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# Spatio-Temporal Attention Graph Neural Network for Remaining Useful Life
Prediction
††thanks: This work is supported within the Digital-Twin-Solar (03EI6024E)
project, funded by BMWi: Deutsches Bundesministerium für Wirtschaft und
Energie/German Federal Ministry for Economic Affairs and Energy.

1st Zhixin Huang  Intelligent Embedded Systems
University of Kassel
Kassel, Germany
<EMAIL_ADDRESS>2nd Yujiang He  Intelligent Embedded Systems
University of Kassel
Kassel, Germany
<EMAIL_ADDRESS>3th Bernhard Sick  Intelligent Embedded Systems
University of Kassel
Kassel, Germany
<EMAIL_ADDRESS>
###### Abstract

Remaining useful life prediction plays a crucial role in the health management
of industrial systems. Given the increasing complexity of systems, data-driven
predictive models have attracted significant research interest. Upon reviewing
the existing literature, it appears that many studies either do not fully
integrate both spatial and temporal features or employ only a single attention
mechanism. Furthermore, there seems to be inconsistency in the choice of data
normalization methods, particularly concerning operating conditions, which
might influence predictive performance. To bridge these observations, this
study presents the Spatio-Temporal Attention Graph Neural Network. Our model
combines graph neural networks and temporal convolutional neural networks for
spatial and temporal feature extraction, respectively. The cascade of these
extractors, combined with multi-head attention mechanisms for both spatio-
temporal dimensions, aims to improve predictive precision and refine model
explainability. Comprehensive experiments were conducted on the C-MAPSS
dataset to evaluate the impact of unified versus clustering normalization. The
findings suggest that our model performs state-of-the-art results using only
the unified normalization. Additionally, when dealing with datasets with
multiple operating conditions, cluster normalization enhances the performance
of our proposed model by up to 27%.

###### Index Terms:

Spatio-Temporal Attention, Remaining Useful Life, Graph Neural Network, RUL
Prediction, Clustering Normalization

##  I Introduction

Predictive Health Management (PHM) stands out as a key tool for reducing
maintenance costs and enhancing system reliability and performance in complex
industrial systems [1]. PHM systems effectively identify component failures,
monitor conditions, and anticipate system breakdowns, assisting or even
independently initiating maintenance strategies [2]. Remaining Useful Life
(RUL) prediction is indispensable for efficacious PHM. The RUL of a system is
denoted as the time from the current time to the end of its useful life [3].
Precise RUL prognostications facilitate smart system maintenance, preemptively
address potential component fault, extend system life, and reduce maintenance
costs [4].

RUL prediction is mainly divided into physical model-based and data-driven
methods [5]. The former emphasizes modeling system degradation trajectories
using mathematical constructs. However, as the system’s complexity increases,
it becomes more and more difficult to establish physical models [6, 3]. As a
result, academic and industrial fields are increasingly leaning towards data-
driven approaches, with a particular emphasis on deep learning models [7, 8,
9, 10, 11]. The foundational premise here entails leveraging sensor recording
data with their corresponding RULs, utilizing these as input-output pairs, and
training a deep neural network. The network reveals complex relationships in
the data with its powerful nonlinear modeling capabilities.

Figure 1: The RUL prediction workflow and the spatio-temporal attention graph
neural network

The data-driven methods encompass Recurrent Neural Networks (RNN),
Convolutional Neural Networks (CNN), Graph Neural Networks (GNN), and hybrid
models. RNNs, and their advanced version Long Short-Term Memory (LSTM), are
popular for their adeptness in modeling temporal correlations, especially in
time series data, rendering them efficacious for RUL predictions [8, 12].
Concurrently, CNNs have garnered recognition in RUL prediction [13, 14],
wherein temporal features are extracted via 1D CNN, often resulting in a more
effective training process compared to LSTM [15, 16, 17]. Beyond the temporal
feature, it’s imperative to acknowledge the graph-based spatial relations to
the multivariate time series engendered by system sensors [4, 10]. In
addition, hybrid models were proposed to extract spatial and temporal
features, respectively by combining GNN and RNN or CNN. spatio-temporal graph
neural networks have many applications, such as traffic forecasting [18] and
activity prediction [19]. In recent research, [4] and [11] proposed to apply
the spatio-temporal graph neural network to RUL prediction and achieved state-
of-the-art (SOTA) performance. In [4], graph convolutional network (GCN) [20]
and temporal convolutional network (TCN) [21] were used in parallel to obtain
temporal features and spatial features, respectively, and the two features
were aggregated through product. Similar to [4], Zhu et al [11] connected GNN
and 1D CNN in parallel as the feature extractors.

Despite the advancements in RUL prediction methodologies, current strategies
exhibit avenues for further refinement. This research identifies three key
questions that will serve as a foundation for subsequent sections, aiming to
explore potential enhancements in RUL prediction accuracy.

Question 1: How does cascade integration of spatial and temporal feature
extractors affect predictive accuracy? SOTA methods, as discussed in
references [4, 11], predominantly parallelize the extraction of spatio-
temporal features, followed by a simple aggregation. An alternative could
involve a cascade arrangement where the spatial feature extractor first
filters the sensor data based on its graph structure. Subsequently, its output
would serve as the input for the temporal feature extractor, thereby allowing
a more nuanced feature extraction process.

Question 2: Can introducing a spatio-temporal attention mechanism amplify the
prediction’s performance? Attention mechanisms have gained traction for their
efficacy in highlighting critical information within neural network
computations, initially finding success in computer vision and later in
natural language processing [22]. Its extension to RUL prediction,
particularly by integrating spatio-temporal dimensions, remains partially
uncharted. Existing studies [7, 12, 9, 23] typically incorporate only one form
of attention, either spatial or temporal. Notably, the SOTA hybrid
methodologies [4, 11] have not yet adopted any form of attention mechanism.
Additionally, analyzing the attention probability matrix can provide insight
into the information that the predictive model deems most critical.

Question 3: How do varying operating conditions impact predictive accuracy? As
an illustration, two subsets of the C-MAPSS dataset [24] comprise 6 distinct
operating conditions. Sensors manifest disparate initial values and variation
ranges under these differing conditions. Examining prevailing literature
reveals a divergence in data normalization approaches during preprocessing.
While certain studies [25, 23, 17] have taken into account the implications of
operating conditions on data, but most data-driven methodologies [13, 7, 8,
12, 9, 4, 26, 11] resort to unified normalization. Unified normalization, as
depicted in Fig 2(a), can inadvertently introduce noise into the normalized
dataset. A comparison between Fig. 2(a) and Fig. 2(b) suggests that clustering
normalization, which factors in operating conditions, might increase
prediction accuracy. However, there are no existing works comparing the
impacts of divergent data preprocessing methodologies on prediction outcomes.

The contributions of this paper are delineated into three aspects: (1) In
addressing questions 1 and 2, we introduce the Spatial-Temporal Attention
Graph Neural Network (STAGNN). As depicted in Fig. 1, our model employs GNN
and TCN as spatial and temporal extractors, respectively. We seamlessly
integrate these two modules through a cascading approach and incorporate a
multi-head spatio-temporal attention mechanism, which enhances predictive
accuracy and increases model explainability. (2) In response to question 3, we
implement two distinct data preprocessing strategies to assess the influence
of operating conditions on predictive outcomes. Beyond the unified
normalization, we also introduce the cluster normalization technique. This
entails clustering the operating conditions and subsequently normalizing each
cluster independently. (3) Leveraging the C-MAPSS dataset, we conducted a
comprehensive experimental evaluation. Our findings demonstrate that, even in
the absence of cluster normalization, the proposed STAGNN model consistently
achieves SOTA results across all four sub-datasets. Notably, when cluster
normalization is incorporated during the data preprocessing phase, there’s an
augmentation in prediction performance by 26% in RMSE and 27% in Score.
Furthermore, to shed light on the model’s explainability, we provide
visualizations of STAGNN’s hidden feature representations and the attention
matrix.

##  II Methodology

###  II-A Problem Statement

In industrial systems, sensor networks typically yield data in the form of
time series signals. For the task of RUL prediction employing data-driven
models, deep learning architectures are utilized to extract degradation
information from these signals. As highlighted in Sec I, it is imperative to
account for the system’s operating conditions, which might encompass extreme
settings such as high or low temperatures and varying humidity levels. Time
series data produced under distinct operating conditions can exhibit
substantial variation. Therefore, an exclusive focus on sensor data might
compromise the model’s prediction accuracy.

In the context of this paper, the task of multi-sensor RUL prediction can be
defined as: given a multi-sensor time series signal, denoted as
$\boldsymbol{X}_{T}$, in conjunction with the system’s operating condition
$\boldsymbol{O}$, we aim to assemble an end-to-end RUL prediction workflow
$f()$, encompassing both data preprocessing and a deep regression model. The
multi-sensor signals, given $T$ observational instances, $\boldsymbol{X}_{T}$,
is formulated as:

$\boldsymbol{X}_{T}=\boldsymbol{[}\boldsymbol{x}_{T}^{1},\boldsymbol{x}_{T}^{2},\dots,\boldsymbol{x}_{T}^{S}]\in\R^{T\times
S},$ (1)

where $S$ represents the count of sensors. Assuming that the system’s
operating conditions remain static after startup, each sensor’s time series
signal $\boldsymbol{X}_{T}$ is generated with a specific operating condition.
The vector $\boldsymbol{O}$ encompasses $d$ conditions and can be denoted as
$\boldsymbol{O}=[o^{1},o^{2},\dots,o^{d}]\in\R^{d}$. Consequently, the RUL
prediction, $y$, at a given time $T$ is expressed as:

$y=f(\boldsymbol{X}_{T};\boldsymbol{O}).$ (2)

Given a dataset,
$\boldsymbol{D}=[(\boldsymbol{X}_{T}^{1};\boldsymbol{O}^{1};y^{1}),\dots,(\boldsymbol{X}_{T}^{N};\boldsymbol{O}^{N};y^{N})]$,
comprising $N$ sequences, our goal is to learn a mapping model to predict the
RUL accurately.

(a) Unified Normalization

(b) Clustering Normalization

Figure 2: Normalization results based on two methods. Sequences with different
colors represent the $N$ records of the 7th sensor.

###  II-B Clustering Normalization based on Operating Conditions

The disparity in the scale of data acquired by various sensors is often
significant. Direct utilization of this data may skew the model’s attention
towards features of larger scales, neglecting those of smaller scales. Such an
oversight can pose challenges during training and compromise prediction
accuracy. Consequently, to counterbalance the influence of disparate feature
scales, most RUL-related research undertakes the scaling of sensor data. Let’s
denote $\boldsymbol{D}^{i}_{\boldsymbol{X}}$ as the all data from the $i$-th
sensor of $\boldsymbol{X}_{T}$ within $\boldsymbol{D}$. The common method is
unified normalization, i.e., the max-min normalization applied individually to
each $\boldsymbol{D}^{i}_{\boldsymbol{X}}\in\R$, which scales the data within
the interval $[0,1]$. However, a single dataset often contains multiple
operating conditions; that is,
$\boldsymbol{O}^{i}\neq\boldsymbol{O}^{j},\exists{i,j}$. As an illustrative
example, the FD002 and FD004 datasets from C-MAPSS contain time series data
spanning six operating conditions.

However, many existing works overlooked the impact of operating conditions
during the preprocessing phase. Consider Fig 2(a), where preprocessing of
FD002 is executed using a unified normalization. We can observe that the time
series, even under identical sensors, appears quite noisy, exhibiting
pronounced fluctuations. Such inconsistencies can potentially impair the
predictive performance of the RUL model.

A promising strategy involves clustering based on operating conditions,
followed by individual normalization for each cluster. Let’s represent the
clustering model as $\mathbb{M}$ and designate $D_{\boldsymbol{O}}$ as
$[\boldsymbol{O}^{1},\dots,\boldsymbol{O}^{N}]\in\R^{d\times N}$, indicating
the unique operating conditions associated with each time series. Upon
clustering $\boldsymbol{D}_{\boldsymbol{O}}$, the output from $\mathbb{M}$
yields a label set $\\{c_{k}^{1},c_{k}^{2},\dots,c_{k}^{N}\\}$, where $c_{k}$
signifies the $k$-th cluster. A cluster, encompassing sequences sharing an
identical cluster label, can be defined as:

$\boldsymbol{D}_{c_{k}}=[(X^{l}_{T},O^{l},y^{l}),...]\subseteq\boldsymbol{D},\forall{l}~{}\mathbb{M}(O^{l})=c_{k}.$
(3)

For each cluster, we perform max-min normalization to scale the data.

$X_{T_{norm}}^{i}=\frac{X_{T}^{i}-\min{D_{c_{k}}^{i}}}{\max{D_{c_{k}}^{i}}-\min{D_{c_{k}}^{i}}},$
(4)

where $i$ denote the $i$-th sensor, $X_{T}^{i}$ and $X_{T_{norm}}^{i}$ stand
for raw and normalized time sequences of sensor $i$. For the $i$-th sensor
within cluster $c_{k}$, the maximum and minimum values are respectively
denoted as $\max{D_{c_{k}}^{i}}$ and $\min{D_{c_{k}}^{i}}$.

The clustering normalized time series is depicted in Fig 2(b). When comparing
with Fig 2(a), it becomes evident that the time series, across varying
operating conditions, manifest more consistent scales. We hypothesize that
employing the clustering normalization approach can increase the RUL
prediction accuracy of the model. Sec III-E delves into the effects of cluster
normalization on prediction, verified by comparative experimental results.

###  II-C Spatio-Temporal Attention Graph Neural Network

The proposed STAGNN is primarily designed to extract the spatio-temporal
relationships inherent in sensor sequences for RUL prediction. As illustrated
in Fig 1, the model comprises three distinct modules: spatial feature
extraction module, temporal feature extraction module, and prediction module.

Spatial Feature Extraction Module: This module employs GCN layers combined
with a spatial attention mechanism. The graph is characterized by the equation
$G=G(\boldsymbol{X}_{T},\boldsymbol{A},\boldsymbol{E})$. In line with Sec.
II-A, $\boldsymbol{X}_{T}\in\R^{T\times S}$ signifies a multivariate sensor
signal series where $T$ represents the time series length, and $S$ accounts
for the number of sensors, considered as nodes within this graph
representation. The set of edges in the graph is represented by $E$. The
adjacency matrix, $\boldsymbol{A}\in R^{S\times S}$, reflects connections
between sensors, with $\boldsymbol{A}{ij}=(v_{i},v_{j})\in E$ denoting a link
between sensors $v_{i}$ and $v_{j}$. Given the absence of prior topological
knowledge in data-driven RUL prediction methodologies, sensor relationships
cannot be provided explicitly. Therefore, the adjacency matrix is formulated
by assessing the covariance between sensors, as described by

$\boldsymbol{A}_{ij}=\left\\{\begin{matrix}1,\text{if~{}}|cov(\boldsymbol{x}_{T}^{i},\boldsymbol{x}_{T}^{j})|>\lambda\\\
0,\text{if~{}}|cov(\boldsymbol{x}_{T}^{i},\boldsymbol{x}_{T}^{j})|<\lambda\end{matrix}\right.$
(5)

where $\lambda$ is the threshold ranging from $0$ to $1$, a higher threshold
implies that connections in the graph are reserved for sensors with closer
relationships. We add a multi-head spatial attention layer after each GCN.
This attention mechanism was first proposed by [27].

The output from a GCN layer, symbolized as $\boldsymbol{H}_{g}\in\R^{n\times
S}$, and $\boldsymbol{H}_{g}^{i}\in\R^{n}$ represents the sequence for $i$-th
sensor. $\alpha_{ij}$ is the attention coefficient on sensors $v_{i}$ and
$v_{j}$, is derived as:

$\alpha_{ij}=\frac{exp\left(\text{LeakReLU}\left(\boldsymbol{W}_{g}^{T}\left[\boldsymbol{H}_{g}^{i}\parallel\boldsymbol{H}_{g}^{j}\right]\right)\right)}{\sum_{u\in\mathbb{N}_{i}}exp\left(\text{LeakReLU}\left(\boldsymbol{W}_{g}^{T}\left[\boldsymbol{H}_{g}^{i}\parallel\boldsymbol{H}_{g}^{u}\right]\right)\right)},$
(6)

where $\boldsymbol{W}_{g}\in\R^{2n}$ is the weight matrix affiliated with the
spatial attention layer, $u\in\mathbb{N}_{i}$ means the neighbors of node $i$,
$\parallel$ stands for the concatenation operation and LeakReLU is a nonlinear
activation function. Based on the calculated spatial attention coefficient,
the $i$-th node’s output of the multi-head spatial attention layer
$\boldsymbol{H}_{g}^{i\text{'}}$ is defined as:

$\boldsymbol{H}_{g}^{i\text{'}}=\frac{1}{M}\sum_{m=1}^{M}\sum_{j\in\mathbb{N}_{i}}\alpha_{ij}^{m}\boldsymbol{H}_{g}^{j}$
(7)

where $m$ enumerates the number of multi-head attention, and $\alpha_{ij}^{m}$
corresponds to the attention coefficient derived by the $m$-th attention head.

Temporal Feature Extraction Module: We incorporate TCN layers, which offer
superior training efficiency compared to LSTM [21]. For temporal feature
extraction, an effective strategy is to emphasize distinct regions of interest
by allocating varied weights to time steps. Therefore, a multi-head temporal
attention mechanism is appended to each TCN layer. This facilitates the
discernment of the significance at different time steps.

Assume that the features procured from the TCN network for a given sample are
represented as $\boldsymbol{H}_{t}\in\mathbb{R}^{n\times S}$. The importance
vector, $\boldsymbol{\beta}\in\mathbb{R}^{n}$, denotes the significance of one
sensor across $n$ time steps and is expressed as:

$\boldsymbol{\beta}=\frac{\exp(\text{Sigmod}(\boldsymbol{H}_{t}\boldsymbol{W}_{t}+\boldsymbol{b}))}{\sum\exp(\text{Sigmod}(\boldsymbol{H}_{t}\boldsymbol{W}_{t}+\boldsymbol{b}))},$
(8)

where $\boldsymbol{W}_{t}\in\mathbb{R}^{S}$ and $b$ are respectively the
weight and bias parameters of the temporal attention layer. The resultant
output from the multi-head temporal attention layer can be delineated as:

$\boldsymbol{H}_{t}^{\text{'}}=\frac{1}{M}\sum_{m=1}^{M}\boldsymbol{H}_{t}\otimes\boldsymbol{\beta^{m}}.$
(9)

Here, $M$ stands for the count of multi-head attentions, and
$\boldsymbol{\beta}^{m}$ embodies the attention coefficients for the $m$-th
head, illustrated as
$\boldsymbol{\beta}^{m}=[\boldsymbol{\beta}_{1}^{m},\boldsymbol{\beta}_{2}^{m},\dots,\boldsymbol{\beta}_{S}^{m}]$.
The operation $\otimes$ is defined as an element-wise multiplication.

Prediction Module: This module fuses the extracted features. It mainly
contains fully connected layers to get the final RUL prediction.

##  III Experiment

###  III-A Dataset Description

The Commercial Modular Aerospace Propulsion System Simulation (C-MAPSS)
dataset serves as the benchmark for assessing the efficacy of the proposed
methodology. This dataset originates from an aircraft engine degradation
simulation tool designed to emulate the engine degradation trajectory across
various operational conditions. As delineated in Tab. I, the dataset contains
four subdatasets covering different operating conditions and is divided into
training and testing subsets. Each record includes 26 features involving
engine ID, operating time in cycles, 3 operating condition settings ($d$=3 in
$\boldsymbol{O}$) and 21 sensors ($S$=21 in $\boldsymbol{X}_{T}$).

TABLE I: The details of C-MAPSS dataset Subsets | FD001 | FD002 | FD003 | FD004
---|---|---|---|---
Number of Training Engines | 100 | 260 | 100 | 249
Number of Test Engines | 100 | 259 | 100 | 248
Number of Operation Conditions | 1 | 6 | 1 | 6

###  III-B Data Preprocess

Data preprocessing in the study is split into two principal stages: data
normalization and data segmentation via sliding windows. The normalization
procedure is described comprehensively in Sec. II-B. As shown in Fig. 2,
unified normalization might influence the prediction accuracy for RUL datasets
across varying operational conditions. To discern the difference between the
unified and clustering normalization, both techniques were employed to
normalize the data. Subsequently, two types of normalized datasets were
respectively trained on our proposed model with identical architecture. A
detailed comparative analysis and results are described in Sec. III-E.

In alignment with existing RUL prediction research, sliding windows are
utilized to segment the primary data [28, 29]. Given a multivariate time
series $\boldsymbol{X}_{T}$ as detailed in Sec. III-A, where the sequence
length is denoted by $T$, we designate the window’s length as $w$ and the
stride of the window as $k$. This segmentation strategy splits
$\boldsymbol{X}_{T}$ into $m$ subsequences, represented as
$\\{\boldsymbol{X}_{m}\\}_{m=1}^{\lfloor\frac{T-w}{k}\rfloor+1}$. Each
subsequence’s RUL, $y_{m}$, is defined as $y_{m}=T-w-(m-1)\times k$. A piece-
wise linear degradation model was implemented to cap the RUL’s maximal value,
with a threshold set at 125, as used by [28, 29]. The operating conditions and
sensor measurements constituted the final input vectors for the prediction
models.

###  III-C Evaluation Metrics

RMSE is widely used for RUL prediction as it is a common evaluation metric in
regression tasks. RMSE is calculated as:

$\text{RMSE}=\sqrt{\frac{\sum_{i=0}^{N-1}(y_{i}-\hat{y}_{i})^{2}}{N}},$ (10)

where $N$ is the number of predicted samples, $y_{i}$ and $\hat{y}_{i}$
represent the actual RUL value and the predicted RUL value for the predicted
sample respectively. In addition, Sateesh et. al [30] proposed a scoring
function and is widely used in RUL prediction research, which is defined as

$\text{Score}=\left\\{\begin{matrix}\sum_{i=1}^{N}(e^{-\frac{\hat{y}_{i}-y_{i}}{13}}-1),\text{if~{}}\hat{y}_{i}<y_{i}\\\
\sum_{i=1}^{N}(e^{\frac{\hat{y}_{i}-y_{i}}{10}}-1),\text{if~{}}\hat{y}_{i}\geqslant
y_{i}\end{matrix}\right.$ (11)

Unlike RMSE which can not distinguish between early and delayed predictions,
the scoring function will bring more penalties to delayed RUL predictions,
which is more in line with actual application scenarios. In this article, we
used these two criteria to evaluate our proposed model’s performance.

###  III-D Experiment Results

(a) Prediction Result of STAGNN* on Test Dataset of FD003

(b) Prediction Result of STAGNN* on Test Dataset of FD004

Figure 3: True RUL and the predicted RUL by the proposed approach on the two
test datasets.

As illustrated in Fig. 1, our model incorporates a two-layer Attention Graph
Neural Network (AGNN) with a hidden feature dimension of 64. The AGNN, which
is responsible for spatial feature extraction, feeds its output into an
attention-based TCN block (ATCN) designed for temporal feature extraction.
This ATCN comprises two layers of TCN-Attention structures with hidden
dimensions of 64 and 10, respectively. The ATCN has a kernel size of two and
employs a dropout rate of 0.5. Its output serves as the feature representation
derived from the hidden layers. To delve into the model’s explainability, we
visualize this hidden feature representation in Sec. III-G. Subsequently, this
representation is channeled through a fully connected layer to get RUL
predictions.

In line with the methodology presented in [12], we employ a sliding window of
length $w=50$ for data segmentation, with a step size $k=1$. The Adam
optimizer is utilized with a learning rate set to $0.001$. Our model is
trained using batches of size 100 over 100 epochs. For each dataset, we
conducted ten trials and subsequently computed the mean values for both RMSE
and Score.

TABLE II: Performance Comparison on C-MAPSS dataset | FD001 | FD002 | FD003 | FD004
---|---|---|---|---
Approaches | RMSE | Score | RMSE | Score | RMSE | Score | RMSE | Score
DCNN [13] | 12.61 | 273.7 | 22.36 | 10412 | 12.64 | 284.1 | 23.31 | 12466
HALSTM [7] | 14.53 | 322.4 | N/A | N/A | N/A | N/A | 27.08 | 5649.1
RNN [8] | 13.58 | 228 | 19.59 | 2650 | 19.16 | 1727 | 22.15 | 2901
AGCNN [12] | 12.42 | 225.5 | 19.43 | 1492 | 13.39 | 227 | 21.50 | 3392
Transformer [9] | 11.27 | N/A | 22.81 | N/A | 11.42 | N/A | 24.86 | N/A
GCN [4] | 12.76 | 266 | N/A | N/A | 12.07 | 278 | N/A | N/A
LSTMBS [16] | 14.89 | 481.1 | 26.86 | 7982 | 15.11 | 493.4 | 27.11 | 5200
STAGNN | 11.50 | 187.2 | 16.92 | 1132.2 | 11.05 | 196.0 | 19.33 | 1443.4
DLSTM* [25] | 12.29 | N/A | 17.87 | N/A | 14.34 | N/A | 21.81 | N/A
HAGCN* [23] | 11.93 | 222.3 | 15.05 | 1144.1 | 11.53 | 240.3 | 15.74 | 1218.6
BDL* [17] | 18.6 | 2774 | 22.9 | 7734 | 27.9 | 19990 | 28.1 | 53295
GAT* [10] | 13.21 | 303.1 | 17.25 | 5338.8 | 15.36 | 507.5 | 21.44 | 2971.9
RGCNU* [11] | 11.18 | 173.5 | 16.22 | 1148.16 | 11.52 | 225.0 | 19.11 | 2215.9
STAGNN* | 11.50 | 187.2 | 13.81 | 826.3 | 11.05 | 196.0 | 14.30 | 1038.5

Tab. II presents the performance outcomes of recent RUL-related studies. As
highlighted in Sec I, previous research has employed various data
preprocessing techniques. To ensure a fair comparison, we have categorized the
results into two primary groups: unified normalization and cluster
normalization. The asterisk (*) denotes methods utilizing cluster
normalization. While RGCNU [11] does not specify its preprocessing approach,
based on its referenced literature, we placed it under the cluster
normalization category. STAGNN and STAGNN* represent the performances of our
proposed method within the respective groups. Notably, STAGNN demonstrates
SOTA performance even in the absence of cluster normalization. Our method’s
RMSE for FD001 is slightly lower than that of Transformer [9] and RGCNU* [11],
but surpasses these two methods in more complex datasets such as FD002 and
FD004 by up to 25.8%. This underscores our method’s robustness in predicting
RUL under complex operating conditions. These comparative insights address
question 1 from Sec. I, confirming that a model incorporating a cascaded
spatio-temporal feature extractor can also achieve SOTA results with a more
streamlined structure than another hybrid model by [4].

###  III-E Analysis the Effectiveness of Clustering Normalization

Utilizing cluster normalization in the data preprocessing phase significantly
enhances the RUL prediction performance of STAGNN* on the FD002 and FD004
datasets, which contain multiple operating conditions. As shown in Tab. II,
RMSE and Score achieved improvements of up to 26% and 27%, respectively. These
findings affirm question 3 from Sec. I: cluster normalization effectively
mitigates the influence of varied operating conditions on model predictions.
Although numerous studies, as indicated in Tab. II, focus on end-to-end models
aiming for performance boost through model refinements, the pivotal role of
feature engineering is often overlooked.

###  III-F Model Ablation Study

TABLE III: Performance Comparison for Model Ablation | FD001 | FD002 | FD003 | FD004
---|---|---|---|---
Approaches | RMSE | Score | RMSE | Score | RMSE | Score | RMSE | Score
GNN | 13.07 | 276.6 | 17.95 | 1545.8 | 11.55 | 207.8 | 20.66 | 2265.4
AGNN | 12.61 | 270.1 | 17.91 | 1244.1 | 11.31 | 204.2 | 20.08 | 2069.1
TCN | 11.63 | 199.8 | 22.09 | 2968.8 | 11.18 | 208.7 | 23.42 | 3020.4
ATCN | 11.75 | 218.4 | 21.88 | 2193.9 | 13.34 | 249.7 | 22.18 | 2436.4
STGNN | 11.55 | 192.6 | 17.63 | 1314.3 | 11.18 | 207.1 | 20.05 | 1771.1
STAGNN | 11.50 | 187.2 | 16.92 | 1132.2 | 11.05 | 196.0 | 19.33 | 1443.4

Ablation analysis of the model aids in evaluating the contribution of each
component to the RUL prediction. We decomposed the proposed STAGNN model into
five distinct sub-models for comparative assessment. These are: a two-layer
GNN dedicated solely to spatial feature extraction; an AGNN, which is a GNN
enhanced with spatial attention; a two-layer TCN focusing exclusively on
temporal feature extraction; an ATCN, which is a TCN augmented with temporal
attention; and a cascaded GNN-TCN model devoid of any attention mechanism,
termed STGNN. Unified normalization was applied during data preprocessing to
assess the model’s adaptability to complex datasets.

We discern three key observations based on Table III. Firstly, while the GNN
surpasses the TCN by a margin of up to 25% in terms of Score on intricate
datasets FD002 and FD004, the outcomes are inversed for FD001 and FD003. This
suggests that spatial features become more influential for RUL predictions on
complex datasets. Secondly, the cascaded STGNN consistently outperforms
standalone GNN and TCN models, particularly in terms of Score. This
underscores the significance of both spatial and temporal features in RUL
prediction. Lastly, comparing models with and without attention mechanisms
reveals that incorporating attention increases RUL prediction accuracy on
complex datasets, addressing question 2 from Sec. I. The attention mechanism
also potentially enhances model explainability, briefly analyzed in Sec.
III-G.

###  III-G Explainability Exploration

Deep neural networks are powerful but often operate as black boxes, making
their decision-making processes opaque to humans. Consequently, model
explainability has gained prominence. Visualizing both the hidden feature
representation and the attention matrix might offer insights into dissecting
model decisions.

(a) STAGNN

(b) STAGNN*

Figure 4: The t-SNE-based visualization of the feature representations for
STAGNN and STAGNN*. The color and size represent the degradation of RUL

Fig. 4(a) and Fig. 4(b) present a t-SNE-based 2D visualization of the feature
representations for STAGNN and STAGNN*, respectively, under the FD002 dataset.
Each plotted point corresponds to the model’s feature representation at a
specific RUL, with the point’s color and size representing the degradation of
RUL. In Fig. 4(a), the feature representation for STAGNN follows a spiral
trajectory as the RUL decreases. Notably, several points deviate from this
trajectory. While STAGNN can adapt to noisy data, it is not entirely immune to
noise-induced prediction. In contrast, the trajectory in Fig. 4(b),
representing STAGNN*, is more distinct. Observing Fig. 4(b) closely, we note
significant jumps when the RUL is approximately 70 and 35. This suggests
potential abrupt changes in a component state around 70 and 35, which could
lead to rapid performance degradation of the entire system.

(a) Spatial Attention

(b) Temporal Attention

Figure 5: Samples of spatial and temporal attention matrix.

Fig. 5(a) and Fig. 5(b) present visualizations of the spatial and temporal
attention matrix, respectively. The spatial attention matrix illustrates the
attention between sensors. A large weight indicates a stronger connection
between the components denoted by the two sensors, suggesting a more
pronounced influence on the RUL. In addition, the temporal attention weight
matrix demonstrates the impact of a sensor on the RUL at specific time steps
in the series. Such insights could guide human experts in identifying root
causes and directing maintenance efforts toward relevant system components.

##  IV Conclusion

In this study, we address three questions related to RUL prediction: 1) the
efficacy of a cascading spatio-temporal feature extractor structure; 2) the
influence of the attention mechanism on prediction accuracy; and 3) the effect
of operating conditions on RUL prediction.

We introduced STAGNN, a cascading spatio-temporal feature extractor that
integrates GNN and TCN. This model also incorporates tailored attention
mechanisms corresponding to different feature extractors. Our results
demonstrated SOTA performance in comparison to existing works. Furthermore, by
employing clustering normalizing based on operating conditions, we
significantly enhanced the RUL prediction accuracy in datasets containing
diverse operating conditions. We visualized the hidden layer feature
representation and the attention matrix to explore the model transparency.

Our future work will encompass evaluating prediction uncertainties using
Bayesian neural networks and delving into techniques for model explainability,
aiming to quantify the explanation result.

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# Dynamics of azimuthal thermoacoustic modes
in imperfectly symmetric annular geometries

Thomas Indlekofer<EMAIL_ADDRESS>Department of Energy and Process
Engineering, Norwegian University of Science and Technology, Trondheim, Norway
Abel Faure-Beaulieu  CAPS Laboratory, Department of Mechanical and Process
Engineering, ETH Zürich  James R. Dawson  Department of Energy and Process
Engineering, Norwegian University of Science and Technology, Trondheim, Norway
Nicolas Noiray<EMAIL_ADDRESS>CAPS Laboratory, Department of Mechanical and
Process Engineering, ETH Zürich

###### Abstract

Using a nominally symmetric annular combustor, we present experimental
evidence of a predicted spontaneous symmetry breaking and an unexpected
explicit symmetry breaking in the neighborhood of the Hopf bifurcation, which
separates linearly-stable azimuthal thermoacoustic modes from self-oscillating
modes. We derive and solve a multidimensional Fokker-Planck equation to
unravel a unified picture of the phase space topology. We demonstrate that
symmetric probability density functions of the thermoacoustic state vector are
elusive, because the effect of asymmetries, even imperceptible ones, is
magnified close to the bifurcation.

###### pacs:

Valid PACS appear here

Symmetry breaking in systems where waves propagate along closed loops or
spheres occur in a wide variety of physical problems. Examples range from
Bose-Einstein condensates in toroidal traps Kolář  _et al._ (2015); Marti  _et
al._ (2015), whispering gallery modes in optomechanical resonators Shen  _et
al._ (2016); Ceppe  _et al._ (2019); M. T. M. Woodley, L. Hill, L. Del Bino,
G.-L. Oppo and P. Del’Haye (2021), counter-propagating modes in optical ring
cavities Megyeri  _et al._ (2018), nonreciprocal acoustic circulators Fleury
_et al._ (2014), intrinsic rotation in Tokamak plasmas Rice  _et al._ (2011),
to sporadic reversal in direction of planetary dynamo waves Pétrélis  _et al._
(2009); Sheyko  _et al._ (2016). Symmetry breaking is also ubiquitous in fluid
mechanics Crawford and Knobloch (1991), for example in von-Kármán swirling
flows with counter-rotating toroidal vortices de la Torre and Burguete (2007);
López-Caballero and Burguete (2013); D. Faranda, Y. Sato, B. Saint-Michel, C.
Wiertel, V. Padilla, B. Dubrulle and F. Daviaud (2017). Recently, symmetry
breaking was observed for azimuthal thermoacoustic modes in annular combustors
typically found in jet engines or power generation gas turbines. First
described by Lord Rayleigh Rayleigh (1878), thermoacoustic instabilities that
arise from a constructive coupling between the acoustic field and the heat
release rate of the flames lead to damaging vibrations. Azimuthal
thermoacoustic modes have been investigated in real engines Krebs  _et al._
(2002); Seume  _et al._ (1998); Noiray and Schuermans (2012a), laboratory
experiments Worth and Dawson (2013a, 2017); Bourgouin  _et al._ (2015a) and
numerical simulations Wolf  _et al._ (2012). Over the last decade, the dynamic
nature of these modes has received significant attention Bauerheim  _et al._
(2016), and the following classification has been established: standing modes,
whose nodal line direction remains constant or slowly drifts around the
annular chamber, spinning modes, whose nodal line spins at the speed of sound,
or a combination of both referred to as mixed modes can exist. Several studies
focused on the effects of clearly identified combustor asymmetries, e.g. Worth
and Dawson (2013b); Bauerheim  _et al._ (2014); Bourgouin  _et al._ (2015b);
Berger  _et al._ (2018); J.-W. Kim, W. Gillman, B. Emerson, D. Wu, T. John, V.
Acharya, M. Isono, T. Saitoh and T. Lieuwen (2021), and several reduced-order
models Evesque  _et al._ (2003); Schuermans  _et al._ (2006); Noiray  _et al._
(2011); Ghirardo  _et al._ (2016); Magri  _et al._ (2016); Hummel  _et al._
(2018); Moeck  _et al._ (2018); Orchini  _et al._ (2019); Yang  _et al._
(2019); Buschmann  _et al._ (2020) were proposed to explain experimental
observations with limited success. Notably, a recently proposed ansatz for
describing the acoustic field Ghirardo and Bothien (2018) was combined with
the wave equation to describe an ideal annular combustor, unifying previous
theoretical frameworks Faure-Beaulieu and Noiray (2020). However, the
transition from linearly stable to self-oscillating azimuthal thermoacoustic
modes has never been investigated. In this letter, we fill this gap with
experimental data showing that the route to a self-sustained quasi-pure
spinning mode passes through stationary states that are characterized by a
statistically prevailing standing mode and intermittent reversals of spinning
direction. We also build upon the model from Faure-Beaulieu and Noiray (2020)
to unravel the complex topology of the phase space  to demonstrate that
symmetrically designed combustors will always display explicit symmetry
breaking of the thermoacoustic modes at the bifurcation.

(a)
---
(b)
---
(c)
---
Cylindrical
---
plenum
---
12 turbulent
---
flames
---
12 burner
---
tubes
---
H2-CH4-Air mixture
---
max
---
min
---
$\dot{Q}$
---
$\Theta$
---
Time
---

Annular

combustion

chamber

Acoustic

pressure

transducers

Acoustic

pressure

Figure 1: (a) Model gas turbine combustor. (b) Acoustic pressure recorded with
6 transducers. During the selected interval, the thermoacoustic limit-cycle is
a quasi-purely spinning eigenmode. (c) Phase-averaged flame chemiluminescence.

Experimental evidence of explicit and spontaneous symmetry
breaking.—Experiments were conducted using an annular combustor with 12
premixed flames operated at atmospheric condition (see Fig. 1) and fueled with
a mixture of $70\text{\,}\mathrm{\char 37\relax}$ H2 and
$30\text{\,}\mathrm{\char 37\relax}$ CH4 by power. The thermal power was fixed
to 72 kW and the equivalence ratio was varied between 0.4875 and 0.575. Twelve
pressure transducers were mounted at six azimuthal positions flush with the
inner wall of the burner tubes sup  and recorded with a sampling rate of
$51.2\text{\,}\mathrm{kHz}$.

800
---
1200
---
1600
---
2000
---
2400
---
Frequency (Hz)
---
Time (s)
---
Time(s)
---
$P$
---
$P$
---
100
---
125
---
150
---

SPL (dBa)
---

1080
---
1115
---
120
---
150
---
-1
---
-1
---
0
---
0
---
1
---
1
---

$p$ (kPa)
---

$p$ (kPa)
---

0
---
1
---

$A$ (kPa)
---

$-\pi/4$
---
$\pi/4$
---

$\chi$
---

0
---
$\pi$
---

$\theta$
---

94
---
96
---
98
---
100
---
$-\pi$
---
$\pi$
---

$\varphi$
---

0.5
---
(a)
---
$\Delta{t}=$2\text{\,}\mathrm{ms}$$
---
(b)
---
(c)
---
(d)
---
I)
---
II)
---
I)
---
II)
---
0.525
---
0.55
---
0.575
---
0∘
---
120∘
---
240∘
---
94
---
96
---
98
---
100
---
Figure 2: (a) SPL for increasing $\Phi$. (b) Selected intervals for
$\Phi=0.55$, during which the azimuthal mode is mixed, with CW and CCW
spinning directions. The angles in the legend correspond to the locations of
the transducers. (c) Evolution of slow-flow variables that define the state of
the azimuthal thermoacoustic mode, during the last 6 seconds of the record for
$\Phi=0.55$, together with their PDFs. (d) PDFs and time traces from the model
with combined resistive and reactive asymmetries.

The natural chemiluminescence of the flames viewed from the exhaust is
recorded with a high speed camera using the same optical system as in Worth
and Dawson (2013b). For each $\Phi$, the combustor was ignited, run for at
least 60 s to reach thermal equilibrium before recording 100 s of acoustic
data while keeping the operating condition constant, and then turned off. The
sound pressure level (SPL) in the annular combustor is plotted in Fig. 2(a)
for four $\Phi$. When $\Phi=0.5$, the thermoacoustic system is linearly
stable. The system’s strongest resonance corresponds to an eigenmode whose
azimuthal distribution exhibits one acoustic wavelength along the annular
chamber circumference, and whose frequency is about 1090 Hz. The resonance
oscillation of the different modes is sustained by the inherent broadband
acoustic forcing from the turbulent combustion process. For larger $\Phi$, the
acoustic energy production resulting from the constructive thermoacoustic
feedback exceeds the acoustic energy dissipation, and the dominant eigenmode
becomes linearly unstable. Under these conditions, thermoacoustic limit cycles
are reached at the aforementioned resonance frequency, e.g. Noiray and
Schuermans (2012b). Their dynamics are governed by the nonlinear coherent
response of the flame to the acoustic field and by the turbulent fluctuations
of the heat release rate. Figure 2(b) presents the acoustic pressure $p$ at
three azimuthal locations during two short intervals (I) and (II) of the 100 s
record for $\Phi=0.55$, for which the self-sustained azimuthal thermoacoustic
mode spins in the counterclockwise (CCW) and in the clockwise (CW) direction.
Recently, a new ansatz for the acoustic pressure field based on quaternions
was proposed to describe azimuthal eigenmodes in annular combustors Ghirardo
and Bothien (2018). A set of four state variables $A,\theta,\chi$ and
$\varphi$, which vary slowly with respect to the fast acoustic time scale
$2\pi/\omega$, and which can be extracted from the acoustic pressure time
series, allows the following well-defined description of the state of an
azimuthal thermoacoustic mode

$\tilde{p}\left(\Theta,t\right)=A\left(t\right)\>e^{i\left(\theta\left(t\right)-\Theta\right)}\>e^{-k\chi\left(t\right)}\>e^{j\left(\omega{t}+\varphi\left(t\right)\right)},$
(1)

where $t$ and $\Theta$ are the time and the azimuthal coordinate, and $i$, $j$
and $k$ are the quaternion units. The acoustic pressure is
$\text{Re}(\tilde{p})$, and $A$ is the real-valued positive slowly-varying
amplitude. The slowly varying angle $\theta$ describes the position of the
anti-nodal line the . The slowly varying angle $\chi$ indicates whether the
azimuthal eigenmode is a standing wave ($\chi=0$), a pure CW or CCW spinning
wave ($\chi=\mp\pi/4$) or a mix of both for $0<|\chi|<\pi/4$. The angle
$\varphi$ stands for slow temporal phase drift. These variables fully describe
the acoustic fields $p$ as a function of the time and azimuthal position.
Figure 2(c) shows the time traces of the extracted slow-flow variables
together with their probability density functions (PDFs) for the last 6 s of
the record, an interval that is representative of the stationary dynamics when
$\Phi=0.55$. The joint PDF of the first three state variables deduced from the
100 s record at $\Phi=0.55$ is also shown in Fig. 3(b), where $A$, $\theta$
and $2\chi$ are taken as spherical coordinates as shown in Fig. 3(a). At the
condition $\Phi=0.55$, the amplitude of the azimuthal mode has a relatively
small standard deviation compared to its mean value. The intermittent changes
of sign of the nature angle $\chi$ correspond to changes of spinning direction
of the mixed mode, as shown in Fig. 2(b). Although the difference is small,
the most probable orientation of the anti-nodal line of the mixed-mode’s
standing component is not the same whether the mode spins CW or CCW, a
phenomenon that had never been reported and that will be explained later.

$A$
---
2$\chi$
---
$\theta$
---
Standing
---
Pure CW
---
spinning
Pure CCW
---
spinning
(a)
---
(b)
---
20%
---
60%
---
Figure 3: (a) Bloch sphere representation of the state of the azimuthal mode.
(b) Experimental joint PDF for $\Phi=0.55$.

We now consider the evolution of the state variable PDFs in the vicinity of
the Hopf bifurcation. These stationary PDFs are shown in the left column of
Fig. 4 for several $\Phi$. The evolution of $P(A)$, shown in Fig. 4(a), is
typical of a supercritical Hopf bifurcation in presence of additive random
forcing Juel  _et al._ (1997). The bifurcation occurs around $\Phi=0.5125$.
This point separates broadband-noise-driven resonances, when the azimuthal
thermoacoustic mode is linearly stable, from limit cycles, when it is linearly
unstable. Next, one can consider how $P(\varphi)$ changes with $\Phi$ in Fig.
4(d). Overall, the slowly varying temporal phase $\varphi$ drifts sufficiently
during these 100 s records to yield rather flat distribution between 0 and
$2\pi$. One can notice that some phases are statistically prevailing for the
higher range of investigated $\Phi$, but these PDF maxima differ from case to
case, and moreover, they wane when longer time traces are considered for
computing the statistics, in agreement with the model which we propose in this
letter.

Figure 4: Stationary PDFs of $A$, $\chi$, $\theta$ and $\varphi$ at several
$\Phi$ in the vicinity of the supercritical Hopf bifurcation. The PDFs are
deduced from the slow-flow variable time traces, which are extracted from the
acoustic pressure time traces following the procedure in Ghirardo and Bothien
(2018). For (a) to (d), the acoustic time traces were obtained experimentally.
For (e) to (h), the acoustic time traces were obtained by simulating 100 s of
Eq. (2). Linear dependencies with respect to $\Phi$ of the parameters ($\nu$,
$c_{2}\beta$, $\kappa$, $\Gamma$) were deduced from the data processing.

More interesting are the statistics of $\theta$ and $\chi$. Let us begin with
$P(\theta)$ in Fig. 4(c): Even though the combustor was designed as
rotationally symmetric as possible with a discrete rotational symmetry of
order 12 (due to the 12 burners), the experimental results show a clearly
preferred and repeatable anti-nodal line position for the different $\Phi$.
Given the long run times and the high-order discrete rotational symmetry of
the system, one would either expect a quasi-uniform distribution of $\theta$
or a new preferred nodal line direction each time the combustor is ignited.
The only explanation for this preferred orientation is the presence of a
rotational symmetry breaking in the thermoacoustic system. Furthermore, in
addition to this explicit symmetry breaking, the system exhibits a spontaneous
symmetry breaking in terms of the nature of the thermoacoustic mode. When
$\Phi<0.55$, $P(\chi)$ is unimodal, with a maximum close to zero, indicating
that the thermoacoustic mode is predominantly standing, which corroborates two
recent theoretical studies Faure-Beaulieu and Noiray (2020); G. Ghirardo and
F. Gant (2021) that predicted the predominance of standing modes when the
normalized stochastic forcing amplitude is large compared to the one of the
limit cycle. For $\Phi=0.55$ and $\Phi=0.5625$, $P(\chi)$ is bimodal: the
spinning component of the mixed mode intermittently changes direction as
illustrated in Fig. 2(c). Now, when $\Phi=0.575$ in Fig. 4(b), these sporadic
reversals of spinning direction do not take place anymore: the mixed mode
spins in the CW direction and the symmetry of its state PDF is spontaneously
broken beyond this threshold. It is worth noting that other runs performed for
$\Phi\geq 0.575$, can also lead to a robust CCW mixed mode. In the remainder
of this letter, we present a model to explain the unexpected explicit symmetry
breaking.

Theoretical model and calibration of its parameters.—In Faure-Beaulieu and
Noiray (2020), the quaternion acoustic field was inserted into the following
wave equation with distributed heat-release-rate fluctuations to model the
thermoacoustic dynamics of an idealized thin annular chamber:

$\dfrac{\partial^{2}p}{\partial t^{2}}+\alpha\dfrac{\partial p}{\partial
t}-\dfrac{c^{2}}{\mathcal{R}^{2}}\dfrac{\partial^{2}p}{\partial\Theta^{2}}=(\gamma-1)\>\dfrac{\partial\dot{Q}}{\partial
t}+\Xi(\Theta,t).$ (2)

In this equation, $\alpha$ is the acoustic damping at the annulus boundaries,
$\mathcal{R}$ the annulus radius, $c$ the speed of sound, $\gamma$ the heat
capacities ratio, $\dot{Q}(\Theta,t)$ the coherent component of the heat
release rate fluctuations and $\Xi$ a spatially distributed white Gaussian
noise representing the effect of turbulent fluctuations of the heat release
rate Clavin  _et al._ (1994); Lieuwen (2003); Lieuwen and Banaszuk (2005);
Noiray and Schuermans (2012b). The coherent heat release rate fluctuations are
then linked to the acoustic field using the expression
$(\gamma-1)\dot{Q}=\beta\,[1+c_{2}\cos(2\Theta)]p-\kappa p^{3}$. The first
term stands for the linear response of the fluctuating heat release rate to
acoustic perturbations. If $\beta>\alpha$ and $c_{2}=0$, the linear growth
rate of the thermoacoustic mode $\nu=(\beta-\alpha)/2$ is positive, i.e. the
mode is linearly unstable, and small acoustic perturbations are exponentially
amplified. This pure exponential growth is not observed in real systems
because of the inherent turbulence-induced stochastic forcing $\Xi$ and
because of the nonlinearity of the flame response to the acoustic field beyond
a certain level. The first term also accounts for spatial non-uniformities
along the annulus, through the second component $c_{2}\cos(2\Theta)$ of the
Fourier expansion of any distribution of heat release rate source, which is
the only component influencing the first azimuthal eigenmode Noiray and
Schuermans (2012a). The second term, $-\kappa p^{3}$, defines the saturation
of flame response, which leads to limit cycle oscillations, and is sufficient
to adequately capture the thermoacoustic dynamics in the vicinity of
supercritical Hopf bifurcations Lieuwen (2003); Noiray and Denisov (2017); Lee
_et al._ (2021), as in the present experiment. Further discussion about this
heat release rate model is given in sup . In Faure-Beaulieu and Noiray (2020),
spatial and temporal averaging are performed to derive a system of coupled
Langevin equations $\dot{\bm{Y}}=\bm{F}(\bm{Y})+\bm{B}(\bm{Y})\bm{N}$,
governing the dynamics of the azimuthal mode state-vector
$\bm{Y}=(A,\chi,\theta,\varphi)^{T}$ via a deterministic contribution
$\bm{F}(\bm{Y})$ and a stochastic forcing term $\bm{B}(\bm{Y})\bm{N}$, where
$\bm{N}=(\zeta_{A},\zeta_{\chi},\zeta_{\theta},\zeta_{\varphi})^{T}$ is a
vector of independent white Gaussian noises of intensity $\Gamma/2\omega^{2}$.
The latter intensity involves the constant $\Gamma$, which is the spatially-
averaged stochastic forcing level originating from the turbulent fluctuations
of the heat release rate. The expressions for $\bm{F}$ and $\bm{B}$ are given
in Faure-Beaulieu and Noiray (2020); sup  with the simplifying assumptions for
their derivation. It is worth noting that this model cannot be used to predict
the stability and slow-flow dynamics of a real combustor from the knowledge of
its geometry and operating condition. However, we show here that its handful
of parameters $[\nu;c_{2}\beta;\kappa;\Gamma]$ can be identified from the
experimental data and then the model can reproduce the thermoacoustic
dynamics. Several methods are available for disentangling deterministic and
stochastic contributions in stochastic trajectories, e.g. Frishman and
Ronceray (2020); Ferretti  _et al._ (2020); Brückner  _et al._ (2020). Here,
we used the method developed in Siegert  _et al._ (1998); Friedrich  _et al._
(2002); Böttcher  _et al._ (2006); Friedrich  _et al._ (2011), which has
already been taken up for investigating, among other areas, thermoacoustic and
aeroacoustic problems, e.g. Noiray and Denisov (2017); Boujo  _et al._ (2020).
First, for each $\Phi$, the transition probabilities of the state variables
are extracted. Then, these probabilities are used to compute the first and
second Kramers-Moyal coefficients that serve as basis for parameter
identification with least square fitting. More details about the robustness of
the parameter identification are given in sup . It was found that simple
linear regression for the dependencies of $\nu$, $c_{2}\beta$, $\kappa$ and
$\Gamma$ upon $\Phi$ allows to unravel the complex topology of the phase
space. Indeed, time domain simulations of Eq. (2) were performed to generate
$100\text{\,}\mathrm{s}$ acoustic time series for each $\Phi$, from which the
state variables and their statistics were extracted. The results are presented
in Fig. 4(e) to 4(h), and are in close agreement with the experiments. This
shows that despite the simplifications invoked to derive the low-order model,
it can quantitatively reproduce the dynamics of the state variables.
Discussion—The two coupled equations for $A$ and $\chi$ recently allowed us to
identify the fundamental mechanisms involved in the spontaneous symmetry
breaking of the thermoacoustic mode state. In fact, the symmetry of the phase
space is not broken, but beyond a certain linear growth rate, the symmetry of
the state may be, if the mode chooses either the CW or the CCW spinning state
and the noise is not sufficiently large to allow sporadic escape from and
return to this attractor Faure-Beaulieu  _et al._ (2021). Here, we go
significantly further by considering the stationary solutions of the Fokker-
Planck equation governing the joint PDF of the state vector in order to
provide a complete picture of the phase space topology (numerical method in
sup ).

$\frac{27\kappa\Gamma}{256\nu^{2}\omega^{2}}$
---
$\frac{c_{2}\beta}{2\nu}$
---
1.7
---
0.17
---
0.017
---

increase $\Gamma$
---

increase $\nu$
---

0
---
1.64
---
0.82
---
increase $c_{2}$
---
increase $\nu$
---
Figure 5: Stationary PDF of the eigenmode state computed with the Fokker-
Planck equation for different noise intensity $\Gamma$ and asymmetry levels
$c_{2}$ (state probability: 20% and 60%). The reference case in the center of
the bottom row corresponds to the parameters of the calibrated model for
$\Phi=0.55$.

We consider the reference case $\Phi=0.55$, from which we arbitrarily change
the linear growth rate $\nu$ and the level of asymmetry $c_{2}$ for fixed
$\kappa$ and $\Gamma$. These stationary solutions are presented in Fig. 5. We
start in the bottom right corner, where the state exhibits a unimodal PDF
corresponding to a statistically prevailing standing mode with fixed anti-
nodal line orientation the . When the rotational asymmetry level is decreased,
the PDF first becomes bimodal, which corresponds to a predominant mixed mode
with intermittent changes of spinning direction. Ultimately, for $c_{2}=0$,
the nodal line cannot lock anymore on a fixed orientation, it slowly drifts as
a random walk and the PDF takes the form of two tori close to the poles. These
results are in line with previous findings from Noiray and Schuermans (2012a),
which gave an incomplete description of the phase space topology. Then, the
effect of noise and of the linear growth rate can be analyzed by moving
vertically in the diagram. For an ideal rotationally symmetric combustor
(first column) and for a fixed turbulent forcing level $\Gamma$ and saturation
constant $\kappa$, the effect of decreasing the linear growth rate is seen by
moving up: the two tori merge when the system is brought close enough to the
Hopf point (top row), which means that the system undergoes a transition from
predominantly spinning states to prevailing standing modes with single torus
PDF. However, a key consequence of the model, which had not been drawn so far,
is that this toroidal PDF will not be observed in real combustors, because the
more the source term gain $\beta$ tends to $\alpha$, and thus
$\nu=(\beta-\alpha)/2\rightarrow 0$, the more $c_{2}\beta/(\beta-\alpha)$ will
rise, which means that such decrease of $\beta$ would be accompanied with a
move towards the right on the diagram. In other words, the effect of a
resistive asymmetry, however minute, will be magnified in the vicinity of the
bifurcation. Furthermore, we can add to the model tiny reactive asymmetries
rea . They also exist in real systems due to inherent geometrical
imperfections, and cause, as shown in Fig. 2(d), the small changes of $\theta$
synchronized with changes of spinning direction and the predominance of CW
spinning modes for $\Phi=0.55$. The spatial phase drift induced by these tiny
imperfections is negligible during a few acoustic periods, but becomes
significant over sufficiently long time scales. We can thus conclude that
manufacturing annular systems free from explicit symmetry breaking at the
bifurcation is out of reach. Possible direct applications of this framework
include the analysis of non-radial stellar pulsations Buchler  _et al._ (1995,
1993); Aerts (2021), or the quest for weakly-dissipative nonreciprocal devices
based on an acoustic analog of the Zeeman effect Fleury  _et al._ (2014); Y.
Ding, Y. Peng, Y. Zhu, X. Fan, J. Yang, B. Liang, X. Zhu, X. Wan, and J. Cheng
(2019), for which decreasing losses in circulators bring them closer to Hopf
points where we show that slight asymmetries have considerable impact on wave
dynamics.

###### Acknowledgements.

This project has received funding from the European Union’s Horizon 2020
program (Grant 765998).

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