Two mixed finite element formulations for the weak imposition of the Neumann boundary conditions for the Darcy flow

Erik BurmanChair of Computational Mathematics, University College London, London, UK. E-mail address<EMAIL_ADDRESS>Riccardo PuppiChair of Modelling and Numerical Simulation, École Polytechnique Fédérale de Lausanne, Lausanne, CH. E-mail address<EMAIL_ADDRESS>
Erik Burman was partially supported by the EPSRC grants EP/P01576X/1 and EP/T033126/1. Riccardo Puppi was partially supported by ERC AdG project CHANGE n. 694515.

We propose two different discrete formulations for the weak imposition of the Neumann boundary conditions of the Darcy flow. The Raviart-Thomas mixed finite element on both triangular and quadrilateral meshes is considered for both methods. One is a consistent discretization depending on a weighting parameter scaling as $\mathcal O(h^{-1})$, while the other is a penalty-type formulation obtained as the discretization of a perturbation of the original problem and relies on a parameter scaling as $\mathcal O(h^{-k-1})$, $k$ being the order of the Raviart-Thomas space. We rigorously prove that both methods are stable and result in optimal convergent numerical schemes with respect to appropriate mesh-dependent norms, although the chosen norms do not scale as the usual $L^2$-norm. However, we are still able to recover the optimal a priori $L^2$-error estimates for the velocity field, respectively, for high-order and the lowest-order Raviart-Thomas discretizations, for the first and second numerical schemes. Finally, some numerical examples validating the theory are exhibited.

§ INTRODUCTION

We consider the finite element approximation for the weak imposition of the Neumann boundary conditions for the Poisson problem in its mixed formulation, also known as Darcy's law in the context of fluid dynamics.

Let us point out that the situation is dual with respect to the standard formulation of the Poisson problem: here the Neumann boundary conditions are essential and to the best of our knowledge it is not clear in the literature how to proceed in order to enforce them by manipulating the weak formulation rather than the functional spaces.

As far as the primary formulation is concerned, a wide variety of techniques have already been proposed and are now well-understood, the most prominent of which are undoubtedly the penalty method introduced in [4], the Lagrange multipliers approach of [3] and, of course, the Nitsche method developed in [15] and later promoted in [17], where the author relates it to the stabilized Lagrange multiplier method of [5].

In this work two different discrete formulations of the Darcy problem for the weak imposition of the Neumann boundary conditions are provided.
Both of them are based on the Raviart-Thomas finite element discretization for triangular and quadrilateral meshes. Let us notice that the two schemes do not add any additional degrees of freedoms and moreover, for simplicity, all dimensionless parameters have been set to $1$.

The first formulation is a consistent discretization of the Darcy system, hence it falls into the class of Nitsche-type methods. A weighting parameter scaling as $\mathcal O(h^{-1})$ needs to be introduced. Let us observe that this formulation had already appeared in the literature in [9] for the lowest-order Raviart-Thomas element, in the context of incompressible flows in fractured media.

The latter, which is inspired by [13] and based on a perturbed variational principle, belongs instead to the family of penalty methods. In this case the penalization parameter scales as $\mathcal O(h^{-k-1})$, $k$ being the order of the Raviart-Thomas discretization, entailing a much more severe ill-conditioning of the resulting stiffness matrix.

We are able to prove that both formulations are stable and give rise to optimal convergent schemes with respect to suitable mesh-dependent norms, which do not scale as the $L^2$-norm as it is customary for the Darcy problem. At this point we are able to demonstrate super convergence results that allow us to find an optimal a priori estimate of the velocity error with respect to the $L^2$-norm, respectively, for any higher order Raviart-Thomas discretization combined with the first method and for the lowest order element with the second formulation.

Note that this work should be considered as a preliminary step towards the much more involved situation of an underlying mesh which is not fitted with the boundary of the physical domain [7].

Let us briefly sketch the outline of the paper. In the next two sections we introduce, respectively, the strong formulation of the Darcy problem and a singularly perturbed formulation of it, parametrized by $\eps\ge 0^{+}$. For the latter we are able to show that its solution is stable independently of $\eps$ and that, for $\eps\to 0^+$, it converges to the solution of the original problem under some extra regularity assumptions on the data and on the boundary. In the third section the Raviart-Thomas finite element is introduced together with our two discrete formulations, both depending on a mesh-dependent weighting parameter $\gamma$. As already mentioned, for the first one, $\gamma = h^{-1}$, while for the other $\gamma=h^{-\left(k+1\right)}$. In section 4 we prove the desired stability estimates, with respect to different mesh-dependent norms, guaranteeing the well-posedness of the associated problems. Then, in section 5, optimal a priori error estimates, in terms of the chosen norms, are demonstrated for the velocity and pressure fields. We demonstrate some super convergent results that enable us, for the two methods, to recover optimality for the $L^2$-error of the velocity field as well. Finally, some two-dimensional numerical examples  are provided in order to corroborate the theory.

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# SiamQuality: A ConvNet-Based Foundation Model for Imperfect Physiological
Signals

Cheng Ding  Department of Biomedical Engineering, Georgia Institute of
TechnologyAtlantaGAUSA<EMAIL_ADDRESS>,  Zhicheng Guo  Department of
Electrical and Computer Engineering, Duke UniversityDurhamNCUSA ,  Zhaoliang
Chen  Department of Computer Science, Emory UniversityAtlantaGAUSA ,  Randall
J Lee  School of Medicine, University of California at San FranciscoSan
FranciscoCAUSA ,  Cynthia Rudin  Department of Computer Science, Duke
UniversityDurhamNCUSA and  Xiao Hu  Nell Hodgson Woodruff School of Nursing,
Emory UniversityAtlantaGAUSA<EMAIL_ADDRESS>
(2018; 20 February 2007; 12 March 2009; 5 June 2009)

###### Abstract.

Foundation models, especially those using transformers as backbones, have
gained significant popularity, particularly in language and language-vision
tasks. However, large foundation models are typically trained on high-quality
data, which poses a significant challenge, given the prevalence of poor-
quality real-world data. This challenge is more pronounced for developing
foundation models for physiological data; such data are often noisy,
incomplete, or inconsistent. The present work aims to provide a toolset for
developing foundation models on physiological data. We leverage a large
dataset of photoplethysmography (PPG) signals from hospitalized intensive care
patients. For this data, we propose SimQuality, a novel self-supervised
learning task based on convolutional neural networks (CNNs) as the backbone to
enforce representations to be similar for good and poor quality signals that
are from similar physiological states. We pre-trained the SimQuality on over
36 million 30-second PPG pairs and then fine-tuned and tested on six
downstream tasks using external datasets. The results demonstrate the
superiority of the proposed approach on all the downstream tasks, which are
extremely important for heart monitoring on wearable devices. Our method
indicates that CNNs can be an effective backbone for foundation models that
are robust to training data quality.

Foundation model, Physiological data, PPG signal, Contrastive learning

††copyright: acmlicensed††journalyear: 2018††doi: XXXXXXX.XXXXXXX††conference:
Make sure to enter the correct conference title from your rights confirmation
emai; June 03–05, 2018; Woodstock, NY††isbn: 978-1-4503-XXXX-X/18/06

##  1\. Introduction

Foundation models, particularly those with transformer architectures as the
backbone (Vaswani et al., 2023), have significantly influenced the landscape
of artificial intelligence in recent years. Their remarkable ability to
capture and generate human language, as well as to process and interpret
complex visual information, has set new standards in language and language-
vision tasks (Min et al., 2021; Zhao et al., 2023; Du et al., 2022; Chen et
al., 2023b). However, these large-scale models need to be trained with massive
high-quality data, which are collected from books, journal articles, and
internet sources. These text data typically follow rules of grammar and
semantics, which allows the model’s outputs to be coherent, contextually
relevant, and linguistically accurate.

However, other real world domains exhibit substantial amounts of poor-quality
data. This gap between the ideal training data quality and the actual quality
of available data makes it difficult to train and deploy foundation models.
This challenge becomes even more daunting in the context of physiological data
analysis (Clifford et al., 2012). The reliability of physiological signals is
often compromised by the presence of artifacts and noise, which can stem from
patient movement, sensor displacement, and physiological variations. These
artifacts can significantly distort the signal, leading to inaccurate
assessments in clinical diagnostics and remote health monitoring. The inherent
noise, incompleteness, and inconsistency, compounded by the complexities of
human physiology and the unpredictable conditions of free-living data
collection environments (Charlton et al., 2023), necessitate robust model
architectures to address these data quality issues.

Figure 1. The proposed SiamQuality to address the data quality issue in
physiological data.

This methodology leverages the power of contrastive learning to extract robust
features from PPG signals, irrespective of their quality. The rationale behind
is, one poor quality PPG signal segments represents the same underlying
physiological information with its good quality neighbors if they are
collected within a limited temporal window from the same patient.

In this study, we propose SiamQuality, a new approach to address the issue of
physiological data quality when developing foundation models. Instead of using
transformers, which are complex and difficult to troubleshoot, we utilize CNNs
as the backbone, given their demonstrated excellent ability to represent
sequences (Tang et al., 2020; Zhao et al., 2017; Liu et al., 2018). If the
goal is solely representational learning, then CNNs provide an efficient
framework to build foundational models. To train and test our approach, we
leverage a large dataset of photoplethysmography (PPG) signals (Charlton et
al., 2023) collected from hospitalized intensive care patients.

As shown in Figure 1, the proposed SiamQuality is based on the SimSiam
architecture (Chen and He, 2021), with a novel signal quality-based pairing
mechanism. This methodology leverages the power of contrastive learning to
extract robust features from PPG signals, alleviating the influence of poor
signal quality. The rationale behind this approach is that representations of
PPG signals from similar physiological states, as learned by CNNs, should be
similar and that a bad-quality PPG signal in a temporal vicinity of a good-
quality one from the same human is likely to be from a similar physiological
state.

As a foundation model, SiamQuality meets three criteria: i) Pre-trained on
large-scale data. In this study, we collected over 36 million 30-second PPG
pairs (600,000+ hours) from 21,000 patients as the training data for
SiamQuality. ii) Accommodates Increasing Complexity. We use CNN, specifically
ResNet (He et al., 2016), as the backbone and successfully observe a
performance improvement of SiamQuality when increasing model size. iii)
Adaptability. We conducted extensive experiments on various tasks, including
investigating different pre-training mechanisms and fine-tuning on six
different PPG-based downstream classification and regression datasets.
Experimental results validate that our method improves performance on all
downstream tasks, achieving state-of-the-art (SOTA) performance across a range
of tasks. Notably, on the tasks of respiration rate and atrial fibrillation
detection, it outperforms state-of-the-art models by 43% and 7% respectively.

This study goes beyond merely presenting an efficient model for PPG signals;
it provides an alternative solution that emphasizes the often-overlooked
aspect of data quality in the development of foundation models. It sets a
precedent for addressing data quality issues more broadly in AI and machine
learning.

##  2\. Related Work

Foundation models for physiological data. There are numerous foundation models
in many areas, including language and computer vision, though only a few of
them focus on physiological data. HeartBeit utilizes masked image modeling to
develop a transformer model based on vision for analyzing electrocardiogram
(ECG) waveforms. This pre-training approach enhances the accuracy and detailed
explainability of the model’s predictions. Another foundation model, the
Biosignal Transformer Model (BIOT) (Yang et al., 2023), learns embeddings for
biosignals, especially for electroencephalogram (EEG) signals. BIOT enables
effective knowledge transfer across different datasets and allows joint
training on multiple sources. In the work (Li et al., 2023), The authors
enhance the ECG encoder by fusing the latent space generated from a frozen
large language model applied to text from ECG reports. Consequently, their
method does not rely on the specific type of annotated data and can be
seamlessly transferred to any new database.

Contrastive learning for time series data. Positive and negative pairs are
often needed in self-supervised contrastive learning. For instance, TS2Vec
(Yue et al., 2022) adopts a hierarchical contrastive learning framework to
handle time series augmentations, where each time step is represented as a
point in an embedding space. CoST (Woo et al., 2022) encodes disentangled
trend and seasonal representations through contrastive learning. The above-
mentioned studies require negative pairs for the self-contrastive learning,
where a negative pair consists of parts from two different time series.
However, SimTS (Zheng et al., 2023a) have conducted experiments showing that
current ways of constructing negative pairs are not efficient for time series
data, meaning that the performance of the model does not benefit from the
negative pairs. Instead, they propose that two consecutive time series can be
combined as one positive pair, where positive pairs are pulled together in
latent space. With SimSiam, which only requires positive pairs, SimTS has
outperformed TS2Vec (Yue et al., 2022) and CoST (Woo et al., 2022) in
experiments of (Zheng et al., 2023a). In our study, we also use the same
SimSiam structure and have included a modified SimTS (Zheng et al., 2023a) as
one of the baselines, discussed in Subsection 3.5.1.

###  2.1. Method

###  2.2. Preliminaries and Problem Formulation

Our dataset $K=\\{x_{i},y_{i},t_{i}\\}_{i=1}^{N}$ consists of PPG strips
$x_{i}$, each associated with a label $y_{i}$ representing the percentage of
artifacts (perturbation in signal) in $x_{i}$ (ranging from 0% to 100%), where
$t_{i}$ is the collection time. Our goal is to learn a robust representation
that is minimally affected by artifacts.

###  2.3. Signal Quality Pairing

Figure 2. The mechanism for signal quality pairing

xxxxx

For our study, we utilized continuous PPG recordings collected from adult
intensive care units (ICUs), generating pairs of varying quality as
illustrated in Figure 2. To categorize each 30-second PPG segment’s quality,
we implemented a quality assessment method previously developed (Chen et al.,
2023a). This method classifies each segment into a spectrum of quality from
good to bad, based on the percentage of artifacts present. For $x_{i}$, the
collection time $t_{i}$ for each strip is also recorded. In our methodology,
segments identified as good quality ($y_{i}=0$) serve as anchor points for
beginning a search for a nearby low-quality signal.

From each anchor point, we search for segments within a 5-minute window that
exhibits bad quality (high $y_{i}$). Among these, we select the segment with
the highest temporal distance from the anchor, ensuring a varied degree of
comparison in terms of artifacts and time. This approach allows us to
construct pairs of PPG segments that are physiologically similar but differ
significantly in their quality, as dictated by the presence of artifacts.
These pairs are then used to train our model, enabling it to learn robust
representations that are minimally affected by artifacts.

In addition, we implemented curriculum learning (Wang et al., 2021a) by
systematically feeding the model training pairs, starting with small
artifacts, and later introducing larger artifacts. The size of the artifact
between a clean and not-clean signal is quantified by a measure
$C(x_{1},x_{2})=|y_{1}-y_{2}|$. After sorting the training data based on C,
the model is initially trained with pairs having lower $C(x_{1},x_{2})$ and
progressively introduced to pairs with higher differences.

###  2.4. Contrastive Learning Framework using SimSiam

Our approach is grounded in the SimSiam architecture (Chen and He, 2021),
adapted for PPG signal processing, as shown in Figure 3. The key components of
our model include an encoder $E$, a projector $P$, and a predictor $D$.

Figure 3. Architecture for SimSiam with quality pairing augmentation

####  2.4.1. Encoder and Projector

The encoder $E$ transforms the input signal x into a representation $h=E(x)$.
The projector P then maps this representation into a lower-dimensional space,
$z=P(h)$, suitable for contrastive learning.

####  2.4.2. Predictor

The predictor D maps the projected representation z to the space where
contrastive loss is applied. For a pair of signals $x_{1}$ (good quality) and
$x_{2}$ (bad quality), their latent representations are $z_{1}=P(E(x_{1}))$
and $z_{2}=P(E(x_{2}))$, respectively. The predictor’s output for $x_{i}$ is
$p_{i}=D(z_{i})$.

####  2.4.3. Contrastic Loss Function

The cosine similarity loss $L_{cosine}$ is used to minimize the distance
between the predicted latent feature vectors within the pair:

$L_{cosine}(x_{1},x_{2})=-\frac{p_{1}\cdot
z_{2}}{\|p_{1}\|_{2}\cdot\|z_{2}\|_{2}}-\frac{p_{2}\cdot
z_{1}}{\|p_{2}\|_{2}\cdot\|z_{1}\|_{2}}$

where $\cdot$ denotes the dot product, and $\|\cdot\|_{2}$ represents the
$L_{2}$ norm.

Algorithm 1 SimSiam-Based PPG Signal Processing with Curriculum Learning

1:Input: Dataset $K=\\{x_{i},y_{i},t_{i}\\}^{N}_{i=1}$ of PPG signals $x_{i}$
with artifact percentage $y_{i}$ and collection time $t_{i}$

2:Output: Specify the output if needed

3:Step 1: Initialization

4:Initialize Encoder $E$, Projector $P$, Predictor $D$, and Classifier

5:Define cosine similarity loss function $L_{cosine}$

6:Step 2: Data Preparation

7:for each PPG signal $(x_{i},y_{i}=0)$ do

8: Find PPG $x_{j}$ that  $j=\max\\{k|(|t_{k}-t_{i}|<5\text{mins})\textrm{ AND
}(y_{k}>0.2)\\}$

9: Pair PPG signals $(x_{i},x_{j},c_{i})$ where $c=y_{j}-y_{i}$

10:end for

11:Step 3: Curriculum Learning and Model Training

12:for each pair $(x_{good},x_{bad})$ do

13: Encode signals: $h_{good}=E(x_{good}),h_{bad}=E(x_{bad})$

14: Project encoded signals: $z_{good}=P(h_{good}),z_{bad}=P(h_{bad})$

15: Predictor: $p_{good}=D(z_{good}),p_{bad}=D(z_{bad})$

16: Compute contrastive loss:
$L=\frac{1}{2}(L_{cosine}(p_{good},z_{bad})+L_{cosine}(p_{bad},z_{good}))$

17: Update model parameters to minimize $L$

18:end for

##  3\. Experiments and Results

###  3.1. Data

The dataset used for model pre-training was obtained from routine patient
monitoring systems in the intensive care units of the University of
California, San Francisco (UCSF) Medical Center. This collection was conducted
with an approved waiver of written patient consent under the UCSF
Institutional Review Board (IRB number: 14-13262). The retrospective use of
this fully de-identified data is conducted according to the terms of a data
use agreement between UCSF and Emory University.

The waveform dataset, which includes physiological signals, cardiac arrhythmia
alarms, and linked electronic health records (EHR) from over 24,100 patients,
was collected at the UCSF Medical Center between March 2013 and December 2018.
The demographics of the patients in this dataset are reported in Table 1. In
the waveform dataset, a total of 2.6 million hours of continuous signals were
collected, including seven-lead ECG (I, II, III, V, AVR, AVL, AVF), one-
channel PPG signal, and one-channel respiration rate. All signals are sampled
at 240Hz.

Table 1. Characteristics of Participants Enrolled in UCSF medical center Characteristics | Total Cohort (N = 28539)
---|---
Sex |
Female | 13203 (46.2%)
Male | 15330 (53.7%)
Others | 6 (0%)
Age |
$\geq$65 yr | 12157 (42.6%)
55-64 yr | 5370 (18.8%)
40-54 yr | 4372 (25.3%)
22-39 yr | 2715 (9.5%)
$<$22 yr | 3925 (13.8%)
Race or ethnic group |
White or Caucasian | 15890 (55.7%)
Black or African American | 2159 (7.4%)
Asian | 4364 (15%)
Unknown/Declined | 1149 (4%)
Others | 4913 (16.9%)
Native Hawaiian | 426 (1.46%)
American Indian | 212 (0.7%)

###  3.2. Data Preprocessing

Continuous PPG recordings were divided into discrete, 30-second segments
without overlap. Each of these segments was first downsampled to a frequency
of 40Hz, followed by min-max normalization to standardize the signal range. To
evaluate the quality of these signals, a previously developed binary PPG
signal quality assessment tool (Pereira et al., 2019) was employed,
categorizing each segment as either ‘good quality’ or ‘bad quality.’ For
segments classified as having bad quality, an additional analysis was applied
using another signal quality segmentation model (Chen et al., 2023a). This
model is specifically designed to segment the locations of artifacts within
each 30-second PPG signal, therefore it can be used to quantify the level of
artifacts present in each bad quality signal.

###  3.3. Downstream Tasks

To test the effectiveness of our method, we selected a range of diverse public
downstream tasks, allowing for reproducibility and validation of our results
by the broader research community. Table 2 provides an overview of the
datasets and their respective statistics.

###  3.4. Performance Evaluation Metrics

In our study, we employ distinct evaluation metrics tailored to the nature of
the tasks. For regression tasks, the Mean Absolute Error (MAE) is used. For
classification tasks, we utilize the F1 score, a balanced metric that
considers both precision and recall, making it suitable for evaluating the
performance of our classification models.

Table 2. Statistics of public datasets for various downstream tasks Task | No. Subjects |
---|---|---
Heart Rate Estimation |  |
Signal Type: ECG, PPG, Accelerometer |  |
TROIKA (Zhang et al., 2014) | 12 |
Dalia (Reiss et al., 2019) | 15 |
WESAD (Schmidt et al., 2018) | 17 |
Blood Pressure Estimation |  |
Signal Type: ECG, PPG, ABP Waveforms |  |
PulseDB (MIMIC-III+VitalDB) (Wang et al., 2023b) | 5361 |
Respiration Rate Estimation |  |
Signal Type: PPG |  |
BIDMC PPG and Respiration Dataset (Pimentel et al., 2016) | 53 |
AF Detection |  |
Signal Type: PPG |  |
Stanford AF dataset (Torres-Soto and Ashley, 2020) | 148 |

Additionally, we introduce an innovative metric, the Artifact Tolerance Curve
(AT-Curve), illustrated in Figure 4. The AT-Curve is designed to assess the
robustness of our models across varying levels of signal quality within the
test data. The level of signal quality is calculated by the method of Chen et
al. (2023a). The horizontal axis represents the upper limit for the signal
quality of each subgroup, with values ranging from 0 to 1.0. A value of 1.0
represents the entire test set, while a value of 0 corresponds to the subset
with the highest signal quality (i.e., the cleanest subset). The height of
each bar on the AT-Curve denotes the sample size for each respective subgroup.

Figure 4. AT-curve. The horizontal axis represents the upper limit of the
signal quality for each subgroup. The height of each bar denotes the sample
size for each respective subgroup. The line plot shows MAE within each
subgroup.

xxxxx

Overlaying the bars, a line plot traces the performance of our models—measured
in terms of MAE for regression tasks and F1 score for classification
tasks—within each of these subgroups. This line plot provides shows how our
models’ performance varies with the quality of the PPG signals. The AT curve
thus serves a dual purpose: it not only illustrates the distribution of signal
quality within our test data but also maps our models’ performance against
this distribution. The AT-curve offers a comprehensive view of our models’
efficacy across a spectrum of signal qualities, highlighting their robustness
and adaptability in real-world scenarios.

###  3.5. Experimental Results

####  3.5.1. Comparison between Different Contrastive Learning Methods

Table 3. Comparison between different contrastive learning methods. Base Encoder: Resnet50 | Augmentation Method | HR (MAE) | RR (MAE) | AF (F1 Score) | BP (MAE)
---|---|---|---|---|---
|  | TROIKA | Dalia | WESAD | BIDMC | Stanford | PulseDB
SimCLR |  | 4.92 | 7.53 | 6.73 | 2.34 | 0.59 | 11.40
SwAV |  | 4.82 | 7.15 | 6.52 | 2.04 | 0.64 | 11.22
MoCo | Conventional | 4.70 | 7.28 | 6.53 | 2.17 | 0.63 | 11.17
BYOL | Augmentation | 4.97 | 7.50 | 6.77 | 2.41 | 0.61 | 11.40
SimSiam |  | 4.71 | 7.34 | 6.65 | 2.33 | 0.63 | 10.71
SimSiam | Temporal Sampling (Modified SimTS) | 4.92 | 7.31 | 6.54 | 2.08 | 0.63 | 10.94
| \+ Quality pairing | 4.73 | 7.15 | 6.35 | 1.96 | 0.65 | 10.65
| \+ Curriculum learning | 4.69 | 7.02 | 6.27 | 1.84 | 0.68 | 10.32

In the first experiment, we conducted a comprehensive evaluation of several
self-supervised learning (SSL) models: SimCLR (Chen et al., 2020), SwAv (Zhu
et al., 2020), MOCO (He et al., 2020), BYOL (Grill et al., 2020), and SimSiam
(Chen and He, 2021). Methods in the top half of Table 3 all used the same
augmentation method, which is to randomly select from (Gaussian noise,
powerline noise, and flipping) within each mini-batch. Original signal and its
augmented counterpart are combined as one positive pair, and signal with
augmentation from another signal are combined as one negative pair. Among the
five SSL approaches, only BYOL and SimSiam do not require negative pairs. We
chose SimSiam for the rest of the experiments that require only positive pairs
since its performance exceeds that of BYOL.

We now investigate different augmentation methods with SimSiam, as reported in
the bottom half of Table 3. Temporal sampling, instead of only using two
consecutive signal as one pair (SimTS (Zheng et al., 2023a)), we modify the
approach that uses one good quality PPG as an anchor point, and randomly
selects another signal within 5 minutes, regardless of signal quality. Then,
we added consideration of signal quality into the temporal sampling, and we
further added curriculum learning as the third augmentation method.

The results show a trend of improvement with each successive augmentation
strategy, for all datasets, for all of the tasks. SimSiam with quality-pairing
and curriculum learning consistently yielded the best results, for all of the
tasks (heart rate prediction, RR prediction, AF detection, and BP prediction).

####  3.5.2. SimSiam with Different Model Sizes

We investigated the scaling laws of SimQuality, as reported in Table 4. The
results reveal a clear trend: as the complexity of the ResNet architecture (He
et al., 2016) increases, there is a marked improvement in performance. This
trend is evident across all metrics, including MAE for Heart Rate, Respiratory
Rate, and Blood Pressure, as well as Atrial Fibrillation detection. ResNet152,
being the most complex model, consistently exhibited the best performance.
However, ResNet152 is more expensive to train than ResNet50. The experiments
in the remaining subsections do not require as much computation as earlier
experiments, so we use ResNet152 from now on.

Table 4. SimSiam with different model sizes Dataset | Resnet50 | Resnet101 | Resnet152
---|---|---|---
TROIKA (MAE) | 4.69 | 4.66 | 4.59
Dalia (MAE) | 7.02 | 6.92 | 6.80
WESAD (MAE) | 6.27 | 6.07 | 5.88
BIDMC (MAE) | 1.84 | 1.53 | 0.89
Stanford (F1) | 0.68 | 0.69 | 0.71
PulseDB (MAE) | 10.32 | 9.64 | 8.60
Table 5. Results compared with SOTA in HR datasets (MAE) Datasets | TROIKA | Dalia | WESAD
---|---|---|---
Deep PPG (Reiss et al., 2019) | 4.00 | 7.65 | 7.47
PPGNet* (A et al., 2019) | 3.36 | - | -
CurToSS* (Zhou and Selvaraj, 2020) | 2.20 | 5.00 | 6.40
TAPIR* (Huang and Selvaraj, 2020) | 2.50 | 4.60 | 4.20
BeliefPPG* (Bieri et al., 2023) | 1.88 | 4.26 | 3.81
Our methods | 4.59 | 6.80 | 5.88

####  3.5.3. Comparing with previous work on each downstream task

We compared our model’s performance against the previous SOTA results for each
specific task. Our model leverages the SimSiam framework, augmented with a
ResNet152 encoder, and employs Quality-Pairing with Curriculum Learning. We
report performance metrics and the AT-curve for each task.

(a) TROIKA Heart Rate

(b) DALIA Heart Rate

(c) WESAD Heart Rate

(d) BIDMC Resipiration Rate

(e) Stanford AF

(f) PulseDB Blood Pressure

Figure 5. AT-Curve for all downstream tasks Table 6. Results compared with SOTA in RR dataset Methods | RR (MAE)
---|---
Multiple auto-regression (Pimentel et al., 2016) | 4.0
EEMD + PCA (Nilsson et al., 2000) | 3.5
EEMD + Kalman filter (Motin et al., 2017) | 1.9
Smart fusion (Karlen et al., 2013) | 5.8
LSTM (Kumar et al., 2022) | 1.51
LSTM+Attn (Kumar et al., 2022) | 1.56
Our methods | 0.89
Table 7. Results compared with SOTA in AF dataset Methods | AF detection (F1 Score) | Artifact Coverage
---|---|---
Shen et al. 2019 (Voisin et al., 2019) | 0.684 | 100
Torres-Soto and Ashley (Deepbeat) (Torres-Soto and Ashley, 2020) | 0.652 | 100
Torres-Soto and Ashley (Deepbeat, Non-poor) (Torres-Soto and Ashley, 2020) | 0.646 | 27.67
Torres-Soto and Ashley (Deepbeat, Excellent only) (Torres-Soto and Ashley, 2020) | 0.58 | 18.77
BayesBeat (Das et al., 2022) | 0.671 | 100
BayesBeat (uncertainty threshold = 0.05) (Das et al., 2022) | 0.754 | 54
Our methods | 0.71 | 100
Table 8. Results compared with SOTA in SBP dataset Method | SBP (MAE)
---|---
CNN-GRUattn (ECG, PPG) (Wang et al., 2023a) | 4.90
UTransBPRNet (ECG, PPG) (Zheng et al., 2023b) | 4.38
Our methods (PPG only) | 8.6
Table 9. Results compared within different fine-tuning configurations Fine Tuning configuration | TROIKA | DALIA | WESAD | BP | AF | RR
---|---|---|---|---|---|---
| (MAE) | (MAE) | (MAE) | (MAE) | (F1 Score) | (MAE)
Fine-tune All | 4.6 | 6.8 | 5.8 | 8.6 | 0.71 | 0.89
Fine-tune Last | 5.78 | 8.55 | 7.3 | 10.5 | 0.61 | 2.2
| (+25.65%) | (+25.74%) | (+25.68%) | (+22.09%) | (-14.08%) | (+147.19%)
In-domain pre-training | 4.84 | 7.15 | 6.1 | 9.3 | 0.64 | 1.6
\+ Fine-tune Last | (+5.22%) | (+5.11%) | (+5.17%) | (+8.14%) | (-9.86%) | (+79.78%)

As reported in Table 5, we observe that discarding poor quality signals
significantly impacts performance (* indicates poor-quality signals were
discarded). Note that it is important to state that poor-quality signals were
omitted because MAE values are not comparable between cases when they are
omitted and when they are not. Comparing DeepPPG and our methods, which did
not discard any signals, we perform better in Dalia and WESAD but are slightly
worse in TROIKA. Through the AT curve shown in Figures 5a – 5c, we can see
that DALIA and WESAD have a more uniform distribution of artifact levels
within each bin, while TROIKA displays a significant skew towards noisy
signals, with a majority of signals exhibiting over 90% artifacts. In
addition, from the AT-curve, we can observe that, in the case of DALIA,
discarding signals containing over 60% of artifacts allows us to surpass the
performance of BeliefPPG. For WESAD, the threshold to exceed BeliefPPG
involves discarding signals with artifact levels above 20%. To outperform
BeliefPPG in the TROIKA dataset, it is necessary to exclude signals with more
than 40% artifacts.

A range of methods are compared for RR prediction, as reported in Table 6.
Among all the previous state-of-the-art (SOTA) methods, the LSTM model
demonstrated strong performance with an MAE of 1.51. However, our method
outperformed all these techniques, achieving the lowest MAE of 0.89,
indicating its superior accuracy in estimating Respiratory Rate. The AT curve
presented in Figure 5d shows that performance does not uniformly improve with
better signal quality.

In the evaluation of AF detection methods, as reported in Table 7, additional
artifact coverage for each method is provided in (Das et al., 2022).
BayesBeat, with an F1 Score of 0.671 at full artifact coverage (without
discarding any signals), significantly improved to 0.754 when applying an
uncertainty threshold, albeit with reduced coverage (i.e., keeping only 54% of
the signals in the test set). Our methods outperformed many others, achieving
an F1 Score of 0.71 with full artifact coverage, indicating a robust and
accurate approach for AF detection across diverse signal conditions. Through
the AT curve in Figure 5e, we can observe that over half of the signals have
more than 90% artifacts. The F1 score can reach up to 90% if signals with over
80% of artifacts are discarded.

In the study of PPG-based SBP estimation tasks, as reported in Table 8, the
CNN-GRUattn method, which employs a blend of Convolutional Neural Networks and
Gated Recurrent Units with attention mechanisms and uses both ECG and PPG
signals, achieved a Mean Absolute Error (MAE) of 4.90. UTransBPNet, also
integrating ECG and PPG data, improved upon this accuracy with an MAE of 4.38.
In contrast, our method, which solely relies on PPG signals, recorded a higher
MAE of 8.6, indicating a lower accuracy in SBP prediction compared to the
multi-signal approaches. Even across the AT curve, the MAEs remain in a
relatively narrow range, from 8.15 to 8.2. Hence, ECG adds value to SBP
estimation. However, PPG is collected passively and can offer continuous
monitoring, while ECG requires subjects to remain stationary and thus is not
nearly as widely available.

##  4\. Pre-training and fine tune configuration

For all the experiments discussed thus far, we fine-tuned all the parameters
for each downstream task (we call this ‘fine-tune All’). We also considered a
‘fine-tune Last’ strategy, where the parameters of the pre-trained encoder are
frozen while only the newly initialized classifier layer is tuned according to
the labels of target tasks). We also tried a third strategy, where we pre-
trained SiamQuality on the downstream datasets and then fine-tuned the
classification layer. (Note again that the pre-training does not require
labels from downstream tasks, since the goal of placing bad- and good-quality
signals requires only simulated labels – it is a self-supervised method.)

As reported in Table 9, fine-tuning all layers of the model yields the best
results across multiple metrics, including MAE in TROIKA, DALIA, WESAD, and BP
predictions, as well as the F1 Score for AF detection. In contrast, fine-
tuning only the last layer significantly diminishes the model’s accuracy,
increasing the MAE by about 25% in most datasets and reducing the F1 Score for
AF detection by 14.08%. The hybrid approach of in-domain pre-training followed
by fine-tuning the last layer offers some improvement over solely fine-tuning
the last layer, but it still falls short of the effectiveness achieved by
fine-tuning all layers.

##  5\. Visualization

Figure 6. Latent space for simulated PPG with various level of artifacts

We explored the latent space of the SiamQuality model by generating simulated
PPG signals with varying heart rates using NeuroKit2 (Makowski et al., 2021).
To assess the model’s robustness to noise, we incrementally introduced
different levels of drift noise, motion amplitude noise, and powerline noise
to the PPG signals, ranging from 0 (clean) to 0.7 (very noisy). After
subjecting these simulated signals to the pre-trained model, we analyzed the
resulting latent space. For dimension reduction and visualization, we employed
PaCMAP (Wang et al., 2021b), with the findings illustrated in Figure 6.
Notably, the results reveal that PPG signals with the same heart rate, despite
varying noise levels, remain clustered together in the latent space. This
observation underscores the efficacy of our proposed quality-pairing mechanism
in mitigating the impact of artifacts in PPG signals on downstream
physiological measurement tasks.

##  6\. Discussion

In this study, we propose SiamQuality, a CNN-based foundation model for PPG
signals. SiamQuality utilizes the SimSiam architecture but with a novel
pairing mechanism, where we pair one good quality signal with its bad quality
temporal neighbor. Curriculum learning was applied, where artifact level
difference was increased over iterations to make the problem more difficult.
After pre-training on 36 million PPG pairs, the model was then fine-tuned on
six downstream datasets. Through extensive experiments, we demonstrate the
capabilities of SiamQuality in terms of its contrastive learning architecture,
scaling ability, and adaptability. The analysis of the results is discussed as
follows:

First, our approach avoids selecting negative pairs (for example, pairs of
signals from different individuals) to train the SimSiam architecture. This is
because identifying helpful negative pairs for PPG model training is
challenging (Zheng et al., 2023a). As PPG signals exhibit strong periodicity,
leading to the recurrence of similar patterns within and across patients,
conventional augmentation methods may result in choosing inappropriate
negative pairs. The same conclusion can be drawn from Table 3: conventional
augmentation for time-series data, such as flipping and adding Gaussian noise,
did not perform as well as temporal sampling and was further outperformed by
quality-based pairing. This conclusion is in line with the results of Zheng et
al. (2023a), who show that using conventional augmentation methods to
construct negative pairs is not sufficient for time series data, and will lead
to performance degradation.

Second, CNNs can serve as an effective backbone for foundation models that are
robust to the quality of training data. Through our experiments, CNNs have
demonstrated notable efficiency in processing and learning from data with
inherent imperfections. This efficiency is attributed to their structural
advantage in capturing local patterns through convolutional filters, which
inherently makes them less sensitive to global noise and artifacts present in
physiological signals.

Third, our previously developed artifact measurement tool (Chen et al., 2023a)
plays a crucial role in this study by providing a quantifiable metric of data
quality, thereby enabling curriculum learning and the generation of the AT-
curve. Curriculum learning enables models to be better equipped to distill
underlying patterns and relationships in the data, even when presented with
signals heavily marred by noise and artifacts. The AT-curve allows a broader
illustration of the model’s performance on physiological datasets with various
artifact distributions.

As an early-stage investigation into foundation models for physiological data,
our work has several limitations.

Zero-shot learning is not available in our setting. The reason is that
SiamQuality focuses on aligning the latent space between good and bad quality
signals, rather than aiming for specific tasks, such as detecting AF or
estimating RR. This means that while SiamQuality excels at generalizing across
different qualities of data and ensuring robustness to artifacts and noise, it
does not inherently possess the ability to recognize or classify conditions it
has not been explicitly trained on.

Class imbalance problem. We encountered a failed case in PPG-based Premature
Ventricular Contraction (PVC) detection where the data provided in (Han et
al., 2020) was small and highly imbalanced (42 30-second PVC-PPG vs. 200
30-second normal-PPG segments). Our approach achieved a 52% F1 Score, while
the original study achieved over 95%. Our model’s struggle with PVC detection
highlights the necessity for strategies that can mitigate the effects of class
imbalance, such as synthetic data generation, oversampling of minority
classes, or the implementation of cost-sensitive learning together with the
fine-tuning stage. Addressing this issue is critical for ensuring that
foundation models for physiological data can accurately and reliably identify
a wide range of conditions, including those that are rare but clinically
significant.

The quality assessment tool is not universal. The tool is specifically
designed for PPG signals, so it requires tailored quality assessment tools for
each type of physiological data, such as ECG and EEG. To make our work more
generalizable and applicable to a broader range of physiological data, it is
important to develop a universal quality assessment tool that can accurately
identify and quantify the quality of signals in their respective domains.

##  7\. Deployment

Our model has been deployed at the following website:
https://www.nursingdatascience.emory.edu/modelmeetsdata. The demo of
ModelMeetsData (M2D) can be found at
https://www.youtube.com/watch?v=yA4Ky7hFdfw. In M2D, users can test various
PPG based models we developed by uploading their own PPG signals, and M2D will
provide the prediction (classification) or estimation (regression) from the
selected model. Any data uploaded by users will not be retained by M2D after
the session. Transition from theory to practical deployment on medical
wearable devices involves significant additional steps. This includes advanced
engineering, extensive testing and clinical trials, and collaboration with a
major manufacturer, which are beyond the scope of this paper.

##  8\. Reproductivity

Due to the nature of the private health data involved, the training dataset is
available upon request to the corresponding author. The six downstream
datasets are publicly available. Additionally, the code associated with this
project is made publicly available in
https://github.com/chengding0713/SiamQuality.

##  9\. Conclusion

SiamQuality addresses the often-neglected issue of data quality in
foundational models. Our focus is on photoplethysmography (PPG) signals — a
type of physiological data where quality concerns are particularly pronounced.
SiamQuality is based on the SimSiam architecture by implementing a unique
pairing mechanism that connects a high-quality signal with a closely following
lower-quality counterpart. After pre-trained on a large-scale dataset with 36
million PPG signal pairs, SiamQuality is further fine-tuned on six downstream
datasets. The outcomes of comprehensive testing highlight effectiveness of
contrastive learning, performance scalability with model size, and its
exceptional performance across a range of applications. This study goes beyond
merely presenting an efficient model for PPG signals; it can be used for many
applications that involve low-quality data, setting a precedent for addressing
data quality issues more broadly in AI and machine learning.

##  10\. Acknowledgement

This work is partially supported by NIH grant award R01HL166233.

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#  $\mathbf{{}^{12}{C}+{}^{12}{C}}$ Fusion $\bm{S^{*}}$-factor from a Full-
microscopic Nuclear Model

Yasutaka Taniguchi<EMAIL_ADDRESS>Department of Information
Engineering, National Institute of Technology (KOSEN), Kagawa College, Mitoyo,
Kagawa 769-1192, Japan  Research Center for Nuclear Physics (RCNP), Osaka
University, Ibaraki 567-0047, Japan  Masaaki Kimura
<EMAIL_ADDRESS>Department of Physics, Hokkaido University,
Sapporo 060-0810, Japan  Nuclear Reaction Data Centre, Hokkaido University,
Sapporo 060-0810, Japan  Research Center for Nuclear Physics (RCNP), Osaka
University, Ibaraki 567-0047, Japan

###### Abstract

The ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ fusion reaction plays a vital role
in the explosive phenomena of the universe. The resonances in the Gamow window
rule its reaction rate and products. Hence, the determination of the resonance
parameters by nuclear models is indispensable as the direct measurement is not
feasible. Here, for the first time, we report the resonances in the
${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ fusion reaction described by a full-
microscopic nuclear model. The model plausibly reproduces the measured low-
energy astrophysical $S$-factors and predicts the resonances in the Gamow
window. Contradictory to the hindrance model, we conclude that there is no
low-energy suppression of the $S$-factor.

Introduction.— The ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ fusion reaction is a
trigger and driving force of the carbon burning in massive stars Rolfs and
Rodney (1988); Woosley  _et al._ (2002) and the X-ray superburst Cumming and
Bildsten (2001); Strohmayer and Brown (2002). Hence, its reaction rate is a
key for understanding these explosive phenomena. In general, the reaction rate
of charged particles is represented by a product of the exponentially damping
factor and the astrophysical $S$-factor Rolfs and Rodney (1988). While the
former represents the Coulomb penetrability, the latter carries the nuclear
structural information consists of the resonant and non-resonant
contributions. Because the direct measurement is quite difficult, the
$S$-factor of the ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ reaction at low-energy
has been extrapolated from the measurement at higher energy. As a result, it
has been the source of considerable uncertainty despite decades of studies
Back  _et al._ (2014); Beck  _et al._ (2020). An estimate by Caughlan and
Fowler (CF88) Caughlan and Fowler (1988), a de facto standard for astrophysics
simulations, assumes constant $S$-factor. On the contrary, Jiang et al.
proposed the hindrance model Jiang  _et al._ (2007), which asserts reducing
the $S$-factor at low energy. It is still controversial which one is
reasonable.

In addition to the uncertainty in the global behavior, the fine structure of
the $S$-factor originating in the resonant contribution is even less
understood. It is well known that low-energy resonances can significantly
affect the evolution of astrophysical phenomena by increasing the reaction
rate in orders of magnitude at specific temperatures Rolfs and Rodney (1988).
For example, Cooper et al. introduced an ad-hoc resonance at 1.5 MeV inside
the Gamow window of X-ray superburst Cooper  _et al._ (2009). In nuclear
physics, such low-energy resonances in the
${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ system have long been discussed Almqvist
_et al._ (1960); Stokstad  _et al._ (1976); Korotky  _et al._ (1979); Freer
(2007). Recently a couple of direct measurements have identified the
resonances just above the Gamow window Spillane  _et al._ (2007); Bucher  _et
al._ (2015); Tan  _et al._ (2020); Fruet  _et al._ (2020). Furthermore, from
the indirect measurement, Tumino et al. reported numerous narrow resonances in
the Gamow window Tumino  _et al._ (2018). Contrary to the hindrance model,
they proposed the reaction rate enhanced at low energy, albeit its absolute
magnitude is still under debate Mukhamedzhanov  _et al._ (2019).

Thus, the resonances in the ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ system are
of particular interest. Since they are unlikely be measurable by the direct
reactions, the study by nuclear models is indispensable. However, the
description of deep sub-barrier resonances is challenging and demanding. The
primary reaction channels,
${}^{12}\mathrm{C}(^{12}\mathrm{C},\alpha){}^{20}\mathrm{Ne}$ and
${}^{12}\mathrm{C}(^{12}\mathrm{C},p){}^{23}\mathrm{Na}$, involve the
rearrangement of many nucleons and the strong channel coupling. To mimic such
complex reaction dynamics, nuclear models usually adopt phenomenological
potentials Gasques  _et al._ (2007); Esbensen  _et al._ (2011); Diaz-Torres
and Wiescher (2018); Chien  _et al._ (2018), undermining their predictability
at low energy. In principle, full-microscopic nuclear models without
phenomenologically adjustable parameters can overcome this problem Godbey  _et
al._ (2019), but the resonant contributions have never been taken into
account. Here, for the first time, we report a plausible description of the
resonances by a full-microscopic nuclear model and provide an evaluation of
the $S$-factor.

Nuclear model and resonance parameters.— To describe the low-energy
resonances, we employ antisymmetrized molecular dynamics (AMD) Kanada-En’yo
_et al._ (2012), which handles the channel coupling by configuration mixing.
Using the Gogny D1S nuclear density functional Berger  _et al._ (1991), it
accurately describes the low-energy resonances of astrophysical interests such
as ${}^{12}\mathrm{C}+{}^{16}\mathrm{O}$ and $\alpha+{}^{28}\mathrm{Si}$ and
their compound systems Kimura and Horiuchi (2004); Taniguchi  _et al._ (2009);
Taniguchi and Kimura (2020); Kimura and Taniguchi (2020). The model wave
function of AMD is a parity-projected Slater determinant of the nucleon wave
packets Kimura (2004),

$\displaystyle\Phi^{\pi}$ $\displaystyle=\frac{1+\pi
P_{r}}{2}{\mathcal{A}}\\{\varphi_{1}\cdots\varphi_{A}\\},\quad(\pi=\pm),$ (1)
$\displaystyle\varphi_{i}$
$\displaystyle=\prod_{\sigma=x,y,z}e^{-\nu_{\sigma}\left(r_{\sigma}-Z_{i\sigma}\right)^{2}}\left(a_{i}\ket{\uparrow}+b_{i}\ket{\downarrow}\right)\eta_{i},$
(2)

where each wave packet $\varphi_{i}$ has the parameters; Gaussian centroid
$\bm{Z}_{i}$, width $\bm{\nu}$ and spin $a_{i}$ and $b_{i}$. The isospin
$\eta_{i}$ is fixed to proton or neutron. Using the constraint on the inter-
nuclear distance Taniguchi  _et al._ (2004), all parameters are determined for
each channel and inter-nuclear distance by the energy variation. In this
study, we have calculated the ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ and
$\alpha+{}^{20}\mathrm{Ne}$ channels by using this method.

|  |
---|---|---
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Figure 1:  The wave functions for the (a)–(c) $\alpha+{}^{20}\mathrm{Ne}$, and
(d)–(f) ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ channels obtained by the energy
variation. Inter-nuclear distance is 6.5, 5.0, and 3.5 fm from left to right.

Figure 1 shows several channel wave functions which have different inter-
nuclear distances and orientations of nuclei. Note that the rotation and
polarization of nuclei depending on the inter-nuclear distance are described
naturally. As explained later, these are essential in describing the low-
energy resonances. In addition, we also include the wave functions of the
compound nucleus ${}^{24}{\rm Mg}$ Kimura  _et al._ (2012). All these channel
wave functions ($\Phi^{\pi}_{i}$) are superposed to describe the resonances,

$\displaystyle\Psi_{M}^{J\pi}=\sum_{iK}c_{iK}P^{J}_{MK}\Phi^{\pi}_{i},$ (3)

where $P^{J}_{MK}$ denotes the angular momentum projector. The resonance
energy and the coefficients $c_{iK}$ are determined by the diagonalization of
the Hamiltonian. The superposition of various channel wave functions
significantly improves the model accuracy. First, it describes the rotational
excitation of nuclei during the reaction process. Therefore, the model can
explain the decays to the excited states such as
$\alpha+{}^{20}\mathrm{Ne}(2^{+}_{1})$. It also accounts for the dynamical
change of the Coulomb barrier height due to the nuclear deformation Hagino and
Takigawa (2012). Second, the channel coupling is described by a full-
microscopic Hamiltonian in a unified manner. Hence, the model is free from
adjustable parameters. These improvements realize an accurate description of
the deep sub-barrier resonances.

To evaluate the decay widths, we calculate the reduced width amplitude (RWA)
Balashov  _et al._ (1959); Chiba and Kimura (2017) which is the overlap
between the decay channel and resonance wave functions,

$\displaystyle
y_{l}^{A_{1}+A_{2}}(a)=\sqrt{\tbinom{24}{A_{1}}}\Braket{\frac{\delta(r-a)}{a^{2}}[\Phi_{A_{1}}\Phi_{A_{2}}Y_{l}(\hat{r})]^{J}_{M}}{\Psi^{J\pi}_{M}},$
(4)

where the ket is the resonance wave function, while the bra is the channel
wave function decaying into two nuclei with masses $A_{1}$ and $A_{2}$
separated by the distance $a$ with the orbital angular momentum $l$. We
consider four decay channels, $p+{}^{23}\mathrm{Na}(3/2^{+}_{1})$,
$p+{}^{23}\mathrm{Na^{*}}(5/2^{+}_{1})$,
$\alpha+{}^{20}\mathrm{Ne}(0^{+}_{1})$ and
$\alpha+{}^{20}\mathrm{Ne^{*}}(2^{+}_{1})$ which are denoted by $p_{0}$,
$p_{1}$, $\alpha_{0}$ and $\alpha_{1}$, respectively. The wave functions of
$\alpha$, ${}^{20}\mathrm{Ne}$, and ${}^{23}\mathrm{Na}$ are also calculated
by AMD. For the later use, the RWAs are given as the ratio to the Wigner
limit,

$\theta^{2}_{A_{1}+A_{2}}(a)=\frac{a}{3}\left|ay_{l}^{A_{1}+A_{2}}(a)\right|^{2},$
(5)

where the channel radii for the ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$,
$\alpha_{0,1}$, and $p_{0,1}$ channels are chosen as 6, 6 and 4 fm so that
RWAs are smoothly connected to the Coulomb wave function.

Table 1: The calculated resonance energies in MeV, isoscalar transition matrix elements in Weisskopf unit, and RWAs of the $0^{+}$ and $2^{+}$ resonances. The RWAs are given as the ratio to the Wigner limit (see text) and multiplied by a factor of a hundred. |  |  | $\theta_{\mathrm{C}}^{2}\times 10^{2}$ | $\theta_{\alpha_{0}}^{2}\times 10^{2}$ | $\theta_{\alpha_{1}}^{2}\times 10^{2}$ | $\theta_{p_{0}}^{2}\times 10^{2}$ | $\theta_{p_{1}}^{2}\times 10^{2}$
---|---|---|---|---|---|---|---
$J^{\pi}$ | $E_{\mathrm{R}}$ | $M_{\rm IS}$ | $l=J$ | $l=0$ | $2$ | $l=0$ | $2$ | $4$ | $l=0$ | $2$ | $4$ | $l=0$ | $2$ | $4$
$2^{+}$ | 0.93 | 1.56 | $1.4$ | — | $3.5$ | $0.061$ | $1.7$ | $6.7$ | $0.47$ | $0.081$ | $0.030$ | $0.20$ | $0.15$ | $0.083$
$0^{+}$ | 0.94 | 0.59 | $7.3$ | $0.20$ | — | — | $7.1$ | — | — | $0.10$ | — | — | $0.69$ | —
$2^{+}$ | 1.50 | 1.04 | $2.9$ | — | $1.1$ | $4.0$ | $0.90$ | $0.51$ | $0.16$ | $0.22$ | $0.001$ | $0.012$ | $0.098$ | $0.005$
$2^{+}$ | 2.18 | 0.51 | $3.4$ | — | $1.0$ | $1.0$ | $0.19$ | $3.4$ | $3.3$ | $0.12$ | $0.010$ | $0.70$ | $0.11$ | $0.23$
$0^{+}$ | 3.02 | 1.05 | $11$ | $0.26$ | — | — | $0.57$ | — | — | $0.99$ | — | — | $0.43$ | —
$2^{+}$ | 3.56 | 0.23 | $1.2$ | — | $0.038$ | $0.056$ | $0.006$ | $0.040$ | $0.66$ | $0.86$ | $0.001$ | $0.029$ | $0.79$ | $0.041$
$2^{+}$ | 3.73 | 0.41 | $8.3$ | — | $0.10$ | $0.066$ | $0.10$ | $0.88$ | $0.24$ | $0.72$ | $0.028$ | $0.043$ | $0.67$ | $0.089$

Table 1 lists the calculated resonance parameters in the energy range of
interest (resonance energy $E_{\mathrm{R}}<4$ MeV). Note that the calculation
yields many deep sub-barrier resonances. In particular, it is remarkable that
some of them plausibly coincide with the observed higher-energy peaks of the
$S$-factor at higher energies. For example, the $J^{\pi}=2^{+}$ resonances
obtained at 2.18 and 3.73 MeV are the candidates for the 2.14 and 3.8 MeV
resonances identified by the direct measurements Kettner  _et al._ (1980);
Spillane  _et al._ (2007); Fruet  _et al._ (2020). Furthermore, the
calculation predicts three resonances within the Gamow window at 0.93, 0.94,
and 1.50 MeV. This result confirms the long-standing conjecture about the
existence of the low-energy resonances in the
${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ system Almqvist  _et al._ (1960);
Stokstad  _et al._ (1976); Korotky  _et al._ (1979) and is in accordance with
the indirect measurement Tumino  _et al._ (2018) which reported many
resonances.

The RWAs of the resonances demonstrate the importance of the microscopic
treatment of channel coupling and rotational excitation. The $0^{+}$ resonance
at 0.94 MeV has a large RWA in the $\alpha_{1}$ channel showing the rotational
excitation of ${}^{20}{\rm Ne}$. Moreover, the $2^{+}$ resonance at 2.18 MeV
has sizable RWAs in the $\alpha_{0}$, $\alpha_{1}$, and $p_{0}$ channels and
in the entrance ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ channel. Therefore, it
should affect the reaction rates in both the $\alpha$ and $p$ channels. The
importance of the channel coupling can also be confirmed differently. If we
perform a single-channel calculation, for example, with only the
${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ channel, we do not obtain any resonance
in the Gamow window. It demonstrates the necessity of the full-microscopic
calculation for quantitative discussion of the sub-barrier resonances.

Figure 2: Calculated and observed Becker  _et al._ (1981); Fruet  _et al._
(2020); Tan  _et al._ (2020); Spillane  _et al._ (2007) modified astrophysical
$S^{*}$-factors in the (a) $p_{1}$ and (b) $\alpha_{1}$ channels. The
calculated $S^{*}$-factors in the $p_{0}$ and $\alpha_{0}$ channels are also
shown for the comparison. Panel (c) compares the calculated and observed total
$S^{*}$-factors with the evaluations Jiang  _et al._ (2007); Caughlan and
Fowler (1988). Open symbols indicate upper limits.

Astrophysical S-factor.— From the resonance parameters, we evaluate the
modified astrophysical $S$-factors ($S^{\ast}$-factors) Patterson  _et al._
(1969). According to the $R$-matrix theory Lane and Thomas (1958);
Descouvemont and Baye (2010), the partial width for the $A_{1}+A_{2}$ decay is
given as,

$\Gamma_{A_{1}+A_{2}}=\frac{2ka}{F_{l}(ka)^{2}+G_{l}(ka)^{2}}\frac{3\hbar^{2}}{2\mu
a^{2}}\theta_{A_{1}+A_{2}}^{2},$ (6)

where the wave number $k$ is related with the decay $Q$-value as
$Q=\hbar^{2}k^{2}/2\mu$ with the reduced mass $\mu$. $F_{l}$ and $G_{l}$ are
the regular and irregular Coulomb wave functions, respectively. The partial
cross section at the center-of-mass energy $E$ is given as the Breit-Wigner
form Lane and Thomas (1958),

$\sigma(E)=\frac{\pi\hbar^{2}(2J+1)}{2\mu
E}\frac{\Gamma_{\mathrm{I}}\Gamma_{\mathrm{A_{1}+A_{2}}}}{\left(E-E_{\mathrm{R}}\right)^{2}+\Gamma^{2}/4},$
(7)

where $J$ and $E_{\mathrm{R}}$ denote the spin and energy of a resonance and
$\Gamma_{\mathrm{I}}$ is the partial width in the entrance
(${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$) channel. The total width $\Gamma$ is
estimated in the same manner as in the latest experimental analysis Tan  _et
al._ (2020); Fruet  _et al._ (2020), i.e., it is estimated from the $p_{1}$
and $\alpha_{1}$ widths and the linear extrapolation of the observed branching
ratio Becker  _et al._ (1981). The $S^{\ast}$-factor is calculated as,

$S^{\ast}(E)=E\sigma(E)\exp(2\pi\eta+0.46~{}\mathrm{MeV}^{-1}E),$ (8)

with the Sommerfeld parameter $\eta=36/137\sqrt{\mu c^{2}/2E_{\text{R}}}$
Patterson  _et al._ (1969).

Figure 2(a) compares the calculated and observed $S^{*}$-factors in the
$p_{1}$ channel. Above the Gamow window ($E\gtrsim 2.0$ MeV), the calculated
$S^{*}$-factor has the contributions from four resonances (2.18, 3.02, 3.56,
and 3.73 MeV) and its magnitude is the order of
$10^{14}$–$10^{15}~{}\mathrm{MeV\cdot b}$, which plausibly agrees with the
experiments Spillane  _et al._ (2007); Tan  _et al._ (2020); Fruet  _et al._
(2020); Becker  _et al._ (1981); Kettner  _et al._ (1980). The peak positions
at 2.2 and 3.8 MeV coincide with the direct
${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ fusion experiments Kettner  _et al._
(1980); Spillane  _et al._ (2007); Fruet  _et al._ (2020). The calculation
predicts that the resonant contributions are also present inside the Gamow
window. In particular, the resonances at 0.94 and 0.95 MeV form a prominent
peak. For the comparison, the calculated $S^{*}$-factor in the $p_{0}$ channel
is also shown in Fig. 2(a). It has a similar shape, but the magnitude is much
larger than the $p_{1}$ channel.

Because of the strong coupling between $p$ and $\alpha$ channels, the same
resonant peaks appear in the $\alpha_{1}$ channel, as shown in Fig. 2(b). The
peaks at 2.2 and 3.8 MeV exist in the $\alpha_{1}$ channel in the calculation
and observed data Kettner  _et al._ (1980); Becker  _et al._ (1981); Spillane
_et al._ (2007). The calculation yields more prominent peak at 3.0 MeV than
that in the $p_{1}$ channel. This peak nicely coincides with the peak
approximately at 3.2 MeV observed in Refs. Kettner  _et al._ (1980); Becker
_et al._ (1981); Spillane  _et al._ (2007). It is impressive that the
calculation predicts pronounced $S^{*}$-factor inside the Gamow window even
stronger than above the window owing to the contributions from the resonances
at 0.93 and 0.94 MeV.

Finally, Fig. 2(c) shows the total $S^{\ast}$-factor as the sum of the
$S^{*}$-factors in all $\alpha$ and $p$ channels. Here, the total
$S^{*}$-factors in the $\alpha$ and $p$ channels are again estimated in the
same manner with Refs. Tan  _et al._ (2020); Fruet  _et al._ (2020). The
figure also shows the estimations by CF88 Caughlan and Fowler (1988) and the
hindrance model Jiang  _et al._ (2007) in addition to the data from Refs.
Becker  _et al._ (1981); Spillane  _et al._ (2007); Tan  _et al._ (2020);
Fruet  _et al._ (2020). Similar to the $S^{*}$-factors in the $p_{1}$ and
$\alpha_{1}$ channels, the calculation plausibly describes the observed global
behavior. Furthermore, as we have already repeated, the calculation predicts
resonance contributions at very low energy. Consequently, the calculated
$S^{*}$-factor has prominent peaks in the Gamow window and does not show the
low-energy hindrance. Hence, the low-energy enhancement of the $S$-factor, for
example, that postulated by Cooper et al. Cooper  _et al._ (2009), is
conceivable enough.

A novel probe for the resonances.— Finally, we propose the isoscalar
transitions as a novel experimental probe to identify the resonances predicted
in this study. The transition matrix element from the ground state to a
resonance is defined as,

$\displaystyle
M_{\mathrm{IS}}=\braket{J^{\pi},J_{z}=0}{\mathcal{M}_{\mathrm{IS}}}{\mathrm{{}^{24}{Mg}(g.s.)}},$
(9)

where the bra is the resonance wave function with the spin-parity $J^{\pi}$,
while the ket is the ground state of ${}^{24}{\rm Mg}$.
$\mathcal{M}_{\mathrm{IS}}$ is either of the isoscalar monopole or quadrupole
operators depending on the spin-parity of the resonance,

$\displaystyle\mathcal{M}_{\mathrm{IS}}=\left\\{\begin{array}[]{l}\sum_{i=1}^{A}r_{i}^{2},\quad
J^{\pi}=0^{+}\\\ \sum_{i=1}^{A}r_{i}^{2}Y_{20}(\hat{r}_{i}),\quad
J^{\pi}=2^{+}.\end{array}\right.$ (12)

In recent years, it has been extensively discussed that the isoscalar
transitions from the ground state to the cluster resonances are considerably
enhanced Kawabata  _et al._ (2007); Kanada-En’yo (2007); Yamada  _et al._
(2008); Chiba  _et al._ (2016). As listed in Table 1, the present calculation
predicts that all the resonances have large transition strengths comparable
with the Weisskopf unit. Therefore, the reactions that induce isoscalar
transitions should strongly populate these resonances. The most promising and
feasible reaction may be the $\alpha$-inelastic scattering, ${}^{24}{\rm
Mg}(\alpha,\alpha^{\prime})^{24}{\rm Mg}^{*}$. Note that this reaction
bypasses the Coulomb barrier, and hence, can easily access the deep sub-
barrier resonances. It is encouraging that a couple of experiments have
already been conducted Youngblood  _et al._ (2009); Kawabata  _et al._ (2013);
Gupta  _et al._ (2015); Adsley  _et al._ (2021), and several candidates of the
resonances are observed in the energy range of our interest, although the
spin-parity and decay branches have not been confirmed firmly. More detailed
analysis and the comparison with the calculations must uncover the resonances
in the Gamow window.

Summary.— We have investigated the low-energy resonances in the
${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ fusion reaction, which is of
astrophysical interest. The AMD, a microscopic nuclear model, firstly realized
a quantitative description of the low-energy resonances by handling the
channel coupling and nuclear rotation without adjustable parameters. It
successfully described observed resonances and behavior of the $S^{*}$-factors
above the Gamow window. Furthermore, it predicted several resonances inside
the Gamow window, which create low-energy $S^{*}$-factor peaks. Consequently,
the calculation suggested no low-energy suppression of the $S^{*}$-factor
supporting the result of the indirect measurement Tumino  _et al._ (2018).
Finally, we propose the ${}^{24}{\rm Mg}(\alpha,\alpha^{\prime})^{24}{\rm
Mg}^{*}$ reaction experiment as the most promising bypass to access the deep
sub-barrier resonances.

###### Acknowledgements.

This work was supported by JSPS KAKENHI Grant Number 19K03859, the
collaborative research program at Hokkaido University, and the COREnet program
in RCNP, Osaka University. Numerical calculations were performed using
Oakforest-PACS at the CCS, University of Tsukuba.

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