# Parameter Estimation for the SEIR Model
Using Recurrent Nets

Chun Fan★▲, Yuxian Meng♣, Xiaofei Sun♣ Fei Wu◆, Tianwei Zhang♠, Jiwei Li◆♣
♣ Shannon.AI, ★Peking University, ▲Peng Cheng Laboratory
◆Zhejiang University, ♠Nanyang Technological University
{yuxian_meng, xiaofei_sun<EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The standard way to estimate the parameters $\Theta_{\text{SEIR}}$ (e.g., the
transmission rate $\beta$) of an SEIR model is to use grid search, where
simulations are performed on each set of parameters, and the parameter set
leading to the least $L_{2}$ distance between predicted number of infections
and observed infections is selected. This brute-force strategy is not only
time consuming, as simulations are slow when the population is large, but also
inaccurate, since it is impossible to enumerate all parameter combinations. To
address these issues, in this paper, we propose to transform the non-
differentiable problem of finding optimal $\Theta_{\text{SEIR}}$ to a
differentiable one, where we first train a recurrent net to fit a small number
of simulation data. Next, based on this recurrent net that is able to
generalize SEIR simulations, we are able to transform the objective to a
differentiable one with respect to $\Theta_{\text{SEIR}}$, and
straightforwardly obtain its optimal value. The proposed strategy is both time
efficient as it only relies on a small number of SEIR simulations, and
accurate as we are able to find the optimal $\Theta_{\text{SEIR}}$ based on
the differentiable objective. On two COVID-19 datasets, we observe that the
proposed strategy leads to significantly better parameter estimations with a
smaller number of simulations.

##  1 Introduction

The SEIR model [19, 2, 14, 18] is a widely-used epidemiological model to
predict the macroscopic behavior of disease spread through a population, e.g.,
the spread of COVID-19 among different populations at different locations over
time [9, 25, 12, 1, 5, 13]. A typical SEIR model models the spread dynamics of
disease using four states of populations: susceptible (S), exposed (E),
infectious (I) and recovered (R), where susceptible individuals can be
transformed into the exposed, and later to the infectious and finally to the
recovered. Each transition is associated with corresponding parameter(s),
forming the parameter set $\Theta_{\text{SEIR}}$ for the model: transmission
rate for $S\rightarrow E$, infection rate for $E\rightarrow I$ and recovery
rate for $I\rightarrow R$. $\Theta_{\text{SEIR}}$ captures the spreading
pattern of the disease, the learning of which is thus crucial for
understanding the disease spread, and developing social-distancing or
interventions policies.

Existing SEIR models relies on grid search to estimate $\Theta_{\text{SEIR}}$
via simulations. Model simulations are performed on different sets of
parameters, and the set that has the smallest $L_{2}$ distance between the
predicted number of infections and the observed number of infections is
selected as the best parameter combination. This is because of the non-
differentiable nature of the $L2$ objective with respect to
$\Theta_{\text{SEIR}}$: predicted number of infections are obtained from
discrete simulations. This learning process is (1) high time-intensive: the
simulation process can be slow when the population is large, which gives a
time complexity of $\mathcal{O}(MNT)$ where $M$ is the number of simulations
performed, $N$ is total number of individuals and $T$ is maximum time steps;
besides, the search space will grow exponentially with respect to the number
of parameters we need to estimate; and (2) inaccurate: because it is
impractical to enumerate all possible combinations of parameters in the
continuous space, the resulting parameter combination would be sub-optimal
given a limited amount of trials.

To address these issues, in this paper, we propose to transform the original
non-differentiable simulation problem into a differentiable one using neural
recurrent nets for parameter estimation in SEIR models. The basic idea is that
neural recurrent nets are differentiable with respect to the input learnable
parameters, so that they are able to automatically learn these parameters via
gradient descent and backpropagation [27]. Another advantage neural recurrent
nets offer is that they intrinsically model the temporal changes of
observations, and have the potentials to make accurate predictions about the
number of individuals at each time step. To train the recurrent net, we first
harvest the training data by running a few SEIR simulations, leading to a
collection of simulated data under different parameter combinations. Next, a
recurrent net is trained to fit these simulations. Then with model parameters
of the the recurrent net $\Theta_{\text{LSTM}}$ fixed, the observation data
are used to train the net with respect to the SEIR parameters
$\Theta_{\text{SEIR}}$ via gradient descent, leading to the optimal parameter
values $\Theta^{*}_{\text{SEIR}}$.

The proposed strategy is both time efficient in that it only relies on a small
number of SEIR simulations, and accurate because we are able to find the
optimal parameters $\Theta^{*}_{\text{SEIR}}$ based on the differentiable
objective. Through experiments on two COVID-19 datasets, we observe that the
proposed strategy leads to more accurate parameter estimates with
significantly better time efficiency.

##  2 Background and Problem Statement

Figure 1: An overview of a standard SEIR model (left) the mobility network
(right: from time $t$ to time $t+1$, person $p_{3}$ moved from location
$l_{1}$ to $l_{3}$, and $p_{1}$ moved from $l_{1}$ to $l_{2}$).

This work focuses on estimating the parameters $\Theta_{\text{SEIR}}$ in the
SEIR mode. A standard SEIR model is formulated in terms of 4 populations of
individuals: the susceptible population ($S$), the exposed population ($E$),
the infectious population ($I$) and the recovered population ($R$), which are
respectively comprised of all individuals susceptible to the infection,all
individuals that have contacted infected patients but are currently during
their incubation period, infected individuals that can transmit the disease to
susceptible population, and recovered individuals that cannot become infected
again and cannot transmit the disease to others. Given a mobility network
$\mathcal{G}_{t}=(\\{\mathcal{P},\mathcal{L}\\},\mathcal{E})_{t}$ at time step
$t(t=1,2,\cdots,T)$ where $\mathcal{P}$ is the set of person nodes,
$\mathcal{L}$ is the set of location nodes and $\mathcal{E}$ is the set of
edges linking persons to locations, the SEIR model needs to predict, what the
numbers of individuals for these populations are in different locations at
time step $t$, according to the preceding populations progress, the current
mobility network and a set of parameters controlling the probability of
transferring from one state of population to the next state.

Formally, we use the superscript l to represent a specific location $l$ and
use the subscript t to represent the time point $t$, e.g., $S^{l}_{t}$ is the
number of individuals for state $S$ in location $l$ at time step $t$. The
number for all populations in location $l$ at time step $t$ is denoted by
$N^{l}_{t}=S^{l}_{t}+E^{l}_{t}+I^{l}_{t}+R^{l}_{t}$, and
$S_{t}=\sum_{l}S^{l}_{t}$ is the population for all locations at time $t$ (the
same for $E_{t},I_{t},R_{t}$). We further assume a constant overall population
$N=N_{t}=S_{t}+E_{t}+I_{t}+R_{t},\forall t$. There are three sets of
parameters $\beta,\kappa$ and $\gamma$ controlling how likely that a person
currently at one particular state would transfer to the next state. $\beta$
controls the probability of transferring from $S$ to $E$, and $\kappa$ and
$\gamma$ are respectively responsible for tranferring from $E$ to $I$ and $I$
to $R$. $\beta$ can be a set of values, representing the transmission rate for
different location categories or cities. $\kappa$ and $\gamma$ can also be a
set of values, representing the infection and recovery rates for different
population groups. With a fixed set of parameters and the mobility network,
the SEIR model can simulate the disease spread process and predict the numbers
of these four populations at each time step for a location. The probability of
an individual transferring from one state to the next at location $l$ for time
step $t$ can be formulated as:

$\begin{cases}p^{l}_{t}(S\to E)=\beta\frac{I^{l}_{t}}{N^{l}_{t}}\\\
p^{l}_{t}(E\to I)=\kappa\\\ p^{l}_{t}(I\to R)=\gamma\\\ \end{cases}$ (1)

During simulations, the state for each individual is sampled. An illustration
is shown in Figure 1. The sum of the $L_{2}$ distances between the observed
infections and the predicted infections over all time steps is the simulation
error for a particular set of parameters, and the best parameter combination
is selected to minimize the simulation error:

$\Theta^{*}_{\text{SEIR}}=\beta^{*},\kappa^{*},\gamma^{*}=\arg\min_{\beta,\kappa,\gamma}\sum_{t}\|I_{t}-\hat{I}_{t}\|^{2}$
(2)

We use the hat notation $\hat{~{}}$ to represent the prediction made by the
model. Towards finding the (near) optimal parameter combination in Eq.(2), the
SEIR models needs to run Eq.(1) over all individuals for $T$ time steps, which
gives a total time complexity of $\mathcal{O}(NT)$.

##  3 Modeling the SEIR Process Using Recurrent Nets

###  3.1 Overview

The core idea of the proposed framework is that, instead of using the brute
force grid search strategy to obtain the parameter set that leads to the least
$L_{2}$ distance between predictions and observations, we train a recurrent
net to fit the time-series data of SEIR based on a small number of simulations
using different sets of parameters. The trained recurrent net can then be used
to directly find the optimal value of $\Theta^{*}_{\text{SEIR}}$, which
denotes the parameter set for SEIR that leads to the optimal predictions. We
employ the widely used Long Short-Term Memory network (LSTM) [17] as the
recurrent net model backbone.

###  3.2 Simulation Dataset Construction

The first step of the proposed framework is to generate smiulation data, which
will be used to train the recurrent LSTM net. We first use different sets of
$\Theta_{\text{SEIR}}$ to perform simulations on the predefined network, as in
Eq.(1). We perform $M^{\prime}$ simulations with $M^{\prime}$ different sets
of $\Theta_{\text{SEIR}}$, denoted by
$\\{\Theta_{\text{SEIR}}^{(m^{\prime})}\\}_{m^{\prime}=1}^{M^{\prime}}$. At
each time step $t$, we iterate over all individuals, assign the previous state
(i.e., $S$, $E$, $I$ or $R$) of an individual, and sample its current state
based on the SEIR model. Then, we sum up all individuals belonging to the same
states, and obtain
$\\{S_{t}^{(m^{\prime})},E_{t}^{(m^{\prime})},I_{t}^{(m^{\prime})},R_{t}^{(m^{\prime})}\\}$
for time step $t$.

###  3.3 Training the Recurrent Nets

Next, we train an recurrent net to fit
$\\{S_{t}^{(m^{\prime})},E_{t}^{(m^{\prime})},I_{t}^{(m^{\prime})},R_{t}^{(m^{\prime})}\\}$.
Our goal is to use LSTM to predict the values of
$\\{S_{t}^{(m^{\prime})},E_{t}^{(m^{\prime})},I_{t}^{(m^{\prime})},R_{t}^{(m^{\prime})}\\}$
for each time step $t$ for each parameter combination in
$\\{\Theta_{\text{SEIR}}^{(m^{\prime})}\\}_{m^{\prime}=1}^{M^{\prime}}$.
Specifically, for each time step $t$, the input to the LSTM model is denoted
by
$\bm{x}_{t}^{(m^{\prime})}=\\{S_{t-1}^{(m^{\prime})},E_{t-1}^{(m^{\prime})},I_{t-1}^{(m^{\prime})},R_{t-1}^{(m^{\prime})}\\}$,
the populations of the four states at the previous time step. The model should
also be aware of the value of $\Theta_{\text{SEIR}}^{(m^{\prime})}$ and the
mobility network structure at time step $t$ because they directly decide the
number of individuals for the four states for the next time step. To serve
this propose, we also feed $\Theta_{\text{SEIR}}^{(m^{\prime})}$ as input to
the LSTM model at time step $t$. For the time-varying mobility network
$\mathcal{G}_{t}$, we map it to a vector representation
$\bm{h}_{\mathcal{G}_{t}}$, which is able to capture its mobility structure
and be conveniently fed to the LSTM network.

#### Mapping $\mathcal{G}_{t}$ to $\bm{h}_{\mathcal{G}_{t}}$

We use the DIFFPOOL model [33] to map the mobility network $\mathcal{G}_{t}$
at time step $t$ to its high-dimensional representation
$\bm{h}_{\mathcal{G}_{t}}$. DIFFPOOL is a differentiable pooling model that
can generate hierarchical representations of graphs by progressively
clustering nodes into a coarser graph. Specifically, at each layer $l$,
DIFFPOOL learns an assignment matrix $\bm{S}^{(l)}\in\mathbb{R}^{n_{l}\times
n_{l+1}}$ to assign each node at layer $l$ to a cluster in the next layer
$l+1$, where $n_{l}$ is the number of nodes (clusters) at layer $l$. Given
node embeddings $\bm{Z}^{(l)}\in\mathbb{R}^{n_{l}\times d}$ and the adjacency
matrix $\bm{A}^{(l)}\in\mathbb{R}^{n_{l}\times n_{l}}$ at layer $l$, DIFFPOOL
generates the new node embeddings $\bm{Z}^{(l+1)}\in\mathbb{R}^{n_{l+1}\times
d}$, the new adjacency matrix $\bm{A}^{(l+1)}\in\mathbb{R}^{n_{l+1}\times
n_{l+1}}$ and the new assignment matrix
$\bm{S}^{(l+1)}\in\mathbb{R}^{n_{l+1}\times n_{l+2}}$ by applying the
following equations:

$\displaystyle\bm{Z}^{(l+1)}=\text{GNN}_{l,\text{embed}}(\bm{A}^{(l+1)},\bm{X}^{(l+1)})$
$\displaystyle,~{}\bm{S}^{(l+1)}=\text{softmax}\left(\text{GNN}_{l,\text{pool}}(\bm{A}^{(l+1)},\bm{X}^{(l+1)})\right)$
(3) $\displaystyle\bm{X}^{(l+1)}={\bm{S}^{(l)}}^{\top}\bm{Z}^{(l)}$
$\displaystyle,~{}\bm{A}^{(l+1)}={\bm{S}^{(l)}}^{\top}\bm{A}^{(l)}\bm{S}^{(l)}$

$\text{GNN}_{l,\text{embed}}$ and $\text{GNN}_{l,\text{pool}}$ are two
distinctly parameterized GNN, and are respectively responsible for generating
new embeddings and producing a distribution over next-layer clusters. Setting
the number of clusters at the last layer $L$ to 1, DIFFPOOL outputs a single
high-dimensional graph representation. We train DIFFPOOL to classify the graph
$\mathcal{G}_{t}$ at time step $t$ to the category label $t$, and use the
extracted graph representation $\bm{h}_{\mathcal{G}_{t}}$ as input to the
LSTM.

#### Training LSTMs $\Theta_{\text{LSTM}}$

Given $\bm{h}^{(m^{\prime})}_{t-1}$, $\bm{x}_{t}^{(m^{\prime})}$,
$\Theta_{\text{SEIR}}^{(m^{\prime})}$ and $\bm{h}_{\mathcal{G}_{t}}$, we are
able to obtain the hidden vector representation $\bm{h}_{t}^{(m^{\prime})}$
for the time step $t$:

$\left[\begin{array}[]{lr}\bm{i}_{t}\\\ \bm{f}_{t}\\\ \bm{o}_{t}\\\
\bm{l}_{t}\\\ \end{array}\right]=\left[\begin{array}[]{c}\sigma\\\ \sigma\\\
\sigma\\\ \text{tanh}\\\
\end{array}\right]\bm{W}\cdot\left[\begin{array}[]{c}\bm{h}_{t-1}^{(m^{\prime})}\\\
\bm{x}_{t}^{(m^{\prime})}\\\ \bm{h}_{\mathcal{G}_{t}}\\\
\Theta_{\text{SEIR}}^{(m^{\prime})}\\\ \end{array}\right]$ (4)
$\bm{c}_{t}=\bm{f}_{t}\cdot\bm{c}_{t-1}+\bm{i}_{t}\cdot\bm{l}_{t}$ (5)
$\bm{h}_{t}^{(m^{\prime})}=\bm{o}_{t}\cdot\text{tanh}(\bm{c}_{t})$ (6)

where $\bm{W}_{i}$, $\bm{W}_{f}$, $\bm{W}_{o}$,
$\bm{W}_{l}\in\mathbb{R}^{K\times(2K+4+|\Theta_{\text{SEIR}}|)}$ where $K$ is
the dimensionality of $\bm{h}_{t}^{(m^{\prime})}$ and
$\bm{h}_{\mathcal{G}_{t}}$. $\bm{h}_{t}^{(m^{\prime})}$ is then passed to a
fully connected layer to obtain $\bm{\hat{h}}_{t}^{(m^{\prime})}$, which is
mapped to scalars to predict
$\\{S_{t+1}^{(m^{\prime})},E_{t+1}^{(m^{\prime})},I_{t+1}^{(m^{\prime})},R_{t+1}^{(m^{\prime})}\\}$:

$\displaystyle\hat{S}_{t+1}^{(m^{\prime})}$
$\displaystyle=\bm{h}_{S}^{\top}\times\bm{\hat{h}}_{t}^{(m^{\prime})}$ (7)
$\displaystyle\hat{E}_{t+1}^{(m^{\prime})}$
$\displaystyle=\bm{h}_{E}^{\top}\times\bm{\hat{h}}_{t}^{(m^{\prime})}$
$\displaystyle\hat{I}_{t+1}^{(m^{\prime})}$
$\displaystyle=\bm{h}_{I}^{\top}\times\bm{\hat{h}}_{t}^{(m^{\prime})}$
$\displaystyle\hat{R}_{t+1}^{(m^{\prime})}$
$\displaystyle=\bm{h}_{R}^{\top}\times\bm{\hat{h}}_{t}^{(m^{\prime})}$

where $\bm{h}_{S}$, $\bm{h}_{E}$, $\bm{h}_{I}$,
$\bm{h}_{R}\in\mathbb{R}^{K\times 1}$. The training objective is minimizing
the distance between simulation outputs
$\\{S_{t+1}^{(m^{\prime})},E_{t+1}^{(m^{\prime})},I_{t+1}^{(m^{\prime})},R_{t+1}^{(m^{\prime})}\\}$
and LSTM predictions
$\\{\hat{S}_{t+1}^{(m^{\prime})},\hat{E}_{t+1}^{(m^{\prime})},\hat{I}_{t+1}^{(m^{\prime})},\hat{R}_{t+1}^{(m^{\prime})}\\}$:

$\small\Theta^{*}_{\text{LSTM}}=\arg\min_{\Theta_{\text{LSTM}}}\sum_{t,m^{\prime}}[\|S_{t}^{(m^{\prime})}-\hat{S}_{t}^{(m^{\prime})}\|^{2}+\|E_{t}^{(m^{\prime})}-\hat{E}_{t}^{(m^{\prime})}\|^{2}+\|I_{t}^{(m^{\prime})}-\hat{I}_{t}^{(m^{\prime})}\|^{2}+\|R_{t}^{(m^{\prime})}-\hat{R}_{t}^{(m^{\prime})}\|^{2}]$
(8)

Eq. 8 can be trained in an end-to-end fashion to obtain optimal
$\Theta_{\text{LSTM}}$.

###  3.4 Finding Optimal $\Theta_{\text{SEIR}}$

The LSTM model with $\Theta^{*}_{\text{LSTM}}$ is able to generalize the
behavior of the SEIR model with a specific value of $\Theta_{\text{SEIR}}$.
When training the LSTM to learn $\Theta^{*}_{\text{LSTM}}$,
$\Theta_{\text{SEIR}}$ is set to a fixed value of
$\Theta_{\text{SEIR}}^{(m^{\prime})}$ and fed as input to LSTMs at each time
step. Due to the fact that $\Theta_{\text{SEIR}}$ can also be viewed as
parameters in LSTMs, i.e., the input to each time step, we can fix
$\Theta^{*}_{\text{LSTM}}$ and relax $\Theta_{\text{SEIR}}$, treating
$\Theta_{\text{SEIR}}$ as learnable parameters, to minimize Eq.(2) which we
write down here for reference:

$\Theta^{*}_{\text{SEIR}}=\beta^{*},\kappa^{*},\gamma^{*}=\arg\min_{\beta,\kappa,\gamma}\frac{1}{T}\sum_{t}\|I_{t}-\hat{I}_{t}\|^{2}$
(2)

where $\hat{I}_{t}$ is the output from LSTM, and $\hat{I}_{t}$ is the
observation data rather than the simulation data used to train the LSTM.
Eq.(2) is differentiable with respect to $\Theta_{\text{SEIR}}$ and can be
trained in an end-to-end fashion based on SGD [20, 27]. To this end, we learn
the optimal values of $\Theta_{\text{SEIR}}$ that minimize the $L_{2}$
distance between predicted infections and observations.

In the case where we have prior knowledge about the values
$\Theta_{\text{SEIR}}$, e.g., all values in $\Theta_{\text{SEIR}}$ should be
larger than 0, the value of $\beta$ is usually smaller than 0.1 based on
clinical observations for COVID-19 [5], we can incorporate regularizers as
side objectives:

$\Theta^{*}_{\text{SEIR}}=\arg\min_{\Theta_{\text{SEIR}}}=\sum_{t}\frac{1}{T}\|I_{t}-\hat{I}_{t}\|^{2}+\lambda||\Theta_{\text{SEIR}}-\text{prior}(\Theta_{\text{SEIR}})||^{2}$
(9)

where $\text{prior}(\Theta_{\text{SEIR}})$ denotes the human prior knowledge
regarding the values of $\Theta_{\text{SEIR}}$, and $\lambda$ controls the
trade-off. We will explore the effects of $\text{prior}(\Theta_{\text{SEIR}})$
and $\lambda$ in experiments.

##  4 Experiments

###  4.1 Datasets and Corresponding SEIR Models

We use two public datasets for evaluations: the infection network of Covid-19
in China (Covid-China) [29, 24], and the infection network of Covid-19 in the
US (Covid-US) [5].

Covid-China consists of infection networks for 31 provinces in China from Apr
2020 to Feb 2021, extracted from action tracking reports of Covid-19 patients.
The network of Covid-China consists of two types of nodes: patient and
location. Time-varying edges are constructed between a patient node and a
location node if the patient visited the location at time $t$. Each location
takes an attribute from 11 categories of locations: households, workplaces,
hotels, supermarkets, banks, restaurants, parks,barber shops/hairdressers,
trains, buses, and airplanes, and the economic city tier (first, second or
third) that it belongs to. A patient node is characterized by features of age
(taking the value of children, youths, adults or seniors) and gender (taking
the value of male and female). Each attribute for gender, age, city-tier and
location type is associated with a specific transmission rate $\beta$. The
transmission rate for a certain person node of gender $s$, age $a$ in location
of type $c$ in a city of tier $t$ is the additive combination of corresponding
$\beta$:

$\displaystyle\beta(s,a,t,c)=\beta_{\text{s}}+\beta_{\text{a}}+\beta_{\text{t}}+\beta_{\text{c}}$
(10)

$\beta=\\{\beta_{\text{s}},\beta_{\text{a}},\beta_{\text{t}},\beta_{\text{c}}\\}$,
along with $\gamma$ and $\kappa$ are parameters to learn. The network for each
city is sliced into consecutive time snippets, with the size of stride set to
two weeks. For each city, we have daily gold number of infections, public by
Chinese CDC. These gold numbers of infections are used to learn
$\Theta_{SEIR}$. Snippets without any infection are removed. Each time step of
each city is labeled with gold number of infections, which is used to train
the SEIR parameters. Snippets are divided to 80%/10%/10% for training, dev and
test.

Covid-US consists of networks that capture hourly visits from each population
group to each location in 10 metro areas in the US. The network is extracted
from mobility data provided by the SafeGraph application. The network consists
of two types of nodes: population group and location. A time-varying edge with
weight $w_{i,j}$ is constructed, if at time $t$, the number of people from
population group $i$ visiting location $j$ is $w_{i,j}$. $w_{i,j}$ is column-
normalized. Each location is associated with a location category (e.g., full-
service restaurants, grocery stores, etc), and each population group is
associated with its race and median income. For the SEIR model, transmissions
can happen within groups or across population groups when two people from two
groups visit the same location. Simulations are performed at the population
group level. Each category location $c$ is associated with a specific
transmission rate $\beta_{c}$, which captures the inter-group transmissions
across groups in the locations. For each population group with race $r$ and
income decile ($i$), the intra-group transmission is set to
$\beta=\beta_{r}+\beta_{i}$. Each area is associated with gold number of
infections published by the The New York
Times111https://github.com/nytimes/covid-19-data. Each snippet consists of the
network for a single city and lasts a week. We divided snippets to 80%/10%/10%
for training, dev and test.

###  4.2 Experimental Details

#### Generating Simulated Data

We first need to sample $\Theta_{\text{SEIR}}$. We limit the value of each
$\beta$ to the scope of [0, 0.1], and we randomly sample its value within the
scope. For $\kappa$ and $\gamma$, based on previous clinical observations [22]
where $\kappa^{-1}$ is around 96 hours and $\gamma^{-1}$ is around 84 hours,
we sample $\kappa$ and $\gamma$ from a normal distribution with expectation
set to 96 and 84. Given a sampled set of $\Theta_{\text{SEIR}}$, we run
simulations on the training datasets to obtain the simulation data, i.e., the
number of individuals for all four states for each time step. For each
episode, we take $K$ samples of $\Theta_{\text{SEIR}}$, leading to a total
number of $K*|\text{train}|$ training sequences, where $|\text{train}|$
denotes the number of training episodes.

#### Learning LSTMs $\Theta_{\text{LSTM}}$ to Fit Simulated Data

We split the simulated data to 90/10 for training and validation. We train a
three-layer LSTM with residual connections [15, 21] to fit the simulated time-
series data using for training, based on Eq.(8). Then size of hidden states is
set to 128. The value of batch size is set to 256, and SGD is used for
optimization. LSTM parameters and embeddings are initialized from a uniform
distribution in [-0.08,008]. Gradient clipping is adopted by scaling gradients
when the norm exceeds a threshold of 1. Dropout rate, learning rate and the
number of training epochs are treated as hyper-parameters to be tuned on the
dev set.

#### Learning $\Theta_{\text{SEIR}}$

We optimize $\Theta_{\text{SEIR}}$ based on Eq.(9) on the number of gold daily
infections, using the daily reported infections as labels. We use AdaGrad [7]
for optimization. Dropout rate, learning rate, the number of training epochs,
and the hyper-parameter $\lambda$ are tuned on the dev set.

| Covid-China | Covid-US
---|---|---
# Simulations | Vanilla | LSTM | Vanilla | LSTM
20 | 34.2 | 22.2 | 1870 | 1250
100 | 30.1 | 13.1 | 1530 | 1130
500 | 18.7 | 9.9 | 1320 | 930
1000 | 15.2 | 9.2 | 1120 | 824
5000 | 13.5 | 8.6 | 970 | 674
Table 1: Average square $L_{2}$ distances for vanilla grid search simulation
and recurrent LSTMs on Covid-China and Covid-US

###  4.3 Results

For baselines, we search the optimal $\Theta_{\text{SEIR}}$ using vanilla grid
search SEIR simulations. For each set of $\Theta_{\text{SEIR}}$, simulations
are performed on all training episodes, and the parameter set that leads to
the minimum $L_{2}$ loss is selected as the final value. Suppose that we
conduct $K$ explorations for $\Theta_{\text{SEIR}}$. This means we need to
perform $K$ simulations on each training episode. The $K$ is here thus
comparable to and the same as the $K$ for simulation data generation.

We report the average of the square of $L_{2}$ distance between the predicted
number and reported number of infections on the test episodes with varying
number of simulations $K$ on the test set. Lower values indicate superior
models. Results are shown in Table 1. Observations are as follows: (1) as the
number of simulations $K$ increases, the performances for both the vanilla
model and the proposed model improve. This is in accord with our expectations:
for the vanilla model, a larger number of simulations means that the model is
able to explore the search space more thoroughly to obtain the optimal value;
for the proposed LSTM model, the model learns better with more training data
and avoids overfitting; (2) with the same number of $K$, the proposed LSTM
model performs significantly better than the vanilla brute-force search model.
This is due to the generalization ability of proposed framework: the vanilla
model can only select the optimal parameters from the set it tries, while the
proposed framework can generalize to the un-tried parameter set; and (3)
notably, the proposed framework is able to achieve comparable performance to
the vanilla search model with significantly smaller number of simulations.
Specifically, for Covid-China, the performance obtained with 100 simulations
(13.1) is comparable to the vanilla model with 5,000 simulations (13.5); for
Covid-US, the performance obtained with 500 simulations (930) is comparable to
the vanilla model with 5,000 simulations (970). This further illustrates the
superiority of the proposed framework.

Figure 2 shows simulations performed on the four test episodes in Covid-China
dataset using parameters learned from the proposed model and the vanilla SEIR
model . As can be seen, the proposed framework offers more accurate
predictions.

Figure 2: Daily reported cases, predictions made by the vanilla SEIR model,
and the predictions made by LSTM for three cities in China – Shijiazhuang,
Harbin and Suihua. Curves are smoothed by 5-day average.

###  4.4 Ablation Studies

In this subsection, we explore the effect of different modules, along with
hyper-paremeters in the proposed framework to explore their influence.

$\log_{10}(\lambda)$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7
---|---|---|---|---|---|---|---|---
Covid-China | 10.4 | 9.9 | 9.6 | 9.4 | 10.1 | 11.4 | 13 | 23.5
Covid-US | 890 | 866 | 861 | 842 | 832 | 824 | 899 | 1423
Table 2: The effect of different values of $\lambda$ on Covid-China and Covid-
US. We show the logarithm with base 10 for $\lambda$. The average square
$L_{2}$ distances are reported.

#### The effect of $\lambda$

The hyper-parameter controls the trade-off between observations and external
prior knowledge. The effect of $\lambda$ is shown in Table 2. As can be seen,
model performance first improves and then declines as the value of $\lambda$
grows. Finding the sweep spot for the balance between observations and prior
knowledge leads to the best performance.

| CN | US
---|---|---
LSTM | 9.4 | 824
GRU | 9.1 | 831
RNN | 12.1 | 890
SRU | 11.5 | 872
Table 3: The effect of recurrent structures.

| CN | US
---|---|---
Each | 9.4 | 824
First | 13.5 | 1007
Hadamard | 9.1 | 821
Table 4: The effect of using different strategies to incorporate SEIR
parameters.

| CN | US
---|---|---
w/o Graph | 24.2 | 2531
Constant | 18 | 2013
Varying | 9.4 | 824
Table 5: The effect of graph representations.

#### The effect of Recurrent Structures

We also conduct experiments using other recurrent net structures, including
vanilla recurrent net (RNN), Gated Recurrent Unit (GRU) [6], and Simple
Recurrent Units (SRU) [23]. Results for different recurrent structures are
shown in Table 5. As can be seen, the LSTM structure performs comparable to
GRU (slightly worse than GRU on Covid-China and slightly better on Covid-US),
better than the vanilla recurrent net and SRU.

#### How to incorporate $\Theta_{\text{SEIR}}$ into LSTMs

We explore the effects of different ways to incorporate $\Theta_{\text{SEIR}}$
in the LSTM model, including the current strategy of (1)
$\Theta_{\text{SEIR}}$ being concatenated with the input $\bm{x}_{t}$ at each
time step (Each); (2) $\Theta_{\text{SEIR}}$ being incorporated only at the
first time step (First); and (3) $\Theta_{\text{SEIR}}$ element-wise
multiplies (Hadamard product) the input for each time step (Hadamard). For
(3), since the dimensionalities of $\Theta_{\text{SEIR}}$ and $\bm{x}_{t}$ are
different, $\bm{x}_{t}$ is first passed to an FFN, the output of which has the
same dimensionality with $\Theta_{\text{SEIR}}$. Results for the three
strategies are shown in Table 5. As can be seen, $\Theta_{\text{SEIR}}$
incorporated only at the first time step significantly underperforms the
strategy that incorporates $\Theta_{\text{SEIR}}$ at every time step. This is
because of the gradient vanishing effect of recurrent nets: reminding the
model of $\Theta_{\text{SEIR}}$ at each time leads to better performances. The
concatenation strategy obtains comparable performances to the Hadamard product
strategy.

#### The Effect of Graph Representation

The time-varying location-population network is captured by the graph
representations through DIFFPOOL. This is critical since the number of
infections highly relies on the spreading network for each time. We explore
its necessity by comparing it with other variants: (1) no network (w/o Graph):
where no time-varying graph embedding is incorporated; (2) constant graph
representation (Constant): the graph embedding is not time-varying, where we
use the graph embedding of the first time step for all time steps; (3) time-
varying graph representation (Varying): the strategy adopted in this work
where time-varying representations are incorporated. Results are shown in
Table 5. As can be seen, when no network information is incorporated, the
model nearly fails to learn anything. Constant networks perform slightly
better than no network, but still significantly worse than the time-vary graph
representations. This is in accord with our expectations since the time-
varying network decides the number of infections and the disease spread
patterns at each time.

##  5 Related Works

The classical compartmental models simplify the mathematical modeling of
disease spread by simulating the population transitions between different
states in the disease spread process. Since the outbreak of Covid-19, the SEIR
model has been widely used to model the Covid-19 spread around the world [9,
12, 13] and provide important insights regarding isolation policies [8, 4, 5]
and vaccine delivery strategies [3, 11]. To achieve a faster simulation speed
and better simulation results, recent works have proposed to leverage deep
neural networks in place of SEIR models to predict pandemic dynamics over
time. For example, [32] used the LSTM model to predict the numbers of new
infections given the contact statistics and the pre-selected
transmission/incubation/recovery/death rates. [10] incorporated sequential
network structures and graph attention to predict the number of infections
upon a temporal and spatial mobility graph. These works aim at taking
advantage of SEIR models to inform effective strategies in response to the
disease spread.

With regard to the estimation of the parameters in SEIR models, a simple
approach is to enumerate parameter combinations, run the SEIR model with each
combination and select the one with the smallest error. An alternative to grid
search is to use approximate Bayesian computation (ABC) [30, 28, 26], a
technique that maintains a small fraction of simulations that are close to the
targt statistics in the light of the computed distance. These simulations are
treated as posterior distributions of the SEIR parameters, which are then used
to infer the optimal parameters. Another approach to estimating parameters is
to view the process of population transitions between states as a problem of
ordinary differentiable equations (ODEs) [19, 16, 14]. However, ODEs can only
give approximate numerical solutions, which could be inaccurate for real-world
modeling.

The most relevant work is from [31], who proposed to make direct use of
existing neural networks to predict the basic reproduction number ($R0$), the
number of secondary cases generated by an infectious individual in a fully
susceptible host population. This work is different from [31] in that (1) they
sought to estimate the basic reproduction number $R0$ and we estimate the
parameters in SEIR (or other SEIR variants) models; and more importantly (2)
they propose to directly output the number to estimate given inputs, whereas
we propose to automatically learn the parameters through neural network
gradient descent and back-propagation. The proposed method can be extended to
other fields that require time-consuming simulations to estimate necessary
parameters.

##  6 Conclusion

In this work, we propose to transform the original non-differentiable
simulation problem of SEIR parameter estimation into a differentiable one by
leveraging neural recurrent nets. The recurrent net is first trained to fit a
small number of simulation data, and then trained on the observation data to
derive the optimal SEIR parameters. This strategy bypasses the needs of time-
consuming simulations, and automatically induces the optimal parameters via
gradient descent, leading to both accuracy and efficiency gains.

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# Trigonal to cubic structural transition in possibly N-doped
$\mathrm{LuH_{3}}$ measured by Raman and X-ray diffraction

Owen Moulding  Institut Néel CNRS/UGA UPR2940, 25 Rue des Martyrs, 38042
Grenoble, France  Samuel Gallego-Parra  European Synchrotron Radiation
Facility, 71 Avenue des Martyrs, 38000 Grenoble, France  Pierre Toulemonde
Institut Néel CNRS/UGA UPR2940, 25 Rue des Martyrs, 38042 Grenoble, France
Gaston Garbarino  European Synchrotron Radiation Facility, 71 Avenue des
Martyrs, 38000 Grenoble, France  Patricia Derango  Institut Néel CNRS/UGA
UPR2940, 25 Rue des Martyrs, 38042 Grenoble, France  Pierre Giroux  Institut
Néel CNRS/UGA UPR2940, 25 Rue des Martyrs, 38042 Grenoble, France  Marie-Aude
Méasson  Institut Néel CNRS/UGA UPR2940, 25 Rue des Martyrs, 38042 Grenoble,
France

###### Abstract

After the reported discovery of room-temperature superconductivity at only 1
GPa in nitrogen-doped lutetium trihydride Dasenbrock-Gammon et al. (2023) and
the resulting heated discussions, there is an urgent need to reproduce the
results and synthesis of this compound. Here, we report the synthesis of
(potentially N-doped) cubic LuH3 starting from pure Lu to produce very stable
trigonal LuH3 which was subsequently pressurised to $\sim$2 GPa with a dilute
N2/He rich pressure medium. Raman spectroscopy and X-ray diffraction were used
to characterise the structure throughout the synthesis process.

††preprint: ver 0.7

##  I Introduction

The Holy Grail of room-temperature superconductivity has been a long sought-
after quest ever since the initial predictions of superconductivity in
metallic hydrogen by Ashcroft in 1968 Ashcroft (1968) shortly after the
publication of BCS theory 1957 Bardeen et al. (1957a, b). Though not pure
hydrogen, many examples of high-temperature superconductivity have been
realised in recent years which have been reliably shattering high-$T_{c}$
records with each new discovery. A notable discovery was the superconductivity
in SH3 at 203 K and 155 GPa Drozdov et al. (2015) as it provided promise for
the field. Subsequent examples continued to push the threshold with the
discovery of superconductivity in YH9 and LaH10 at 243 K and 260 K
respectively both at approximately 200 GPa Kong et al. (2021); Somayazulu et
al. (2019); Drozdov et al. (2019). Clearly these syntheses require extremely
high pressures that few groups are able to reach, and this has been the
primary technical challenge to overcome.

Hence why the very recent claim of room-temperature superconductivity (294 ∘K
in N-doped lutetium hydride and at such a low pressure of 1 GPa Dasenbrock-
Gammon et al. (2023) has drawn so much attention, as not only it is a new
record $T_{c}$ for superconductivity, but it also brings superconductivity
into the domain of practicably achievable at near-ambient conditions.
Furthermore, the samples are said to be metastable at ambient pressure which
further adds to the wishful properties of such a material. In a very short
period of time, an impressive number of groups have already tried to replicate
the results both theoretically and experimentally Shan et al. (2023); Huo et
al. (2023); Li et al. (2023); Sun et al. (2023); Xie et al. (2023); Ming et
al. (2023); Xing1 et al. (2023); Zhang et al. (2023); Hilleke et al. (2023).
Two experimental papers started from pure Lu and NH4Cl and CaH2 precursors
Xing1 et al. (2023); Ming et al. (2023) which decompose to provide the
required N2 and H2. Dasenbrock et al. Dasenbrock-Gammon et al. (2023) and a
very recent report by Cai et al. Cai et al. (2023) used pure Lu with H2/N2 gas
mixture loading. Here we choose another process, by first synthesising pure
LuH3 and then load it with a mixture of dilute N2 gas and helium in a diamond
anvil cell (DAC). We then characterise the obtained compound thanks to Raman
spectroscopy and X-ray diffraction (XRD).

##  II Methods

###  II.1 Experimental Methods

Lutetium (Alfa 3N) was characterised by EDX before polishing it, whereupon
oxygen was clearly identified in Lu2O3 deposits with atomic concentrations
between 20-50 $\%$. A small amount of tantalum (atomic concentrations between
0.2-1 $\%$) was identified (Cf. SI). We then polished a 150 mg piece of
lutetium in air until the surface became shiny instead of black in order to
remove the oxide from the surface.

We then synthesised LuH3 by hydrogen absorption using the Sievert method. We
used a HERA C2-3000 device to measure the quantity of hydrogen absorbed (or
desorbed) by the piece of lutetium as a function of time. This is calculated
by measuring the hydrogen pressure variation in a sample holder of known
volume. The measurement of the hydrogenation rate is performed out of
equilibrium. A piece of polished lutetium (147.67 mg) was placed in the
sample-holder of the reaction chamber. The sample-holder and compensation
chambers were pumped for one hour at ambient temperature to remove
contaminating gases. The temperature was then increased to a maximum
temperature of 500°C at 10-5 mbar and kept stable during 4000 s. The
temperature was then decreased to 200 ∘C, and H2 gas at 4 MPa was injected
into the chamber. The percentage of absorbed H2 was measured to be 1.7 $\%$ in
mass which corresponds to the expected composition LuH3, as shown in figure 1.
After the synthesis, the sample-holder was closed and transferred into a glove
box (under argon) where it was opened to recover the LuH3 powder.

Figure 1: Synthesis of LuH3 from lutetium: weight percentage of absorbed H2 by
lutetium as a function of time. After 2 hours at 200 ∘C, 1.7$\%$ of absorbed
H2 is reached showing the successful synthesis of LuH3.

A thin sample of LuH3 was prepared in a diamond anvil cell (DAC with culets of
800 $\mu$m diameter) by pressing the synthesised powder between the two
diamonds until the sample was $\sim$ 5-10$\mu$m thick. A stainless steel
gasket was indented to a thickness of 80 $\mu$m and a hole of 400 $\mu$m was
drilled for the pressure chamber. A ruby sphere and a small piece of silicon
were placed inside the pressure chamber. Prior to loading the DAC, the LuH3
sample was characterised by Raman spectroscopy inside the unloaded DAC.

We then used a gas loader (Top Industrie) to load a mixture of nitrogen and
helium. After purging with helium, the system was filled with 10 bar of N2 and
then 1500 bar of helium. We estimate that the quantity of N2 in the pressure
chamber was 4 nmol whilst the quantity of LuH3 was 11 nmol. The DAC was then
sealed at 0.1 GPa and then we applied 1.92 GPa and proceeded to characterise
the sample by Raman spectroscopy and XRD.

X-ray powder diffraction of the starting LuH3 was performed after the
hydrogenation of Lu using a D5000T diffractometer (Cu K$\alpha$ radiation), at
ambient pressure (and outside the DAC). The measurement was repeated several
times (up to 9 days after the first measurement) to determine the effect of
air exposure on LuH3. The Rietveld refinements were done with Fullprof
software. The X-ray powder diffraction after loading and at 1.9 GPa in DAC was
performed on the ESRF beamline ID15B at $\lambda$=0.411 Å. Calibration of the
detector-to-sample distance, beam orientation, detector tilt with respect to
the omega rotation axis, and the used wavelength is determined by a Si powder
standard (‘NIST 640 C’ from NIST). The X-ray beam is focused to 4 x 3$\mu$m2
using Be compound refractive lenses. 2D images were collected with a six
degrees oscillation of the DAC using an Eiger 2X CdTe 9M photon counting
detector from Dectris and integrated into 1D diffractogram using the Dioptas
software Prescher and Prakapenka (2015). Le Bail refinements (lattice
parameter, peak profile and background) on the loaded DAC at 2 GPa were done
using the gsas2 package Toby and Von Dreele (2013).

Polarised Raman scattering was performed in quasi-backscattering geometry at
300 K with an incident laser-line at 532 nm from a solid state laser. The DAC
was placed in a vacuum to avoid measuring the Raman-response of air. We used a
laser power between 2.5mW-10mW with a typical size spot of 25 $\mu$m. The
scattered light was analysed by a single grating and a triple grating
subtractive spectrometer, both were equipped with liquid nitrogen-cooled CCD
detectors. The crossed and parallel polarisation dependence was measured by
changing the orientation of the polariser on the collection path. We measured
the Raman signal of pure LuH3 just before loading in the DAC and after loading
at 1.9 GPa in the same DAC.

In the following, we will consider various space groups: cubic
$Fm\overline{3}m$, trigonal $P\overline{3}c1$, tetragonal $P\overline{4}m2$;
and additionally hexagonal $P6_{3}cm$, $P6_{3}$, and $P6_{3}/mmc$, and
$Ia\overline{3}$ for Lu2O3 impurity. The groups respectively correspond to Oh,
D3d, D2d, C6v, C6, and D6h point groups for LuH3-based compound and $T_{h}$
for Lu2O3. The polarised Raman response was measured in crossed and parallel
configurations on non-crystalline samples. The only clear expected selection
rules concern the $A$ modes that are mainly Raman active in parallel
polarisations (except for C6).

##  III Experimental results

###  III.1 Imaging of the sample

Images of the sample in the DAC before (300 K, 1bar) and after (300 K, 1.92
GPa) loading are presented in Fig. 2. A white light was used to illuminate the
sample in reflection and in transmission. A ruby ball and a piece of silicon
were placed inside the pressure chamber. The sample appears translucent with a
red color at 1 bar and seems to become opaque at high pressure; however, this
could be due to the majority of the sample raising up off of the diamond
during loading. After loading with the mixture of He/N2 and pressurising to
1.92 GPa, the surface becomes reflective and blue. On the image c), we can
also see a red region. This part remained flat against the diamond and Raman
spectroscopy on it, so performed after loading and at 1.92 GPa, presents
trigonal structure of LuH3.

Figure 2: Image of the sample before (a and b) and after (c and d) loading; in
transmission (a and c) and in reflection (b and d), using a white light.

###  III.2 X-ray diffraction

Fig 3 presents the Rietveld fit of the XRD pattern measured on the trihydride,
which we kept protected from the air between two pieces of cellotape. We
identify the structure to be trigonal $P\overline{3}c1$ as expected for LuH3
at ambient pressure. The lattice parameters are a=6.1680(8)Å and c=6.422(1)Å
which a yield Lu-Lu distance in the range of 3.537(10)-3.828(6) Å. For
comparison, Tkacz et al. Tkacz and Palasyuk (2007) analysed that LuH3 has a
hexagonal ($P6_{3}/mmc$) phase with a= 3.57±0.01 Å and c = 6.41±0.02 Å at
ambient conditions. This yielded a Lu-Lu distance of 3.570 Å.

The powder is primarily LuH3 at 96.9(1)$\%$, and the rest was identified to be
impurities of Lu2O3. This is likely to originate from the deposits on the
lutetium surface that were not removed before hydrogenation. The space group
of Lu2O3 is $Ia\overline{3}$ and the refined lattice parameter is 10.380(8) Å
in agreement with the literature Lin et al. (2010).

Figure 3: Powder X-ray diffraction on the trigonal LuH3 (a) sample kept in
glove-box and sealed between two pieces of tape before and during the
measurement. (b) After up to 9 days of exposure to air. The quantity of
impurity Lu2O3 did not significantly change with time.

After the first XRD measurement, we left a fraction of the powder exposed to
air and measured the XRD periodically over the course of 9 days to check its
stability. The rest of the powder was immediately stored under vacuum or in an
argon glove box. As shown Fig.3 b), the oxide Lu2O3 content is the same within
the error bar, i.e. 3.4(1)$\%$ for 3.2(1)$\%$ before.

X-ray diffraction in the loaded DAC at 1.9 GPa is shown in figure 4. It was
measured at the ID15B line of the ESRF in five different spots with a size of
4x3 $\mu$m and separated by 20 $\mu$m in a cross-shape. The results on the
different spots are remarkably similar and indicate that the sample is
homogeneous in this region.

Figure 4: (a) X-ray diffraction pattern on the synthesised (N-doped) LuH3 at
300 K and 1.92 GPa measured at the ESRF (beamline ID15B). Inset: XRD results
on the 5 different spots. (b) XRD pattern of the trigonal LuH3 powder measured
on the same beamline at ambient pressure.

From the XRD results, the transformation to a cubic phase is clear. More
precisely, a first refinement at 1.9 GPa yielded two different cubic phases
with fcc $Fm\overline{3}m$ structure and lattice parameters of a2=4.9800(2) Å
(75-80$\%$ of the sample) and a1=5.128(1) Å(20-25$\%$ of the sample). Further
refinements considering single but more complex space groups are ongoing.

Tkacz et al. Tkacz and Palasyuk (2007); Kong et al. (2012) reported a
structural transition of LuH3 at room temperature from hcp to fcc at a
moderate pressure of $\sim$ 12 GPa, but much above our present pressure. The
transition to a cubic phase is unlikely due to pure LuH3 transformation. It
was previously shown in YH3 that a similar transition from hcp to fcc could be
induced via mechanical milling Kataoka et al. (2021). We are however confident
that our thinning process of repeatedly compressing the LuH3 between the
diamonds does not have a similar effect as milling. Indeed, Raman spectroscopy
after the thinning process clearly showed trigonal hcp structure. The
transformation to cubic happened only after the loading and application of 2
GPa.

We will now compare our XRD results with known phases in the Lu-H-N landscape
at room temperature and $\sim$ 2 GPa. It consists of $Fm\overline{3}m$ ammonia
NH3 Gauthier et al. (1986, 1988), fcc CaF2-type LuH2 ($Fm\overline{3}m$), fcc
rock-salt (NaCl-type $B_{1}$, $Fm\overline{3}m$) LuN, hexagonal LuHx
($P6_{3}/mmc$), oxide Lu2O3 ($Ia\overline{3}$). Firstly, ammonia is unlikely
to be detected here by XRD so none of the phases can originate from ammonia.
Nevertheless phonons of ammonia could be present in the Raman spectra; this
will be discussed in the next section. LuN has a lattice parameter of
a=4.7563(4) Å Suehiro et al. (2004) at ambient conditions which is then
unlikely to explain one of the two phases. The lattice parameters of
$Fm\overline{3}m$ LuH2 is reported to be a=5.033 Å at ambient conditions Yang
and An (2016); Bonnet and Daou (1977); Ming et al. (2023). This phase could
possibly explain the XRD pattern of the phase with the smallest lattice
parameter.

LuHx with various concentration up to x=0.2 are reported to be hexagonal
$P6_{3}/mmc$ with a=3.52 Å and c=5.6 Å Bonnet (1976); Daou and Bonnet (1974).
Our attempt to reproduce any of the two phases by this hexagonal compound
failed. Note that above x=0.2, cubic LuH2 is synthesised.

Lin et at. Lin et al. (2010) report that Lu2O3 transforms to monoclinic
structure under high pressure, above 17 GPa. As we have shown, this cubic
phase impurity is minor in our XRD pattern at ambient pressure, even less than
3$\%$ at the ESRF beamline measurement. Besides, as will be explained later,
we do not see any Raman signature of this phase so we rule out the possibility
that Lu2O3 is responsible for one of the two cubic phases observed at 1.9 GPa.

We now compare our XRD results with the published ones by Dasenbrock et al..
After high-pressure synthesis and releasing the pressure, Dasenbrock et al.
determined the ambient pressure lattice parameters of two distinct
$Fm\overline{3}m$ phases (named A and B) to be aA=5.0298(4) Å and aB=4.7529(9)
ÅDasenbrock-Gammon et al. (2023). Similar XRD patterns are obtained by us,
i.e. two cubic phases with slightly different lattice parameters.
Nevertheless, we note that our majority phase is the one with a smaller
lattice parameter. More surprisingly, the extracted lattice parameters for
both of our phases are larger than the ones of Dasenbrock et al., despite our
compound being under higher pressure. A tempting explanation might rely on the
synthesis process which, starting from pure LuH3, would tend to produce
compounds with higher hydrogen content, near to the trihydride.

Unfortunately, Powder XRD is largely insensitive to lighter atoms like
hydrogen (and nitrogen). In our case, this means that XRD determines the
sublattice of the heavy lutetium. Raman spectroscopy is then complementary as
a tool since although it does not directly provide numerical structural
values, the excitations depend on the global symmetry of the lattice and the
occupied Wyckoff positions. These excitations manifest themselves as phonons
that can originate from light or heavy atoms. Thus the number of observable
phonons can be used to predict viable global symmetries of the crystal.

###  III.3 Raman spectroscopy

We first recall the nature of the $\Gamma$-point phonons expected in the
various space groups under consideration. From the literature on YH3 and LuH3,
Kong et al. (2012); Palasyuk and Tkacz (2005); Kierey et al. (2001) the
crystal structure could correspond to $Fm\overline{3}m$ or $P\overline{3}c1$
space groups. Additional possibilities such as $P6_{3}cm$, $P6_{3}$, and
$P6_{3}/mmc$ are discussed in the literature on YH3 van Gelderen et al.
(2001); Kierey et al. (2001).

The Wyckoff positions and the Raman active modes for each space group are
written in table 1, and from these we expect a total of 5A1g\+ 12Eg Raman
active phonon modes in the trigonal $P\overline{3}c1$ phase, or a single
Raman-active T2g in the cubic structure of LuH3. In the latter structure,
three infrared-active modes T1u are predicted which could become Raman-active
with enough disorder . In the cubic phase, the T2g mode is associated with the
displacement of the hydrogen atoms which occupy the 8c sites.

| $P\overline{3}c1$ | $Fm\overline{3}m$ | $P6_{3}cm$ | $P6_{3}$ | $P6_{3}/mmc$
---|---|---|---|---|---
Lu | $6f$ | $4a$ | $6c$ | $6c$ | $2c$
H | $2a$ | $8c$ |  $6c$ X 2 |  $6c$ X 2 | $2a$
H | $4d$ | $4b$ | $4b$ |  $2b$ X 2 |
H | $12g$ |  | $2a$ | $2a$ |
|  5A1g + |  |  7A1 + |  11$A$+ |  A1g+
|  12Eg |  1 T2g |  11E1+12E2 |  11E1+12E2 |  E1g+2E2g
Table 1: Wyckoff positions of the specified atoms within the considered space
groups for LuH3 compound. Below are the total number of Raman active modes for
each space group.

Fig 5 shows the polarisation dependent Raman spectra of the ambient pressure
trigonal LuH3 below 955 $\mathrm{c}\mathrm{m}^{-1}$; at higher energies we do
not identify any excitations that clearly originate from LuH3. Within the
aforementioned range, we observe 13 features (marked by arrows) which could
account for most of the expected 17 phonons for the $P\overline{3}c1$ trigonal
structure. Overall, we do not observe any significant differences between the
different polarisations. The inset shows the low energy spectra down to 20
$\mathrm{c}\mathrm{m}^{-1}$ where we do not see any more notable features.

Figure 5: Raman susceptibility of LuH3 at 300 K and 1 bar, measured in the
unloaded DAC in cross and parallel configurations. Arrows point to the
phonons. Below 175 $\mathrm{c}\mathrm{m}^{-1}$, data are measured in triple-
stage with a 2400gr/mm grating and are scaled to overlay with the 2400gr/mm
single-stage data measured at higher energy. The inset shows the unscaled
triple-stage data at low energy. The raw data at ambient pressure from ref
Dasenbrock-Gammon et al. (2023) are shown in grey and are scaled to aid
comparison. Figure 6: Raman spectra of a) trigonal LuH3 in the unloaded DAC
and b) Synthesised (N-doped) LuH3 in the loaded DAC at 1.92 GPa. Data on two
different spots are reported which are shown in the inset image (for
additional data Cf. SI). Below 175 $\mathrm{c}\mathrm{m}^{-1}$, triple-stage
data are overlaid on the high-energy spectra. c) For comparison, the raw Raman
spectra of part $A$ of the sample from Dasenbrock et al. Dasenbrock-Gammon et
al. (2023) at ambient pressure and at 2.17 GPa are presented. d) and e) Scaled
data on the peak at 250 $~{}\textrm{cm}^{-1}$ with background correction, to
help the comparison, at $\sim$ 2GPa from our results and from Dasenbrock et
al. Dasenbrock-Gammon et al. (2023). Scaling is the same in d) and e).

Figures 6.a and 6.b show wide-range Raman spectra on the ambient pressure
trigonal LuH3 and the high-pressure structure of (N-doped) LuH3. Firstly, we
note that all of the phonons observed at ambient pressure have disappeared
which indicates a structural transition. In particular, for the cubic
$Fm\overline{3}m$ structure we only expect one Raman-active mode for the more
symmetric structure, hence the loss of modes is consistent.

We also observe a large increase in the background by a factor of $\sim$ 10
which seems to be intrinsic as we observe it in multiple regions of the
sample, as shown in figure 6.b. Despite this, we observe a couple of features
at high pressure. Most notably, we observe two peaks that consistently appear
at approximately 1240 $\mathrm{c}\mathrm{m}^{-1}$ and 250
$\mathrm{c}\mathrm{m}^{-1}$ which were not present at ambient pressure. We
note that the large increase of the background Raman response does not apply
to the peaks, i.e. the mode at 1240 $~{}\textrm{cm}^{-1}$ has similar
intensity (5-10 counts/s.mW) as the mode at 550 $~{}\textrm{cm}^{-1}$ at 0
GPa. This tends to show that the change in the Raman response when applying
pressure might be of electronic origin.

In the cubic phase of LuH3, a single T2g Raman active mode is expected. An
additional IR-active T1u is predicted at lower energy. Sun et al. calculate
the phonon spectrum of cubic LuH3 at 25 GPa Sun et al. (2023), and although
the pressure is different from the pressure observed here, they predict high
energy mode at 1334 $\mathrm{c}\mathrm{m}^{-1}$ and the other mode at lower
energy 467 $\mathrm{c}\mathrm{m}^{-1}$ (14 Thz). Though higher in energy, the
high energy modes probably correspond to the phonon we observe at 1240
$\mathrm{c}\mathrm{m}^{-1}$; this in turn corresponds to the Raman-active T2g
mode that we expect for this structure based on other work in rare-earth
trihydrides. The 14THz mode could correspond to our mode at 250
$\mathrm{c}\mathrm{m}^{-1}$, however, the energies are significantly different
and the remaining phonon modes for this cubic structure are T1u, which are
Raman-inactive. It is possible that these become Raman-active if a sufficient
amount of disorder is present in the system Granville et al. (2009). Most
importantly, we observe other excitations at an energy below 250
$~{}\textrm{cm}^{-1}$ that shall be discussed shortly.

For completeness, we shall discuss potential impurity phases. We recall that
Lu2O3 remains cubic until 12GPa Lin et al. (2010); Jiang et al. (2010). The
most intense Raman-active mode is measured at 390 $~{}\textrm{cm}^{-1}$ at
ambient pressure and hardens slightly up to 400 $~{}\textrm{cm}^{-1}$ at 2 GPa
Jiang et al. (2010). All the other modes are much less intense. Therefore we
are confident that our Raman-active modes are not due to Lu2O3 impurities
neither at ambient or high pressure. Yang et al. Yang and An (2016) calculated
the Raman active mode of LuH2 to be a single T2g mode between 960-1170
$~{}\textrm{cm}^{-1}$ at ambient pressure. We cannot rule out that the LuH2
phase explains our high-energy Raman mode, but such a scenario would be
unlikely. Indeed we know that we loaded trigonal LuH3. Thus to form a
measurable quantity of LuH2 would require a significant change in chemical
composition and such a transformation then depends on the thermodynamic
landscape between the two compositions, both of which are stable Huo et al.
(2023); Liu et al. (2023); Sun et al. (2023); Hilleke et al. (2023). Overall,
this seems like an unlikely scenario. Besides, it is clear that LuH2 cannot
explain the low energy modes.

At room temperature and 2 GPa, pure N2 may form either a fluid or a solid
$\beta$ phase. The $\beta$-phase crystallises in the $P6_{3}/mmc$ space group
Ouillon et al. (1990). A single mode is expected at $\sim$
2330$~{}\textrm{cm}^{-1}$ , which we observe as a narrow peak in this range of
energy. N2 gas has a similar vibron mode at high energy but also other peaks
at low energy below 150 $~{}\textrm{cm}^{-1}$ Ohno et al. (2021). Some modes
we measured might be ascribed to N2 gas but not the ones at 195
$~{}\textrm{cm}^{-1}$ and 166 $~{}\textrm{cm}^{-1}$ or clearly our dominant
modes at 1240 $\mathrm{c}\mathrm{m}^{-1}$ or 250 $\mathrm{c}\mathrm{m}^{-1}$.

At 2 GPa and ambient temperature, ammonia NH3 is expected to be in the
$Fm\overline{3}m$ space group or still fluid. Raman scattering under pressure
Gauthier et al. (1988) show that only modes at energy higher than 3100
$~{}\textrm{cm}^{-1}$ are measured; the spectra are similar in the liquid
phase. So we exclude that ammonia is responsible for the Raman modes we
measure.

The primary nitride that could form is LuN with a $Fm\overline{3}m$ rock-salt
(RS) structure. This is in principle Raman-inactive as only the 4a and 4b
sites are occupied. Despite this, a strong excitation was observed at 582
$~{}\textrm{cm}^{-1}$ but this was ascribed to be caused by strong disorder
Granville et al. (2009) and we do not identify such mode in our Raman data.
Note that the synthesis of LuN compound is challenging and previously required
heating pure lutetium and nitrogen at 1600°C Suehiro et al. (2004). Since we
have not laser heated our sample, we do not expect the formation of this
compound.

Finally, we compare our spectra to the raw data of Dasenbrock et al.
Dasenbrock-Gammon et al. (2023) (part A of the sample), as shown Fig. 5 at
ambient pressure. Their sample is measured after high-pressure synthesis,
therefore all comparisons are against the reported cubic, $Fm\overline{3}m$,
N-doped LuH3. We note that very few peaks agree with our data, such as the
ones at 100 $\mathrm{c}\mathrm{m}^{-1}$ and 148 $\mathrm{c}\mathrm{m}^{-1}$ as
shown in the inset. Most notably, the intense broad peak at 550
$\mathrm{c}\mathrm{m}^{-1}$ of trigonal LuH3 is absent in Dasenbrock et al.
data. Therefore we conclude that it is unlikely that trigonal LuH3 is present
in the Dasenbrock sample.

Nevertheless, when we compare our pressurised samples against one another
(figures 6.b and 6.c) we see that our high-energy peak at 1240
$\mathrm{c}\mathrm{m}^{-1}$ is clearly present in the Dasenbrock sample.

We also see that when we scale the two data on the 250
$\mathrm{c}\mathrm{m}^{-1}$ peaks (figure 6.d), the 1240 $~{}\textrm{cm}^{-1}$
modes have the same shape and energy. Besides, still with the same scaling, we
also see several peaks at low energy with similar intensities that are shared
by both samples. The energies of some modes are similar (Cf. SI) and some are
different (Cf. SI). It is then possible that we synthesise similar sample as
Dasenbrock et al. with slightly different stoichiometry.

The origin of these low energy Raman peaks is unclear. Dasenbrock et al.
reported that the primary impurity phase could be LuN in either an RS
structure or a zinc-blend (ZB) structure. As discussed earlier, the RS
structure is in principle Raman-inactive and is synthesised at high-
temperatures. The $F\overline{4}3m$ ZB structure has occupied 4a and 4c sites
that contribute one T2 Raman-active phonon each. This structure has been
predicted to form under pressures above 250 GPa Singh and Verma (2015);
Chouhan et al. (2011), and experimentally the RS structure has been shown to
form preferentially when synthesised at 30 GPa and 2000 K Niwa et al. (2009).
Overall we do not expect either structure of LuN to be present, and even if
they were present, neither structure could explain the vast number of low
energy peaks in the Dasenbrock sample and in our sample at 2 GPa.

##  IV Discussion

The low energy Raman modes in Dasenbrock et al. and in our sample at 2 GPa
appear to be a fundamental puzzle. Indeed, they cannot be explained by the
known structures in this temperature-pressure region, especially since the Lu
sublattices were identified to be cubic by XRD. Nor can these modes be
explained by the primary impurity phases either, as discussed above. As stated
earlier, X-rays preferentially scatter off of heavy atoms and interact far
less with light atoms like hydrogen and nitrogen compared to lutetium. As
clearly stated by Hilleke et al. Hilleke et al. (2023), the fcc sublattice of
the lutetium atoms provides a constraint about which we should search but it
does not necessarily describe the entire structure.

The Raman spectra clearly show this based on the shear number of low energy
peaks that were observed previously Dasenbrock-Gammon et al. (2023); Huo et
al. (2023) as well as by us. One way to observe a large number of phonon modes
from a structure with a global $Fm\overline{3}m$ symmetry is if the lower
symmetry sites (24e, 32f, 48g…) become occupied. Another way is that the
overall symmetry distorts whilst the fcc lutetium sublattice is preserved. It
is indeed possible if substitutions between Hydrogen and Nitrogen atoms
happen. Consequently the space group will have to be revisited. Further work,
both experimentally and theoretically, is necessary to unveil this space
group.

##  V Conclusion

We synthesise a compound, which presents structural similarities with the
sample of Dasenbrock et al. Dasenbrock-Gammon et al. (2023), starting from
pure trigonal LuH3 loaded in a DAC with a mixture of N2/He gas. From X-ray
diffraction, we clearly see a structural transformation from trigonal to cubic
under pressure. Similarly, with Raman spectroscopy we observe the loss of the
modes associated with the trigonal structure and see the appearance of a
strong mode at 1240 $\mathrm{c}\mathrm{m}^{-1}$ that we associate with the T2g
Raman-active mode of the cubic structure. However, we (and others) observe
other more excitations than is possible for a simple cubic structure. Overall
we believe that it is unlikely that these excitations come from impurity
phases since they are either not visible in XRD, chemically unlikely to form,
or simply their excitations do not occur in the energy range. Thus we conclude
that although the sublattice of the lutetium atoms is cubic, with potentially
two cubic phases in the sample, this does not describe the global symmetry of
the entire structure as a whole. It is then possible that a compound with
substituted H-N atoms was formed and could explain both the XRD and Raman
results. If the compound we synthesised is superconductor has still to be
studied.

###### Acknowledgements.

We thanks Sebastian Pairis for EDX measurements, Abdellali HADJ-AZZEM and
Elise PACHOUD for Lutetium preparation. We thanks Laetitia Laversenne for
fruitful discussions and Eva Zurek for stimulating exchanges of information.

## Competing interests

The authors declare no competing interests.

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