and discharge
the battery, and these actions are either rewarded or penalized by the
environment.

However, we also want to derive useful interpretations and explanations from
the model which is a black-box model. Initial steps include interpreting the
workings of the model, e.g. by obtaining a ranking of features that have an
impact on the predictions with the help of break-down profiles (bdplot, ).
Thus, we can formulate our research question as

> Can we use RL to solve an optimization problem in a small-scale energy
> system (balancing power of a single building) and how can we improve trust
> in such a black-box model by explaining its learned policies and actions?

In summary, our research contributes in the following way:

  * •

Analysing the applicability of RL in solving optimization problems in energy
systems, like minimizing the cost of purchasing electricity from the grid by
maximizing the consumption of PV panels’ generated power

  * •

Having a comparative analysis of different state-of-the-art RL methods

  * •

Having explanations of the black-box models to build trust in the outcomes of
these models and analysing how features impact the predictions

Later on, we will look for full explanations of why the model took a certain
action and why it did not take a different one (counterfactual explanations),
and ideally, generate these explanations dynamically to explain the rationale
of each step and every point in time. The goal is to increase trust and
transparency so that these systems can be implemented at large scales and
experts and laypeople feel confident while adapting such systems.

##  5\. Conclusion

Energy systems, as critical infrastructure and with many intrinsic
optimization problems, are excellent systems to apply explainable ML and RL
techniques. Unfortunately, XAI is still used too reluctantly for supervised
problems and even more rarely for RL. Hence, we propose to bridge this gap and
develop suitable explainable reinforcement learning methods for energy
systems.

##  6\. Acknowledgements

We gratefully acknowledge funding from the Helmholtz Association under grant
no. VH-NG-1727 and the Networking Fund through Helmholtz AI.

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# CoSense3D: an Agent-based Efficient Learning Framework for Collective
Perception

Yunshuang Yuan  Institute of Cartography and Geoinformatics
Leibniz University Hannover, Germany
0000-0001-5511-9082  Monika Sester  Institute of Cartography and
Geoinformatics
Leibniz University Hannover, Germany
0000-0002-6656-8809

###### Abstract

Collective Perception has attracted significant attention in recent years due
to its advantage for mitigating occlusion and expanding the field-of-view,
thereby enhancing reliability, efficiency, and, most crucially, decision-
making safety. However, developing collective perception models is highly
resource demanding due to extensive requirements of processing input data for
many agents, usually dozens of images and point clouds for a single frame.
This not only slows down the model development process for collective
perception but also impedes the utilization of larger models. In this paper,
we propose an agent-based training framework that handles the deep learning
modules and agent data separately to have a cleaner data flow structure. This
framework not only provides an API for flexibly prototyping the data
processing pipeline and defining the gradient calculation for each agent, but
also provides the user interface for interactive training, testing and data
visualization. Training experiment results of four collective object detection
models on the prominent collective perception benchmark OPV2V show that the
agent-based training can significantly reduce the GPU memory consumption and
training time while retaining inference performance. The framework and model
implementations are available at https://github.com/YuanYunshuang/CoSense3D

###### Index Terms:

Collective Perception, Object Detection, Efficient Training

Figure 1: CoSense3D: Agent-based training framework. Black arrows indicate the
instruction passing direction, green arrows show the data passing direction.

##  I Introduction

Reliability and safety are pivotal factors in autonomous driving. Relying
solely on the perception capabilities within the view range of a single
vehicle is problematic due to potential occlusions. Leveraging collective
perception, connected autonomous vehicles (CAV) can mutually share
environmental information, enabling observations from multiple vantage points.
This approach significantly mitigates occlusions, elevating perception
accuracy and enhancing overall safety.

In addition to safety benefits, autonomous driving offers improved efficiency
through optimized trajectory planning. The extended observation range enables
autonomous vehicles (AVs) to anticipate and plan more efficiently. However,
achieving high accuracy in long-range perception within a single vehicle is
challenging because of the limited perception ranges of sensors and increasing
sparsity at distant regions. Collective perception markedly enhances accuracy
in distant regions by merging denser observation points in these regions from
other AVs.

Past studies on collective perception have demonstrated notable success in
enhancing the overall accuracy of perception tasks, such as object detection
[1, 2, 3] and map semantic segmentation in Bird’s Eye View (BEV) [4, 5]. Yet,
developing deep neural networks for collective perception demands substantial
computational power. This requirement involves loading and processing a
significantly larger volume of data compared to conventional single-agent-
based perception frameworks. To mitigate this challenge and optimize training
efficiency while minimizing GPU memory usage, employing more efficient models
like EfficientNet [6] for image processing and leveraging Sparse 3D
convolutions [7, 8, 9] for point cloud data can be instrumental. Another
effective strategy involves utilizing Tensors with lower precision for
training and inference [10]. In addition to these methods, we investigate the
impact on perception performance of reducing gradient calculations for
specific portions of the input data during training.

To this end, we design a novel framework that optimizes the model development
efficiency from two perspectives. Firstly, existing frameworks for autonomous
driving perception such as mmdetection3d [11] and OpenCOOD [2] may encounter a
CPU bottleneck and slow down the training process during pre-processing, data
transformation and augmentation on a lower-end CPU, particularly evident in
collective perception scenarios. In such cases, a single frame of data may
contain dozens of images and point clouds, potentially reaching several
hundred if the sequential data in temporal dimension is considered. To
alleviate this bottleneck without necessitating hardware upgrades, our
approach involves migrating the transformation and augmentation processes to
the GPU. Secondly, computing gradients for all incoming data significantly
burdens GPU memory, notably impacting the development of collective perception
pipelines, especially with larger models like transformer-based architectures
and resource-intensive image processing backbones. To increase development
efficiency, we introduce CoSense3D, a specialized framework tailored for
collective perception. CoSense3D provides an API for easy definition of data
processing pipeline of CAVs, enables flexible control over gradient
computation for each CAV, optimizing GPU memory usage and reducing the
training time. Regarding each CAV as an agent, we call this agent-oriented
framework that defines the behaviour of each agent separately as agent-based
framework and the training processing under this framework as agent-based
learning.

To assess the effectiveness and performance of our framework in reducing
training time and GPU memory usage, we conducted a series of experiments using
state-of-the-art models—F-Cooper [12], FPVRCNN [1], EviBEV [4] and AttnFusion
[2]—for object detection. These experiments are carried out on the collective
perception benchmark OpenV2V [2].

In summary, the contributions of this paper are two fold. Firstly, we proposed
and implemented an interactive training, test and visualization framework
specially tailored for collective perception of CAVs. Based on this framework,
we are able to flexibly define the data processing pipeline and gradient
calculation for each CAV. Secondly, based on the above mentioned four models,
the results of two groups of comparative experiments with gradient computation
for all CAVs versus a limited number of CAVs show that, if the fusion method
such as attention fusion with efficient gradient back-propagation is used,
reducing the gradient computation for some of the CAVs significantly reduces
the GPU memory consumption and training time while maintaining model
performance.

Figure 2: Workflow of CoSense3D Central Controller.

##  II CoSense3D Framework

###  II-A Formalization of Collective Perception

Consider a set of CAVs denoted as $\textbf{C}={C_{1},C_{2},...,C_{N}}$ engaged
in a collective perception scenario. Each CAV is equipped with local sensors.
The CAVs $\textbf{C}_{\text{nbr}}=\\{C_{j}|d(C_{i},C_{j})<R\\}$ in the
communication range $R$ of CAV $C_{i}$ are able to share information to
$C_{i}$, where $d$ is the distance between two vehicles. Assuming $C_{i}$ is
the ego vehicle, at each frame, it first sends its pose $\textbf{P}_{i}$ as a
request to all neighboring vehicles $\textbf{C}_{\text{nbr}}$, once these
vehicles have received the request, they project their local data to the ego
vehicle’s coordinate system and process these data, which are then sent back
to the ego vehicle as a response (CPM). The ego vehicle then fuses all
response data from $\textbf{C}_{\text{nbr}}$ and generates the final object
detection result.

###  II-B Framework Structure

The overall framework is illustrated in figure 1. It contains four main
modules, namely Dataloader, Graphical user interface (GUI), Runner and Central
Controller. The Central Controller is the core module of the CoSense3D
framework which contains four sub-modules: CAV manager, Data manager, Task
manager and Forward runner. Black arrows indicate the instruction flow, green
arrows show the data flow. The framework can run either with or without
visualization in the GUI.

Dataloader: The framework standardizes the data loading API for collective
perception with a predefined dictionary format to store the meta information
in JSON files. With this API, a new dataset can be easily converted to the
CoSense3D format without rewriting the PyTorch Dataloader and coping the large
media files, such as point clouds and images, to a new data structure. Only
the meta information such as scenarios, frames, timestamps, parameters of
sensors and the annotations are parsed and saved to CoSense3D format in JSON
files. This standardized Dataloader is able to load images, point cloud data,
2D annotations for images, 3D local annotations for perception without CAV
cooperation and 3D global annotations for collective perception.

(a) GLViewer

(b) CoSense3D GUI OutputViwer.

Figure 3: CoSense3D GUI.

GUI: The graphical user interface visualized with the red box in figure 1 is
designed to visualize the training and test data and check the training and
test outcomes by one click. This is helpful for loading new datasets and
developing new models. Before training on a new dataset, it is necessary to
check if the data is converted and loaded correctly. During and after
training, visualizing the model output is also helpful to identify the
drawbacks and problems of the model and then refine or modify the model
accordingly.

As shown in figure 1, the GUI can send commands to the runner to start, stop
or step the runner process. After each runner step, it updates the
visualization modules, 3D GLViewer, ImgViewer, ImgAnno3DViewer and
OutputViewer. As shown in figure 3(a), GLViewer is a OpenGL-based visualizer
for 3D data, annotations (green boxes) and predictions (red boxes). ImgViewer
shows image data and the corresponding 2D bounding boxes. ImgAnno3DViewer is
used to visualize if the transformations and augmentations of images and 3D
annotations are correctly loaded and processed. Each row in ImgViewer and
ImgAnno3Dviewer shows the images of a single CAV. After training the model,
the OutputViewer can be used to visualize the test result. The OutputViewer
can contain multiple canvases which can be customized by the user. Figure 3(b)
is an example that shows the BEV segmentation (top) and object detection
(bottom) result.

Runner: In this framework, three types of Runners are available, namely,
TrainRunner, TestRunner and VisRunner. The user can launch these runners with
or without GUI. They are used for training, testing and input data
visualization, respectively. Runners manage the frame-wise data and orders
dispatching to Central Controller, which then process the orders with the
provided frame data accordingly.

Central Controller: As shown in figure 1, the Central Controller is the core
module of the CoSense3D framework, it communicates with the order-dispatcher
(Runner) and the CAVs through its CAV manager. The Data manager is responsible
for data gathering and scattering between the central controller and the CAVs.
Similarly, the Task manager gathers pseudo tasks generated by CAVs, batches
these tasks and dispatches them to the forward runner, which contains all
shared deep learning modules for all CAVs, for implementation. In this
framework, a standardized CAV prototyping API is provided to allow the user to
define the customized workflow for collective perception, including the data
augmentations, CAV coordinate transformations, CPM sharing strategies, the
forwarding order of the shared neuron network modules and gradient computation
strategies of these modules.

Based on the CAV prototype, the central controller will then implement a
standardized pipeline based on the tasks generated by the CAV prototypes as
shown in figure 2. Once the Central Controller receives the order and frame
data from the Runner (step 0), the CAV manager will update the CAVs according
to the meta information in the frame data and the provided prototype of CAV
(step 1). Then the Data manager distributes the input frame data to the
updated CAVs (step2). Upon receiving the input data, the CAVs then pre-process
the input data and generate tasks and send them back to the Central Controller
for processing (step3). To increase the efficiency of the forward process, the
Task manager will first summarize the tasks from all CAVs and batch them in
two forward steps, one requires gradients, and one without gradient
computation, for parallel processing in the Forward Runner (step 4 and 5).
After finishing these tasks, the generated results are then distributed back
to individual CAVs.

##  III Experiments

Figure 4: Collective perception pipeline with agent-based training. Blue and
gray blocks show the data flow with and without gradient calculation,
respectively. Sketched blocks with blues strips are the shared deep learning
models.

To evaluate the efficacy of agent-based training on reducing the GPU memory
usage and training time, we conduct two groups of experiments with the state-
of-the-art models on OPV2V benchmark. In group one, all models are trained
with all gradients enabled. Group two only enables a limited number
$\textbf{\small N}_{\textit{grad}}$ of CAVs for gradient calculation. As shown
in figure 4, we take three CAVs as an example and set $\textbf{\small
N}_{\textit{grad}}=1$. Only CAV1 has gradients enabled during training, the
features of CAV 2 and 3 are generated with the shared feature extractor with
gradient calculation disabled and sent to the CAV 1, where all features are
then fused and fed to the task head.

###  III-A State-of-the-art Networks

For the above mentioned two groups of experiments, we select the following
four representative models with different backbones and fusion methods for
cooperative object detection to explore the how these methods influence the
detection accuracy and the training efficiency with reduced gradient
calculation:

AttnFusion [2] network stands out as one of the top-performing models within
the OPV2V benchmark. It uses Voxelnet [13] as the backbone network to encode
point cloud data, and then convert the sparse Voxels into dense 3D grid to be
further encoded with several layers of dense 3D convolutions, which is highly
GPU memory demanding. The 3D voxel features are then concatenated to obtain
the final BEV features map with shape $(H,W,d)$ and shared to the ego vehicle,
where $H$ and $W$ represent the height and width of the feature map, and $d$
signifies its depth. Subsequently, this feature map is shared with the ego
vehicle. To fuse these feature maps, as implied by its name, AttnFusion
incorporates an attention module which consolidates the BEV feature maps from
all $M$ neighboring CAVs, generating a 4D Tensor with shape $(M,H,W,d)$. The
attention mechanism functions by attending to and blending features along the
first axis among the $M$ CAVs. At last, it uses an anchor-based detection head
to generate the bounding box classification and regression result.

F-Cooper is the earliest model applied to feature-level cooperative
perception. It first encode the voxels into 2D BEV feature maps, and then fuse
these maps by a Maxout operation, which takes the maximum feature values among
all feature maps coming from CAVs as the fused features. Afterwards, an
anchor-based detection head is used to predict the objects in the scenario.

FPVRCNN is a two-stage detector that requires the least communication
bandwidth with adequate performance within the OPV2V benchmark. As the name
indicates, this framework is built based on PVRCNN [14]. However, in order to
get better proposal detections for sharing, it uses a spatial-semantic feature
aggregation (SSFA) module proposed in CIASSD [15] to enhance the BEV feature
encoding for the first-stage detection. Based on these detections, FPVRCNN
samples some keypoints within the detected bounding boxes, and aggregate deep
features of different resolutions for each keypoints using the Voxel Set
Abstraction (VSA) Module of PVRCNN, the keypoints features along the proposal
bounding boxes are then shared and fused in the second stage to obtain refined
bounding boxes.

EviBEV is a fully sparse and highly efficient framework [4] for LiDAR-based
cooperative BEV semantic segmentation and 3D object detection. It encodes the
points with a Minkowski-convolution-based Unet. To enhance the connectivity
between the sparse voxels during the convolutions, it additionally augments
the original point clouds with free space points along the LiDAR casting ray
before feeding the points to the Unet. In addition, several layers of
coordinate-expandable sparse convolutions a utilized after the Unet to
compress the features along the vertical axis and dilate the sparse
coordinates so that they can have a better coverage over the centers of
visible object and then perform the final detection. Different from the
networks mentioned above, this work uses the anchor-free center-point [16]
head for object detection over the sparse BEV coordinates.

###  III-B Dataset

In this study, we utilize the simulated dataset OpenV2V [2] to assess the
efficiency of our framework. OpenV2V serves as a prominent benchmark
specifically curated for collective perception tasks. It comprises 44 scenes
and 6765 training frames, along with 16 scenes and 2170 frames designated for
testing purposes. These scenes encompass diverse driving scenarios, spanning
urban environments, rural areas, and highways across nine simulated cities.

Adhering to the official settings, our parameters are configured as follows:
the communication range $R$ is set to $70$ meters, the maximum number of
cooperative vehicles $\textbf{C}_{\text{nbr}}$ is set to 7, and the detection
ranges span $[-140.8,140.8]m$, $[-38.4,38.4]m$, and $[-3.0,1.0]$ along the x-,
y-, and z-axes, respectively.

###  III-C Experiment Settings

All models are trained on the a single Nvidia RTX3090Ti GPU and an Intel Core
i7-8700 CPU for 50 epochs. Batch size is set to 2. All models use Adam
optimizer with starting learning rate of $0.002$ and weight decay of $1e-4$.
The learning rates reduce at epoch $15$ and $30$ with the multiplication
factor of $0.1$. During training, the input data are augmented with rotation
along z-axis with a random angle in the range of $[-90,90]^{\circ}$, random
flipping along x- and y-axis, and scaling with a ratio in the range of
$[0.95,1.05]$. Additionally, the ground truth bounding boxes with less than
two points are removed during training process.

##  IV Result and Evaluation

Model | $\textbf{N}_{\textit{grad}}$ |<EMAIL_ADDRESS>|<EMAIL_ADDRESS>|<EMAIL_ADDRESS>|<EMAIL_ADDRESS>
---|---|---|---|---|---
F-Cooper | all | 82.2 | 89.9 | -13.5 | -6.4
2 | 68.7 | 83.5
FPVRCNN | all | 84.0 | 87.3 | +0.9 | +0.5
1 | 84.9 | 87.8
EviBEV | all | 84.1 | 91.1 | -4.6 | -2.0
1 | 79.5 | 89.1
AttnFusion | all | 87.6 | 92.3 | -0.5 | +0.3
1 | 87.1 | 92.6
TABLE I: Average precision of object detection on OPV2V benchmark with
different gradients configurations. AP$@iou$: average precision at IoU
threshold $iou$. AP-$@iou$: the AP drop of reduced gradient computation in
comparison to the full gradient computation.

###  IV-A Average Precision (AP) of Object detection

The AP of object detection of the selected models are listed in Table I. Two
commonly used Intersection over Union (IoU) thresholds are used for
calculating the APs, namely $0.5$ and $0.7$. They are notated with<EMAIL_ADDRESS>and<EMAIL_ADDRESS>respectively. AP- in the last two columns of table I means the
performance drop of reduced gradient computation in comparison to the full
gradient computation. $\textbf{\small N}_{\textit{grad}}$ is the number of
CAVs that require gradient calculation during training. $\textbf{\small
N}_{\textit{grad}}=\text{all}$ means all CAVs require gradients. Different
from the other three models, we set $\textbf{\small N}_{\textit{grad}}$ of
F-Cooper to $2$ in the second group of experiments. This adjustment was made
as $\textbf{\small N}_{\textit{grad}}=1$ in this case exhibited notably poor
performance. In general, F-Cooper performs worse than the other models and is
very sensitive to the number of gradient-enabled CAVs. This is caused by the
Maxout operation, which filters out some of the BEV features coming from a
gradient-enabled CAV and keeps the non-gradient features. In this way, the
downstream modules after Maxout are not able to back-propagate all gradients
back to the upstream modules that encode point clouds into BEV features.

In comparison, EviBEV has significantly smaller performance drops than
F-Cooper when using ego-agent-based training. However, its performance is
worse than FPVRCNN and AttnFusion. This is because it uses naive fusion to
merge the sparse BEV features from all CAVs, which is similar to Maxout that
requires no gradient calculation. Specifically, it involves mixing all BEV
feature points from all CAVs and randomly sampling one feature point at a
particular BEV position in cases where multiple CAVs have overlapping feature
points at that BEV position. This operation also weakens the back-propagation
for upstream modules because some of the gradient-enabled features are
dropped.

Differently, both FPVRCNN and AttnFusion use gradient-enabled modules for
fusing the features coming from all CAVs. FPVRCNN merges all keypoint features
with a RoI grid pooling layer while AttnFusion fuses BEV maps from different
CAVs by an attention module that attends the features along all CAVs. The
performance drops of these two models with only the gradient calculation for
ego agent are not noticeable. In some cases, the performance is even enhanced.
For example, the APs at the IoU threshold of $0.5$ for FPVRCNN and AttnFusion
with ego-agent-based training both surpasses that with all-agent-based
training by a small margin.

From the experiments result shown in table I, we conclude that agent-based
training may deteriorate the detection performance if the model uses a non-
learnable module that partially drops the upstream features from gradient-
enabled CAVs. However, if a more effective learnable fusion module is used to
merge all upstream features without dropping gradient-enabled features, we are
able to use the more efficient agent-based training framework to accelerate
the training and model development process without noticeable performance
drop.

###  IV-B Efficiency of Agent-based Learning

Figure 5: Correlation between object detection performance and GPU memory
usage (left) as well as training time (right) with different gradient
configurations.

To quantitatively evaluate the efficiency of our proposed agent-based training
framework, we illustrate the detection AP versus the memory usage and training
time as shown in figure 5. Yellow markers show the results of training with
all gradients enabled, and green markers the results of agent-based training.
The results of different models are illustrated with different shape of
markers. It is readily evident that agent-based training can significantly
reduce the GPU memory usage and the training time, especially for larger
models. For example, AttnFusion consumes about 16.7$G$ GPU memory with
gradient enabled for all CAVs, while agent-based training only uses 7.3$G$,
and thus reduces about 56% of GPU memory usage.

The reduced load of gradient calculation directly impacts the training time.
This can be identified by the similar patterns of the yellow and the green
markers between the left and the right plot in figure 5. The reduction ratio
in GPU memory usage closely mirrors that of the training time except F-Cooper.
The memory usage of F-Cooper is nearly halved, whereas the training time keeps
nearly unchanged. This result testifies again the conclusion drawn from the
failure case of F-Cooper discussed in section IV-A which is caused by Maxout
operation. The memory usage is reduced because the gradients for non-learnable
CAVs are not cached during the forward run. However, for all configurations of
$\textbf{\small N}_{\textit{grad}}$, F-Cooper only back-propagates $m\leq
h\cdot w\cdot d$ gradients to the upstream modules before Maxout, where $m$ is
the number features with gradients back-propagated and $h$, $w$, $d$ is the
height, width and depth of the BEV feature map of a single CAV, respectively.
Increasing $\textbf{\small N}_{\textit{grad}}$ for F-Cooper only makes $m$ to
be closer to the maximum value $h\cdot w\cdot d$. In contrast, gradients of
learnable fusion method such as attention fusion used by AttnFusion are fully
propagated back to the learnable CAVs. Mathematically, $m=\textbf{\small
N}_{\textit{grad}}\cdot h\cdot w\cdot d$ gradients are back-propagated, where
$m$ is linearly related to the number of gradient-enabled CAVs. Therefore,
decreasing $\textbf{\small N}_{\textit{grad}}$ of AttnFusion can linearly
reduce the gradients calculation hence reducing the training time. This effect
can hardly be observed by F-Cooper because it only reduces a very small
portion of gradients of a single BEV feature map. Similarly, EviBEV only
reduces the training time by less than one hour because of the random sampling
of the naive fusion module. In addition, the memory reduction of EviBEV by
agent-based learning is noticeable less than the other three models because it
is a fully-sparse model that is already very efficient with respect to the
increasing number of CAVs.

Therefore, agent-based training can significantly reduce the memory usage and
training time without noticeable performance drops if an appropriate fusion
method without dropping learnable features is used. In this case, the memory
usage and training time of dense models is roughly positively proportional to
the number of the learnable CAV $\textbf{\small N}_{\textit{grad}}$. However,
for sparse models, this relationship is non-linear and related to the actual
size of the sparse tensors.

##  V Conclusions

We proposed an efficient agent-based training framework that is specifically
designed for multi-agent collective perception. It aims to reduce the GPU
memory usage and the training time for the development of collective
perception models, which usually needs to process enormous image and point
cloud data for multiple CAVs. This framework explicitly separates the data
management of different CAVs and deep learning modules to provide a clean
workflow to flexibly define the data processing pipeline and adjust the
gradient configuration for each CAV. Two groups of comparative experiments
with full gradient calculation versus reduced number of CAVs for gradient
calculation have shown the training efficiency and performance with our
proposed framework. The result shows that reducing the number of CAVs for
gradient calculation can significantly save GPU memory and training time
without noticeable performance drop, if an appropriate fusion module without
dropping learnable features is used. On the OPV2V dataset, the training
resources needed has been reduced by more than a half for the best-performing
model AttnFusion. In addition, we also showed two failure cases, F-Cooper with
Maxout fusion and EviBEV with Naive mix-up fusion, and discussed the reason
behind. Similar operations can be avoided by model design if our proposed
agent-based training is used.

## Acknowledgment

This work is supported by the projects DFG RTC1931 SocialCars.

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# Supplemental material for: “A new room-temperature equation of state of Bi
up to 260 GPa”

Daniel J. Campbell  Daniel T. Sneed  E.F. O’Bannon III  Per Söderlind  Zsolt
Jenei  Lawrence Livermore National Laboratory, 7000 East Avenue, Livermore, CA
94550, USA

##  I Comparison of pressure scales

As mentioned in the main text, we used the results of Run B, where Bi, Cu, and
Ne were all present, to derive a Ne equation of state calibrated to Cu that
could be used for Run C, where only Bi and Ne were present. In doing this we
ensure that, despite using two different materials, our calibrations
ultimately derive from the same source, namely Ref. [1]. By reducing the
number of steps between the original calibration we reduce compounding error.
Furthermore, our three experiments were all performed with similar setups and
DACs. The resulting Ne EOS is in fact very similar to that of Ref. [2], which
is the highest pressure Ne calibration available. Table S-I compares our
calibration with that one as well as another commonly used set of parameters
from Fei et al. [3]. However, the maximum pressure in the latter measurement
was 115 GPa, which would require extending it to more than twice that value
for our data sets, and its values are further from the other two.

Despite the similarities in the values from our data and those of Ref. [2], by
the highest pressure of Run C they still differ by 10 GPa, or about 4% [Fig.
S1]. Since the same $V_{0}$ was used and the two sets of $K_{0}$ and
$K_{0}^{\prime}$ values are very similar, the difference between the two Vinet
equations grows essentially linearly with pressure, underscoring the
importance of consistency between calibrations for experiments far beyond 1
Mbar. We note that the calibration through our own data results in a lower
volume for a given pressure. This is further support for using our
calibration, as the lower volume indicates reduced uniaxial stress.

Table S-I: Three different equation of state parameters for Ne. The 0 GPa
atomic volume was fixed to 22.234 Å3 for the Run B data to match the value of
Ref. [2] (In Ref. [3] it is 22.241 Å3).
$K_{0}$ (GPa) | $K_{0}^{\prime}$ |  $P_{max}$ (GPa) | Ref.
---|---|---|---
1.046 | 8.38 | 184 | Run B
1.070 | 8.40 | 209 | [2]
1.16 | 8.23 | 115 | [3]
Figure S1: A comparison of two different pressure scales for the volume of Bi
in the Run C data. On the left axis, the calibration determined by comparing
to Cu in the Run B data (green squares) and that of Dewaele et al. [2] (blue
circles). The right y-axis shows the difference in pressure between the two
calibrations (red triangles)

##  II Relative Uniaxial Stress Component

Figure S2(a) reproduces Fig. 5 from the main text, with error bars removed for
clarity. The lower panel shows those same data relative to the total pressure.
It can be seen that only for a few data points do the values exceed $\pm$1% of
the total pressure as measured by either Cu or Ne.

Figure S2: (a) The absolute value of $\alpha{}t$ for the three experiments,
the same data presented in the main text. (b) The same values as a percentage
of the total pressure measured by the Cu or Ne lattice parameter. The region
within $\pm$1% is highlighted in gray.

##  III Pressure-Volume data for Bi-V

Table S-II: Lattice parameters (in Å) of Bi and (when present) Cu and Ne
during the three experiments, as well as calculated values for Bi. As in the
main text, pressures (in GPa) are calculated with the Cu EOS [1] for Runs A
and B, and our Ne EOS for Run C. The missing Ne value in one case for Run B is
because a Ne peak could not be found in that pattern. Data are presented only
above 10 GPa, when no peak from the Bi-III phase are observed.
Run | P | $a_{\textrm{Cu}}$ | $a_{\textrm{Ne}}$ | $a_{\textrm{Bi-V}}$
---|---|---|---|---
A | 10.07 | 3.5386 | – | 3.7732
A | 10.15 | 3.5382 | – | 3.7709
A | 10.15 | 3.5382 | – | 3.7714
A | 10.37 | 3.5368 | – | 3.7688
A | 10.69 | 3.5347 | – | 3.7642
A | 11.54 | 3.5294 | – | 3.7527
A | 13.61 | 3.5170 | – | 3.7278
A | 14.18 | 3.5137 | – | 3.7217
A | 14.73 | 3.5105 | – | 3.7177
A | 15.10 | 3.5084 | – | 3.7142
A | 15.38 | 3.5068 | – | 3.7112
A | 15.86 | 3.5042 | – | 3.7052
A | 15.88 | 3.5041 | – | 3.7037
A | 15.94 | 3.5037 | – | 3.7058
A | 15.95 | 3.5037 | – | 3.7052
A | 16.10 | 3.5028 | – | 3.7022
A | 16.37 | 3.5014 | – | 3.6999
A | 16.85 | 3.4987 | – | 3.6943
A | 17.64 | 3.4945 | – | 3.6839
A | 18.47 | 3.4901 | – | 3.6759
A | 19.29 | 3.4859 | – | 3.6668
A | 20.16 | 3.4814 | – | 3.6581
A | 21.21 | 3.4763 | – | 3.6484
A | 22.30 | 3.4709 | – | 3.6380
A | 23.58 | 3.4648 | – | 3.6276
A | 24.98 | 3.4583 | – | 3.6166
A | 26.30 | 3.4524 | – | 3.6062
A | 27.71 | 3.4462 | – | 3.5958
A | 29.54 | 3.4383 | – | 3.5827
A | 30.70 | 3.4335 | – | 3.5751
A | 31.89 | 3.4286 | – | 3.5678
A | 33.26 | 3.4231 | – | 3.5606
A | 34.46 | 3.4184 | – | 3.5550
A | 34.67 | 3.4176 | – | 3.5541
A | 35.17 | 3.4156 | – | 3.5514
A | 35.92 | 3.4128 | – | 3.5468
A | 36.99 | 3.4087 | – | 3.5403
A | 38.18 | 3.4043 | – | 3.5328
A | 39.47 | 3.3996 | – | 3.5251
A | 40.76 | 3.3950 | – | 3.5173
A | 41.88 | 3.3911 | – | 3.5110
A | 42.78 | 3.3879 | – | 3.5062
A | 44.35 | 3.3826 | – | 3.4978
A | 47.02 | 3.3737 | – | 3.4847
A | 49.27 | 3.3665 | – | 3.4737
A | 51.30 | 3.3601 | – | 3.4639
Run | P | $a_{\textrm{Cu}}$ | $a_{\textrm{Ne}}$ | $a_{\textrm{Bi-V}}$
---|---|---|---|---
A | 52.70 | 3.3558 | – | 3.4574
A | 54.38 | 3.3508 | – | 3.4503
A | 56.67 | 3.3440 | – | 3.4409
A | 58.72 | 3.3381 | – | 3.4319
A | 60.80 | 3.3323 | – | 3.4242
A | 62.72 | 3.3270 | – | 3.4165
A | 64.92 | 3.3211 | – | 3.4084
A | 66.86 | 3.3160 | – | 3.4014
A | 69.57 | 3.3090 | – | 3.3951
A | 69.62 | 3.3089 | – | 3.3963
A | 70.87 | 3.3057 | – | 3.3923
A | 72.98 | 3.3005 | – | 3.3909
A | 74.84 | 3.2959 | – | 3.3793
A | 77.45 | 3.2897 | – | 3.3712
A | 80.22 | 3.2832 | – | 3.3627
A | 80.57 | 3.2824 | – | 3.3615
A | 82.84 | 3.2773 | – | 3.3544
A | 85.63 | 3.2711 | – | 3.3466
A | 88.59 | 3.2646 | – | 3.3385
B | 16.80 | 3.4990 | 3.4333 | 3.6918
B | 17.48 | 3.4953 | 3.4240 | 3.6865
B | 14.34 | 3.5127 | 3.4154 | 3.6820
B | 17.99 | 3.4926 | 3.4144 | 3.6807
B | 18.76 | 3.4886 | 3.4079 | 3.6772
B | 19.50 | 3.4848 | – | 3.6768
B | 18.55 | 3.4897 | 3.4062 | 3.6752
B | 18.61 | 3.4894 | 3.4036 | 3.6751
B | 18.74 | 3.4887 | 3.4024 | 3.6743
B | 19.29 | 3.4859 | 3.3949 | 3.6675
B | 21.73 | 3.4737 | 3.3616 | 3.6452
B | 28.94 | 3.4408 | 3.2814 | 3.5880
B | 31.31 | 3.4310 | 3.2608 | 3.5728
B | 36.10 | 3.4121 | 3.2230 | 3.5433
B | 38.01 | 3.4049 | 3.2082 | 3.5273
B | 39.83 | 3.3983 | 3.1946 | 3.5162
B | 42.93 | 3.3874 | 3.1737 | 3.4993
B | 47.37 | 3.3726 | 3.1463 | 3.4769
B | 51.71 | 3.3588 | 3.1229 | 3.4568
B | 54.81 | 3.3495 | 3.1072 | 3.4432
B | 56.01 | 3.3459 | 3.0990 | 3.4371
B | 58.88 | 3.3377 | 3.0879 | 3.4269
B | 59.98 | 3.3346 | 3.0830 | 3.4232
B | 60.92 | 3.3320 | 3.0785 | 3.4192
B | 62.62 | 3.3273 | 3.0728 | 3.4134
B | 67.68 | 3.3138 | 3.0554 | 3.3973
B | 71.07 | 3.3052 | 3.0432 | 3.3849
B | 74.35 | 3.2971 | 3.0321 | 3.3744
B | 79.14 | 3.2858 | 3.0182 | 3.3603
B | 81.81 | 3.2796 | 3.0096 | 3.3516
B | 83.91 | 3.2749 | 3.0027 | 3.3450
B | 85.79 | 3.2707 | 2.9981 | 3.3405
B | 86.62 | 3.2689 | 2.9943 | 3.3370
B | 89.34 | 3.2630 | 2.9874 | 3.3302
B | 91.46 | 3.2585 | 2.9813 | 3.3244
B | 94.55 | 3.2521 | 2.9737 | 3.3164
B | 96.14 | 3.2489 | 2.9729 | 3.3135
B | 97.79 | 3.2455 | 2.9673 | 3.3084
B | 100.14 | 3.2409 | 2.9623 | 3.3037
B | 101.25 | 3.2387 | 2.9591 | 3.3006
B | 104.04 | 3.2333 | 2.9531 | 3.2946
B | 105.08 | 3.2313 | 2.9501 | 3.2915
Run | P | $a_{\textrm{Cu}}$ | $a_{\textrm{Ne}}$ | $a_{\textrm{Bi-V}}$
---|---|---|---|---
B | 107.16 | 3.2274 | 2.9450 | 3.2864
B | 109.10 | 3.2238 | 2.9405 | 3.2820
B | 110.05 | 3.2221 | 2.9375 | 3.2796
B | 111.87 | 3.2187 | 2.9337 | 3.2754
B | 114.62 | 3.2138 | 2.9277 | 3.2691
B | 118.05 | 3.2077 | 2.9194 | 3.2614
B | 120.58 | 3.2034 | 2.9138 | 3.2558
B | 123.37 | 3.1986 | 2.9077 | 3.2494
B | 127.59 | 3.1916 | 2.8994 | 3.2410
B | 131.25 | 3.1857 | 2.8918 | 3.2333
B | 134.88 | 3.1799 | 2.8843 | 3.2256
B | 139.01 | 3.1735 | 2.8762 | 3.2176
B | 140.70 | 3.1709 | 2.8733 | 3.2140
B | 142.13 | 3.1687 | 2.8695 | 3.2099
B | 144.91 | 3.1645 | 2.8634 | 3.2048
B | 148.26 | 3.1596 | 2.8575 | 3.1986
B | 151.00 | 3.1556 | 2.8508 | 3.1918
B | 151.79 | 3.1545 | 2.8483 | 3.1897
B | 154.59 | 3.1505 | 2.8442 | 3.1849
B | 158.24 | 3.1454 | 2.8377 | 3.1787
B | 161.35 | 3.1411 | 2.8328 | 3.1740
B | 164.35 | 3.1370 | 2.8272 | 3.1686
B | 167.86 | 3.1323 | 2.8214 | 3.1627
B | 171.20 | 3.1279 | 2.8167 | 3.1574
B | 174.22 | 3.1240 | 2.8121 | 3.1526
B | 176.92 | 3.1206 | 2.8076 | 3.1484
B | 178.40 | 3.1187 | 2.8046 | 3.1453
B | 179.91 | 3.1168 | 2.8032 | 3.1434
B | 182.22 | 3.1139 | 2.7993 | 3.1399
B | 184.47 | 3.1111 | 2.7955 | 3.1362
C | 93.85 | – | 2.9762 | 3.3080
C | 94.69 | – | 2.9739 | 3.3093
C | 99.18 | – | 2.9618 | 3.2972
C | 103.54 | – | 2.9505 | 3.2856
C | 111.21 | – | 2.9318 | 3.2664
C | 119.39 | – | 2.9132 | 3.2484
C | 125.29 | – | 2.9005 | 3.2378
C | 132.56 | – | 2.8857 | 3.2226
C | 138.05 | – | 2.8750 | 3.2120
C | 147.03 | – | 2.8585 | 3.1981
C | 149.00 | – | 2.8549 | 3.1940
C | 155.60 | – | 2.8435 | 3.1830
C | 164.50 | – | 2.8289 | 3.1665
C | 171.46 | – | 2.8180 | 3.1540
C | 179.40 | – | 2.8061 | 3.1435
C | 184.15 | – | 2.7992 | 3.1374
C | 191.41 | – | 2.7890 | 3.1276
C | 197.48 | – | 2.7807 | 3.1189
C | 200.78 | – | 2.7764 | 3.1139
C | 205.96 | – | 2.7697 | 3.1071
C | 208.98 | – | 2.7658 | 3.1032
C | 213.89 | – | 2.7597 | 3.0961
C | 218.83 | – | 2.7537 | 3.0903
C | 224.22 | – | 2.7472 | 3.0839
C | 229.06 | – | 2.7416 | 3.0779
C | 233.69 | – | 2.7363 | 3.0725
C | 237.62 | – | 2.7319 | 3.0677
C | 241.96 | – | 2.7271 | 3.0625
Run | P | $a_{\textrm{Cu}}$ | $a_{\textrm{Ne}}$ | $a_{\textrm{Bi-V}}$
---|---|---|---|---
C | 246.25 | – | 2.7225 | 3.0581
C | 249.09 | – | 2.7195 | 3.0543
C | 252.30 | – | 2.7161 | 3.0507
C | 256.12 | – | 2.7121 | 3.0471
C | 258.69 | – | 2.7095 | 3.0439
Th. | -1.08 | – | – | 2.90186
Th. | -0.29 | – | – | 3.99808
Th. | 0.57 | – | – | 3.97657
Th. | 1.50 | – | – | 3.95508
Th. | 2.51 | – | – | 3.93357
Th. | 3.60 | – | – | 3.91208
Th. | 4.79 | – | – | 3.89058
Th. | 6.07 | – | – | 3.8691
Th. | 7.45 | – | – | 3.8476
Th. | 8.95 | – | – | 3.82609
Th. | 10.56 | – | – | 3.80461
Th. | 12.31 | – | – | 3.78312
Th. | 14.19 | – | – | 3.76162
Th. | 16.23 | – | – | 3.7401
Th. | 18.42 | – | – | 3.71863
Th. | 20.79 | – | – | 3.69715
Th. | 23.35 | – | – | 3.67566
Th. | 26.10 | – | – | 3.65417
Th. | 29.05 | – | – | 3.63268
Th. | 32.25 | – | – | 3.61118
Th. | 35.69 | – | – | 3.58968
Th. | 39.39 | – | – | 3.56819
Th. | 43.38 | – | – | 3.54669
Th. | 47.67 | – | – | 3.52515
Th. | 52.29 | – | – | 3.50367
Th. | 57.27 | – | – | 3.4822
Th. | 62.64 | – | – | 3.46068
Th. | 68.41 | – | – | 3.43922
Th. | 74.63 | – | – | 3.41773
Th. | 81.33 | – | – | 3.39619
Th. | 88.54 | – | – | 3.37473
Th. | 96.29 | – | – | 3.35322
Th. | 104.63 | – | – | 3.33174
Th. | 113.61 | – | – | 3.31022
Th. | 123.27 | – | – | 3.28873
Th. | 133.65 | – | – | 3.26726
Th. | 144.83 | – | – | 3.24576
Th. | 156.86 | – | – | 3.22423
Th. | 169.81 | – | – | 3.20273
Th. | 183.73 | – | – | 3.18127
Th. | 198.72 | – | – | 3.15978
Th. | 214.86 | – | – | 3.13827
Th. | 232.26 | – | – | 3.1168
Th. | 250.97 | – | – | 3.09531
Th. | 271.09 | – | – | 3.0738
Th. | 292.73 | – | – | 3.05227
Th. | 316.50 | – | – | 3.03079
Th. | 342.01 | – | – | 3.0093
Th. | 368.83 | – | – | 2.9878
Th. | 397.68 | – | – | 2.96629
Th. | 428.75 | – | – | 2.94484
Th. | 462.36 | – | – | 2.92332

## References

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  * Dewaele  _et al._ [2008] A. Dewaele, F. Datchi, P. Loubeyre, and M. Mezouar, High pressure–high temperature equations of state of neon and diamond, Phys. Rev. B 77, 094106 (2008).
  * Fei  _et al._ [2007] Y. Fei, A. Ricolleau, M. Frank, K. Mibe, G. Shen, and V. Prakapenka, Toward an internally consistent pressure scale, Proc. Natl. Acad. Sci. U.S.A. 104, 9182 (2007).