# Asteroseismology applied to constrain structure parameters of $\delta$ Scuti
stars

Subrata Kumar Panda  Tata Institute of Fundamental Research, Colaba, Mumbai
400005, Maharashtra, India  Siddharth Dhanpal  Tata Institute of Fundamental
Research, Colaba, Mumbai 400005, Maharashtra, India  Simon J. Murphy  Centre
for Astrophysics, University of Southern Queensland, Toowoomba, QLD 4350,
Australia  Shravan Hanasoge  Tata Institute of Fundamental Research, Colaba,
Mumbai 400005, Maharashtra, India  Timothy R. Bedding  Sydney Institute for
Astronomy (SIfA), School of Physics, University of Sydney, NSW 2006, Australia

###### Abstract

Asteroseismology is a powerful tool to probe the structure of stars. Space-
borne instruments like CoRoT, Kepler and TESS have observed the oscillations
of numerous stars, among which $\delta$ Scutis are particularly interesting
owing to their fast rotation rates and complex pulsation mechanisms. In this
work, we inferred model-dependent masses, metallicities and ages of 60
$\delta$ Scuti stars from their photometric, spectroscopic and asteroseismic
observations using least-squares minimization. These statistics have the
potential to explain why only a tiny fraction of $\delta$ Sct stars pulsate in
a very clean manner. We find most of these stars with masses around
$1.6~{}{\rm M_{\odot}}$ and metallicities below $Z=0.010$. We observed a
bimodality in age for these stars, with more than half the sample younger than
30 Myr, while the remaining ones were inferred to be older, i.e., hundreds of
Myrs. This work emphasizes the importance of the large-frequency separation
($\Delta\nu$) in studies of $\delta$ Scuti stars. We also designed three
machine learning (ML) models that hold the potential for inferring these
parameters at lower computational cost and much more rapidly. These models
further revealed that constraining dipole modes can help in significantly
improving age estimation and that radial modes succinctly encode information
pertaining to stellar luminosity and temperature. Using the ML models, we also
gained qualitative insight into the importance of stellar observables in
estimating mass, metallicity, and age. The effective surface temperature
$T_{\mathrm{eff}}$ strongly affects the inference of all structure parameters
and the asteroseismic offset parameter $\epsilon$ plays an essential role in
the inference of age.

##  1 Introduction

Asteroseismic observations from TESS (Ricker et al. 2014) and Kepler (Borucki
et al. 2010) have shed light on the dynamics and interiors of thousands of
pulsating stars (Paparó 2019; Bowman 2020; Aerts 2021; Kurtz 2022). A sizeable
fraction of the pulsating class from these missions are $\delta$ Scuti stars
(Bowman et al. 2018; Guzik 2021), which lie at the junction of the instability
strip and the main sequence (MS) band in the Hertzsprung-Russell (HR) diagram
(Uytterhoeven, K. et al. 2011), providing direct views of both these classes.
These are low-to-intermediate mass ($1.5-2.5M_{\odot}$) main-sequence
variables with spectral types ranging from A-F (Bowman et al. 2018; Murphy et
al. 2021b; Kurtz 2022). High amplitude delta Scuti stars (HADS) are used as
standard candles (McNamara et al. 2007), and in assessing the metallicities
and ages of stellar clusters (Murphy et al. 2022), in turn enabling Galactic
archaeology.

Asteroseismology is a powerful tool which can be used to estimate various
structure parameters such as mass, composition, and age on different pulsating
classes, e.g., stochastic oscillators (Chaplin et al. 2013; Hekker &
Christensen-Dalsgaard 2017; Vrard, M. et al. 2016; Hon et al. 2017, 2018;
Dhanpal et al. 2022), $\gamma$-Doradus stars (Li et al. 2019; Mombarg, J. S.
G. et al. 2021) and high-mass coherent oscillators (Hendriks et al. 2019),
among others. Some efforts have been applied to similar studies of $\delta$
Sct stars (Suárez, J. C. et al. 2014a; García Hernández et al. 2017; Barceló
Forteza, S. et al. 2020; Pamos Ortega et al. 2022). In this paper, we develop
a methodology to measure structure parameters - mass, metallicity, and ages of
$\delta$ Sct stars. These measurements can in principle constrain the
metallicities and ages of the host open clusters from which these stars formed
and were dispersed before escaping to become field stars.

$\delta$ Sct stars predominantly exhibit low radial-order pulsations, with
high-frequency pressure and low-frequency gravity modes, roughly separated
around $5~{}\mathrm{d}^{-1}$, although this depends on $T_{\mathrm{eff}}$ to
leading order (Moya et al. 2017). The pressure modes are mainly driven by the
$\kappa$ mechanism (Chevalier 1971) in the Helium ionization zone. They
primarily propagate in the stellar envelope and probe the near-surface
regions.

Although we have precise estimates of the luminosity ($L$), effective
temperature ($T_{\mathrm{eff}}$) and pulsation frequencies, the seismology and
the parameter inference of $\delta$ Sct stars is challenging because of the
following reasons: (a) they have low radial order p-modes and g-modes, where
asymptotic theory fails, (b) many stars exhibit fast rotation, leading to
ellipsoidal deformation (Reese, D. et al. 2006), (c) complex mode-selection
mechanisms (Dziembowski et al. 1990) influencing the observed spectra, and (d)
appearance of island modes (Reese, D. et al. 2006), chaotic modes (Barceló
Forteza, S. et al. 2017) etc. making it very difficult to characterize
$\delta$ Sct spectra.

Finding regular patterns in $\delta$ Sct spectra is very helpful for carrying
out asteroseismology. Although some $\delta$ Sct stars had been previously
reported to show regular frequency spacings (Matthews 2007; García Hernández,
A. et al. 2009; Zwintz, K. et al. 2011; Paparó, M. et al. 2013; Suárez, J. C.
et al. 2014b), a larger ensemble of such stars was found by Bedding et al.
(2020), who identified 60 $\delta$ Sct stars from TESS and Kepler exhibiting
regular pulsation patterns. They were able to identify some modes, label their
radial orders and deduce the large-frequency separation ($\Delta\nu$) and
p-mode offset ($\epsilon$). Many more examples of high-frequency patterns in
$\delta$ Sct stars have subsequently been identified (e.g., Murphy et al.,
2020, 2021b; Hasanzadeh et al., 2021; Le Dizès et al., 2021; Kahraman Aliçavuş
et al., 2022; Murphy et al., 2022). Here, we use seismic parameters to infer
stellar structure through the application of neural networks and other
techniques. As the relationships between observables such as luminosity,
temperature, pulsation frequencies and structure parameters can be highly non-
linear, neural networks can potentially be useful in building a model
connecting them all. In addition, a well-trained neural network infers these
structure parameters substantially faster than conventional methods such as
MCMC, making it a capable method for ensemble studies.

At the core of this work lies the simulated models of $\delta$ Sct stars,
which we briefly describe in section 2. Following that we elaborate on
developing three different machine learning methods in Section 3, which in
principle could have inferred mass, metalicity, and ages of $\delta$ Sct
stars. However, none of the above methods worked uniformly well over all the
stars in our sample because of various limitations. Yet, the networks helped
us in gaining qualitative insights into importance of different observables,
which we present alongwith each method. Since our goal was to characterize the
60 $\delta$ Scutis from Bedding et al. 2020, we finally deployed a grid-search
based least-squares minimization technique in Section 4, with which we were
able to infer ($M,Z,\tau$) parameters for all stars. We end the article with
presenting the statistics of these inferences. While we have presented all the
machine-learning based methods for the sake of completeness, we found that the
least-squares fitting method performed best and therefore regard its output as
our final result.

##  2 Grid of stellar models

We have used the model grid presented in Murphy et al. (2023) which contains
more than 800,000 stellar models. However since we aimed for training the
networks on $\delta$ Sct stars, we extracted 524,247 models based on a simpler
criterion of 6500 K $\leq T_{\mathrm{eff}}\leq$ 10000 K. MESA inlists used for
the grid computation and the corresponding output models are available as
supplementary files in Murphy et al. 2023. This grid covers a wider span of
mass ($M$) and metallicity ($Z$) over the ranges $[1.3-2.2]~{}M_{\odot}$ and
$[0.002-0.026]$, with $0.1M_{\odot}$ ($M$) and 0.002 ($Z$) resolution
respectively.

These models were evolved using the stellar evolution code mesa (Paxton et al.
2010, 2013, 2015, 2018, 2019) and corresponding pulsation frequencies were
calculated using gyre (Townsend et al. 2013, 2017; Goldstein et al. 2020) –
which in turn provides the large-frequency separation ($\Delta\nu$) and offset
parameter ($\epsilon$).

The computed eigenmodes comprise p, g and mixed modes oscillating at
frequencies between 0 to 95 $d^{-1}$ and at harmonic degrees of $\ell$ = 0 and
1. Since the majority of $\delta$ Sct stars pulsate in radial and dipole
modes, and higher-degree modes are not observable due to geometric
cancellation, the availability of modes with degree $\ell$ = 2 or higher is
not necessary for our purpose. All the computed modes correspond to non-
rotating $m=0$ components. $\Delta\nu$ and $\epsilon$ were calculated by
fitting the asymptotic relation to radial ($\ell=0$) p-modes with order
($n_{\rm pg}$) larger than 5.

We used this grid to train the neural networks to infer structure parameters
from observables and asteroseismic quantities.

The HR diagram (Figure 1) shows the stellar models, including the potential
$\delta$ Sct models ($3.83\leq\log T_{\mathrm{eff}}\leq 4$) on this HR
diagram. For reference, we mark the target stars of this work using ‘+’
symbols, which are taken from the observed $\delta$ Sct stars in Bedding et
al. (2020). Figure 2 contains the same models as Figure 1, but showing the
asteroseismic $\Delta\nu-\epsilon$ diagram. Both figures demonstrate that the
phase space of our stellar grid is broad enough to represent the wide range of
observed $\delta$ Sct stars.

Figure 1: HR diagram showing the positions of all synthetic stellar models,
with $\delta$ Sct stars present in the solid patch. The $\delta$ Sct targets
of this paper are denoted using ‘+’ symbols. Figure 2: $\Delta\nu-\epsilon$
diagram denoting the positions of all synthetic stellar models, with $\delta$
Sct stars present in the solid patch. The $\delta$ Sct targets of this paper
are denoted using ‘+’ symbols.

##  3 Machine learning based methods

The application of machine learning (ML) algorithms to the fitting of stellar
models has shown great promise (Verma et al. 2016; Hendriks et al. 2019;
Dhanpal et al. 2022; Scutt et al. 2023). For the work presented in this
article, we have developed three different versions of neural networks which
perform regression to infer the values of $M$, $Z$ and age ($\tau$). Our
networks were developed using the Python libraries tensorflow (Abadi et al.
2015) and keras (Chollet et al. 2015).

In each set of ML experiments, we randomly split the $\delta$ Sct model grid
into ‘training’ and ‘test’ (validation) data at a $95:5$ ratio. The network
used the training data to learn different features and compute the output. The
training aimed to minimize the loss function by reducing the mean-squared
error between the actual and predicted outputs (equation 1) through stochastic
gradient descent.

$\mathrm{MSE}=\sum_{i=1}^{N}\dfrac{(y^{i}_{\mathrm{predicted}}-y^{i}_{\mathrm{true}})^{2}}{N}$
(1)

The training takes place through a back propagation technique which gradually
optimizes the network parameters using the Adam optimizer (Kingma et al.
2014). To avoid the situation where networks memorize the dataset instead
learning the inherent features, we equipped the output layer of each network
with an L2 regularizer having regularizing parameter $\lambda=10^{-6}$.

Once the training is over, we validated network accuracy on the unseen test
data and also compared the network prediction against ground truth. If the
accuracy and the comparison were deemed acceptable on the validation data, we
concluded that the network was successfully trained without overfitting.

We used the cross-validation method (Mosteller et al. 1968) to yield a
distribuion of 40 instances of inferences from trained network of which
median, 16th and 84th percentiles were calculated to obtain most probable
result, lower and higher uncertainties.

However, these three networks were not able to yield meaningful inferences
over real stars due to several limitations discussed below. Nonetheless, they
have the potential to provide qualitative insights on relative importance of
different observables in the way (Bellinger et al. 2016) assessed for main
sequence stars. We performed this analysis for all the three networks.
Physical interpretation of such analysis are also presented.

###  3.1 Method 1: ML using seismic indices

In this method we deployed three regression networks to infer $M$, $Z$ and
$\tau$ from four input parameters: {$L,T_{\mathrm{eff}},\Delta\nu,\epsilon$}.
Although radius ($R$) is dependent on $L$ and $T_{\mathrm{eff}}$,
supplementing it as an additional input helped the networks train quickly and
achieve higher accuracy.

For the network to infer age ($\tau$), it typically requires prior knowledge
of $M$ and $Z$: this is because each physical parameter ($\theta$) is a
functions of $M$, $Z$ and $\tau$, which is why age inversion requires ($M$,
$Z$) to be supplied as inputs.

$\theta=f(M,Z,\tau)\implies\tau=\tilde{f}^{-1}(M,Z,\theta)$

The network architecture comprised of 1 input layer (with 5 to 7 neurons), 10
intermediate layers (of 400 neurons each), and 1 single-neuron output layer
(Figure 3). With the exception of the output layer, which has a tanh
activation function, all other neurons were activated using the rectified
linear unit (reLU) function. Since we developed a few alternative methods, we
label this procedure as Method-1 for easy reference.

Figure 3: Architecture of the neural-network models. Figure 4: Plots showing
network predictions for validation samples against their corresponding true
values. Shown for (a) $M$, (b) $Z$, and (c) $\log\tau$. Uncertainties
associated with predicted values are also shown. Figure 5: Distribution of
errors between network predictions for validation samples and their
corresponding true values: shown for (a) $M$, (b) $Z$, and (c) $\log\tau$.

We trained the networks and compared the network-predicted values for the
validation data with the corresponding true values (Figure 4). The network was
able to achieve high accuracy in predictions of $M$, and $Z$. Figure 5 shows
the distributions of errors in our predictions on the validation data.

From Figure 4 (c), it is evident that while age inferences are accurate for
younger ($\log\tau/\rm{Myr}<1$) and older stars ($\log\tau/\rm{Myr}>2.5$), the
network systematically over-predicts for the stars of intermediate age
($\log\tau/\rm{Myr}\sim 1.3-2.5$). Possible reason for this discrepancy is
that no observable evolves as a monotonic function of age. All structure and
seismic parameters cross their pre-main-sequence values sometimes within the
main sequence, as shown in Figure 6. Hence, none of our inputs may be used to
uniquely distinguish between two possible ages. This fact was also discussed
in Murphy et al. 2021a.

Figure 6: Evolution with time of stellar properties for a model with mass 2.2
$M_{\odot}$ and $Z=0.026$. Most structural and seismic quantities cross their
pre-MS values during MS. Hence, values of these parameters are degenerate at
two different ages. Crossing effects are shown for (a) $\rho$, (b) $\log g$,
(c) $\log T_{\mathrm{eff}}$, (d) $\Delta\nu$, (e) $\epsilon$ and (f) $\nu_{1}$
(frequency corresponding to $n=1,\ell=0$).

We implemented our trained networks on 43 $\delta$ Sct stars from Bedding et
al. (2020), for which $T_{\mathrm{eff}},L,\Delta\nu,\epsilon$ measurements
were simultaneously available. While investigating the results, we noticed
that most of the inferences have unusually high metalicity of $Z$ = 0.026,
which is also the upper boundary of our model grid.

It is well-known that what the neural networks learn is not easily
interpretable (interpretml). This makes it impossible to investigate and
mitigate the shortcomings with the method.

####  3.1.1 Feature Importance

We determined the qualitative importance of various inputs to determine how
strongly they influence parameter estimation. To measure the independent
contributions arising from the inputs, we perturbed each input quantity by
0.5% (without changing other inputs) and measured the relative differences in
outputs. We expected the contribution of the input to be proportional to the
relative difference in output. We carried out this process for all input
quantities to determine their qualitative importance. Figure 7 shows the
average relative differences, which can be taken as a proxy for feature
importance.

Figure 7: Different inputs contribute unequally towards overall inferences of
(a) $M$, (b) $Z$ and (c) $\log\tau$. This figure qualitatively depicts the
contribution strengths corresponding to the input parameters when perturbed by
0.5%.

Figure 7 indicates that $\log T_{\mathrm{eff}}$ contributes the most towards
the inference of all parameters. $T_{\mathrm{eff}}$ is strongly influenced by
the conditions at the stellar core ($T_{c}$) through various transport and
mixing processes. Hence stellar age, which strongly depends on core hydrogen
abundance ($X_{c}$), can be expected to correlate with $T_{c}$ and hence,
indirectly with $T_{\mathrm{eff}}$.

$M$ and $Z$ are necessary inputs for the inference of age, which is
straightforward from the inversion relation discussed above. Despite being an
offset parameter that controls the shift in the $\ell=0$ frequencies,
$\epsilon$ plays an important role in the inference of $\tau$. This was also
emphasized in Bedding et al. 2020, where the $\Delta\nu-\epsilon$ diagram was
shown to encode information about age ($\tau$).

Apart from $T_{\mathrm{eff}}$, $M$ significantly depends on $L$ which is
likely due to the well known “mass-luminosity” relation. Its dependence on
$\Delta\nu$ is not surprising owing to the fact that $M$ is related to density
($\rho$) and $\Delta\nu\propto\sqrt{\rho}$.

$\Delta\nu$ plays an important role in the inference of $Z$. It is highly
correlated with the square root of mean stellar density – which is tightly
related to metallicity ($Z$) because the higher the metallicity, the lower the
stellar density.

###  3.2 Method 2: ML using radial modes

Although we used $L$ as an input quantity in Method-1, the reliability of its
measurements depends on the accuracy of stellar distance, as $L$ is calculated
using magnitude and distance. Since structure parameters are very sensitive to
$\log T_{\mathrm{eff}}$ (evident from section 3.1.1), errors in measuring the
latter propagate into inferences of the former.

In this section, we show that if we use first seven eigenfrequencies of
radial-mode oscillations instead of the observables
($L,T_{\mathrm{eff}},\Delta\nu,\epsilon$), we still achieve similar results
while overcoming existing issues. Reason behind such selection is that (i) all
the synthetic models at least seven overtones of radial modes and (ii) radial
modes are far more easily identifiable than dipole modes using period ratios
(Petersen & Christensen-Dalsgaard (1996)) and multicolor photometry (Garrido
et al. 1990) like techniques.

Architecture and training of this network is similar to that of Method-1,
except its input layer comprising of seven neurons to accept the seven radial
modes.

In this formalism, neural networks could learn all the parameters, even
without needing {$L,T_{\mathrm{eff}},\Delta\nu,\epsilon$}, as seen in Figure
8. We compare the average learning accuracy ( = 100% - average% error)
associated with each parameter inference using the current method and Method-1
of Section 3.1. This method is as accurate as Method-1 in inferring $M$ and
$Z$. However, it infers age at a significantly higher accuracy (Table 1). We
designate this procedure as Method-2 in order to distinguish it from Method-1.

| Method-1 | Method-2
---|---|---
| (Section 3.1) | (Current Method)
Inputs: | ($L,T_{\mathrm{eff}},\Delta\nu,\epsilon$) | {$\nu_{1}-\nu_{7}$}
$M$ | 99.9% | 99.5%
$Z$ | 99.7% | 99.5%
$\log\tau$ | 93% | 96%
Table 1: Comparing performances as quantified using average learning accuracy
of the robust method discussed in this section and Method-1 of Section 3.1.
While both perform similarly, Method-2 works without
{$L,T_{\mathrm{eff}},\Delta\nu,\epsilon$}. Figure 8: Network predictions for
validation samples for (a) $M$, (b) $Z_{i}$ and (c) $\log\tau$, on taking
eigenfrequencies {$\nu_{1},\dots,\nu_{7}$} as inputs (without $L$,
$T_{\mathrm{eff}}$, $\Delta\nu$, $\epsilon$). This implies that these non-
asymptotic modes are helpful for parametric inference, particularly useful
when $L$ or $T_{\mathrm{eff}}$ are not available or reliable.

This method highlights the importance of the $\ell=0$ eigenfrequencies of
radial orders $n=1-7$. It shows that the non-asymptotic modes succinctly
preserve the representation of $L$ and $T_{\mathrm{eff}}$, although how this
is so is unclear.

Despite its robustness, we were unable to apply this method to the observed
stars since accurately identifying continuous radial overtones
($\nu_{1}-\nu_{7}$) is difficult. Additionally, it was harder to automate this
method for multiple stars because identifying radial orders requires
significant human intervention. Identifying radial modes crucially depends on
choosing the correct $\Delta\nu$, without which échelle diagrams cannot be
constructed. Even if this method worked efficiently, we were unable to resolve
the age discrepancy (similar as Method-1) over the synthetics, as evident from
Figure 8 (c). In Section 3.3, we propose another method that improves over
this issue.

####  3.2.1 Feature Importance

We investigated the strengths of contributions arising from each mode involved
in the inference routine. We measured their importance in a manner similar to
that in Section 3.1.1. The attendant measures are shown in Figure 9. It may be
understood from the figure that all frequencies are not equally important.
$\nu_{6}$ has the highest contribution, followed by $\nu_{2}$ and $\nu_{3}$.
Hence, both non-asymptotic and asymptotic modes seem to be important.

Figure 9: Importance of first 7 eigenfrequencies in the inference of (a) $M$,
(b) $Z$ and (c) $\log\tau$.

It is surprising that the network depends so significantly on $\nu_{6}$,
robust over a range of different $\Delta\nu$ and $\epsilon$. The probable
cause is that the transition to asymptotic nature ($n\gg\ell$) from low-order
non-asymptotic modes occurs around $\nu_{6}$. It is possible that $\nu_{6}$
therefore contains information from both regimes.

Having emphasized the salient features of $\nu_{6}$, it is interesting that
the network does not equally depend on modes of radial order 4, 5 and 7 -
which altogether constitute the asymptotic series and hence carry similar
$\Delta\nu$ characteristics. This may be similar to principal component
analysis (PCA) or dimensionality reduction algorithms, and the network is
trying to rely on as few inputs as possible, leaving others redundant.

###  3.3 Method 3: ML using radial and dipole modes

In this section, we present the final follow-up experiment to study the
importance of dipole modes. Every experiment carried out so far has dealt with
radial ($\ell=0$) modes, or quantities such as {$\Delta\nu,\epsilon$} that
depend on radial modes. However, since dipole modes are often seen in the
échelle diagrams (Bedding et al. 2020) of $\delta$ Sct stars, it is useful to
conduct these experiments in order to study their contribution.

In this method, we built a model that took observables {$L,T_{\mathrm{eff}}$}
and eigenfrequencies with $n\in[4-7]$ from each of the $\ell=$ 0 and 1 ridges,
and produced as output the structure parameters. For age inference, we
supplemented pre-inferred $M$ and $Z$ to the frequencies and
{$L,T_{\mathrm{eff}}$}. We refer to this procedure as Method-3.

Figure 10 and Table 2 indicate that the networks were able to learn $M$,$Z$,
and $\log\tau$ from synthetics and the accuracy of age inference has improved.
Since the current method adequately constrains age, this emphasizes the
importance of dipole modes.

| Method-1 | Method-2 | Method-3
---|---|---|---
| (Section 3.1) | (Section 3.2) | (Current Method)
Inputs: | {$L,T_{\mathrm{eff}},\Delta\nu,\epsilon$} | {$\nu_{1}-\nu_{7}$} | {$L,T_{\mathrm{eff}},$ 8 modes}
$M$ | 99.9% | 99.5% | 99.8%
$Z$ | 99.7% | 99.5% | 99.6%
$\log\tau$ | 93% | 96% | 98.1%
Table 2: Comparing performances (average learning accuracy) of Methods-1 and 2
(Section 3.1, 3.2) with Method-3 (this section), where we use
$T_{\mathrm{eff}},L$, 4 radial and 4 zonal dipole modes as inputs. Figure 10:
Improved network prediction after inputting radial-dipole ($\ell=0,1$) modes
of overtone $n\in[4-7]$ along with {$L,T_{\mathrm{eff}}$}. The age inference
is much more accurate than the approaches described earlier.

With more number of inputs to neural network, it is supposed to gain higher
accuracy. We also measured $\chi^{2}/N$ for all the three experiments to
assess which method is more efficient. Here $\chi^{2}$ mean average squared
differences between true and machine predicted values and $N$ represents the
number of inputs fed to the networks. Table 3 summarizes these facts.

| Method-1 | Method-2 | Method-3
---|---|---|---
| (Section 3.1) | (Section 3.2) | (Current Method)
Inputs: | {$L,T_{\mathrm{eff}},\Delta\nu,\epsilon$} | {$\nu_{1}-\nu_{7}$} | {$L,T_{\mathrm{eff}},$ 8 modes}
$M$ | $7\times 10^{-5}$ | $10^{-5}$ | $2\times 10^{-6}$
$Z$ | $10^{-7}$ | $5\times 10^{-8}$ | $9\times 10^{-9}$
$\log\tau$ | $3\times 10^{-3}$ | $6.5\times 10^{-4}$ | $10^{-3}$
Table 3: Comparing $\chi^{2}/N$ for all the methods.

We observe that Method-2 consistently outperforms Method-1. Method-3 excels
among all of these but for age inference it could not outperform the Method-2.
This may be due to the relatively larger (yet tolerable) spread at larger ages
compared to Method-2. Global statistics such as the $\chi^{2}$ average out-
small scale information and hence cannot be used to assess the performance
across all the ages. However a performance-wise comparison after studying
Figures 8 and 10 suggest that the outlier level is lower for age inference in
Method-3. Hence, we conclude that results from Method-3 are preferred.
Nonetheless, Method-1 remains practically applicable until definitive mode
identification becomes possible in $\delta$ Sct stars.

At present, we were unable to apply this method to measure structure
parameters in observed stars as it is challenging to accurately label the
dipole modes.

####  3.3.1 Feature Importance

To understand the importance of the dipole and radial modes, we carried out a
feature-importance experiment for these parameters and show the plots in
Figure 11.

Figure 11: Qualitative contributions arising from input quantities for
inferences of (a) $M$, (b) $Z$ and (c) $\log\tau$.

An important conclusion is that while $M$ and $Z$ do not significantly depend
on radial and dipole frequencies, the exact opposite is seen when inferring
age. Due to their global nature, $L$ and $T_{\mathrm{eff}}$ appear to encode
enough information of global parameters $M$ and $Z$, rendering the oscillation
measurements less useful. In contrast, age depends on the deep interior more
than the shallow layers since the core hydrogen fraction ($X_{c}$) governs the
evolution. Since pulsation frequencies are ideal probes of the stellar
interior, age determination strongly relies on them.

##  4 Method 4: Least square fitting of stellar parameters

Each method described above has its own set of advantages and disadvantages.
In this section, we employed a grid-search algorithm to minimize the least-
square loss functions in order to identify the model best matching to the
observation. This method requires interactive inspection to obtain accurate
fits, provided that there exists a close fit model to the observation. A
similar approach was used by Steindl, T. et al. 2022 to fit stellar parameters
to a small number of $\delta$ Sct stars.

One of the advantages of this method is it can be applied to the 60 stars from
Bedding et al. 2020 to achieve good results. Even for stars with missing
values of $L$ or $T_{\mathrm{eff}}$, this method can obtain a good fit to
observed spectra.

###  4.1 Method

The method of least squares is a common technique to fit photometric ($L$),
spectroscopic ($T_{\mathrm{eff}}$) and seismic quantities (eigen-frequencies)
between observations and models. The underlying principle involves assigning a
$\chi^{2}$ value (as shown in equation 4.1) to each sample in the model, which
represents a weighted combination of the squared differences between the
observed and model values. The objective is to find the model sample that
results in the minimum value of $\chi^{2}$. The values of $\sigma_{L}$,
$\sigma_{T_{\rm eff}}$, and $\sigma_{\nu_{i}}$ are critical parameters in this
approach, as they represent the magnitude of the errors that can be tolerated.
This methodology can be highly effective if the appropriate $\sigma$ values
are used.

$\displaystyle\chi^{2}=$ $\displaystyle\left\\{\dfrac{(L_{\rm obs}-L_{\rm
model})^{2}}{\sigma_{L}^{2}}+\dfrac{(T_{\rm eff;obs}-T_{\rm
eff;model})^{2}}{\sigma_{T_{\rm eff}}^{2}}\right.$
$\displaystyle\left.+\sum_{i=1}^{N}\dfrac{(\nu_{i,{\rm obs}}-\nu_{i,{\rm
model}})^{2}}{\sigma_{\nu_{i}}^{2}}\right\\}$ (2)

The last part of equation 4.1 indicates the requirement of determining both
the observed ($\nu_{i,{\rm obs}}$) and model frequencies ($\nu_{i,{\rm
model}}$) of the same radial order and angular degree. However, accurately
identifying and labeling the observed modes in $\delta$ Sct stars is
challenging due to the high density contamination, missing modes as well as
presence of modes with unknown origin. Identification of genuine modes
requires iterative pre-whitening followed by elimination of frequency
combinations – which adds to the complexity of the fitting routine.

In our approach, we have made slight modifications to a similar method. Our
primary objective is to search for a model that can reprodcue as many observed
modes as possible without having additional spurious peaks. We have applied
constraints to the models using the observed values of $\Delta\nu$ (which are
obtained from Bedding et al. 2020) to prevent selection of erroneous fits -
since a model with much lower $\Delta\nu$ can accurately match a relatively
large number of significant observed modes. Finally, we introduce a $\chi^{2}$
term to compare few observed frequencies to their nearest model frequencies
($\nu^{\rm closest}_{\rm model}$). Our modified $\chi^{2}$ looks like equation
4.1.

$\displaystyle\chi^{2}=$ $\displaystyle\left\\{\dfrac{(L_{\rm obs}-L_{\rm
model})^{2}}{\sigma_{L}^{2}}\right.$ $\displaystyle+\dfrac{(T_{\rm
eff;~{}obs}-T_{\rm eff;~{}model})^{2}}{\sigma^{2}_{T_{\rm eff}}}$
$\displaystyle+\dfrac{(\Delta\nu_{\rm obs}-\Delta\nu_{\rm
model})^{2}}{\sigma^{2}_{\Delta\nu}}$
$\displaystyle\left.+\sum_{\nu_{1}}^{\nu_{N}}\dfrac{(\nu_{\rm obs}-\nu^{\rm
closest}_{\rm model})^{2}}{\sigma^{2}_{\nu}}\right\\}$ (3)

The $\chi^{2}$ formula therefore has similarity to that of Steindl, T. et al.
(2022) and the likelihood function of Scutt et al. (2023).

Values of $\sigma_{L}$ have been taken from Bedding et al. (2020). Following
the same, we have set the uncertainty in $T_{\rm eff}$ to be 2% and that of
$\Delta\nu$ to be 0.02 ${\rm d^{-1}}$. Rayleigh frequency resolution criterion
has been taken as the uncertainty in mode frequencies, similar to Steindl, T.
et al. (2022). Since we fitted three parameters simultaneously ($M,Z,\tau$),
$1\sigma$ uncertainties (or 68% confidence interval) associated with the best-
fit parameters correspond to 3.5 increase from $\chi^{2}_{\rm min}$ (Table 1
of Avni (1976)). Hence we assembled all the model parameters corresponding to
$\chi^{2}\in[\chi^{2}_{\rm min},\chi^{2}_{\rm min}+3.5]$ and used their spread
to calculate the uncertainties.

Regarding the frequency component of the $\chi^{2}$, we neither calculated
eigen-frequenies from iterative pre-whitening nor assigned any possible
identifications to the peaks. Instead, we visually inspected the spectrum and
started by selecting four peaks of relatively high frequency and significant
amplitudes. We selected these modes through trial and error. After begining
with these arbitrary set of modes we searched for their closest eigen-
frequencies ($\nu^{\rm closest}_{\rm model}$) across all the model samples
disregarding their $n$ or $\ell$. Treating these frequenies as inputs to
equation 4.1, we calculated $\chi^{2}$ across all the models and inspected for
$\chi^{2}_{\rm min}$. We repeatedly selected different sets of input
frequencies as well as varied the number of fitted modes (N = 1, 2, 3, 4,
rarely 5) and carried out the entire fitting process afresh until we achieved
a minimum possible $\chi^{2}_{\rm min}$. Finally, we also constructed an
èchelle diagram to ensure that the $N$ selected modes were genuine $m=0$
frequencies and the obtained solution was able to fit other significant radial
and dipole modes as well. Hence, we used the èchelle diagrams to confirm that
we are not comparing model modes with incorrect peaks such as rotational
splits, combination frequencies etc.

The reason we favoured selecting very few modes is as follows. It is not
always possible to obtain exact fits to all the observed modes simultaneously.
In such scenarios, fitting larger numbers of modes would demand a highly
sophisticated mode selection process since we are picking up the modellable
modes through trial and error. Inappropriate mode selection usually leads to a
solution where none of the modes are comparable to the observed modes.
However, for each star, we were successful to identify a smaller number of
genuine modes using which we could fit most of the observed modes.

This fitting routine takes us $\sim$ 10 seconds given that we vectorized this
operation across 112 cpu cores using NumPy. Otherwise, this would correspond
to $\sim$ 20 cpu minutes (per single star) without any core-level
parallelization. However the mode selection process (being manual and
interactive) is the most difficult one and it took us around 10 minutes per a
single star.

In Figure 12, we present an example of our successfully fitted results using
this methodology. It was challenging for us to find a close-fit model where
lower-order model modes would precisely align with the observed modes.
Additionally, dipole modes of the fitted models were sometimes seen to be
present in the vicinity of the observed ones but not precisely. This
discrepancy could possibly be due to the unequal splitting of dipole modes and
the rotation-induced shift of $m=0$ components, even in radial modes.

Figure 12:  Obtaining a best-fit model to observation of HD 40317. The dotted
lines in the right side spectrum shows the 3 modes we selected for the fitting
routine. These 3 modes are also shown in the échelle diagram (on left) as
filled rectangle symbols. After determining the best-fit model, we show its
radial and dipole modes as open circles and triangles over both the échelle
diagram and the oscillation spectrum. Some of these open symbols do not
exactly fall on the observed modes because the fitted frequencies fall within
the spectral windows in the vicinity of the sharp peaks. We also report the
observed and best-fitted values of $L,T_{\mathrm{eff}}$ and $\Delta\nu$ in the
title of the spectrum plot. The ($M,Z,\tau$) values of the best fitted model
are presented in the same title.

###  4.2 Results

In this section, we present the results of our method on 60 stars from Bedding
et al. 2020. For 3 of these stars, either one or both of
($L,T_{\mathrm{eff}}$) inputs were missing. For such stars, we ignored the
corresponding contributions of $\chi^{2}$. As the best fit model has an
inherent value of missing parameters, one can have a crude estimate of the
parameters for these stars.

In Table 4 and Figure 14 \- 16 of appendix A, we have presented the fitted
structure parameters of the individual stars. Additionally, Figure 13
summarizes the statistics, indicating that most of the fitted masses are
approximately $\sim 1.6~{}\text{M}_{\odot}$, while the dominant metallicity
values are distributed around $Z=0.010$, corresponding to [Fe/H] $\sim-0.146$.
We also observed a bi-modal age distribution among the stars, where more than
half of them are young (around $\sim 10$ Myr) and a few are very old (over
$100$ Myr). However, we emphasize that age inferences are not highly precise
due to the degeneracy effect, which means that at two different ages, the star
can have similar physical and seismic structure.

Figure 13: Statistics of the fundamental parameters of 60 $\delta$ Sct stars,
inferred using the least-squares minimization method. The dotted vertical
lines represent the boundary of the model grid.

Table 4: Inferred values of mass, composition, and age of 60 $\delta$ Sct stars taken from Bedding et al. (2020). These are obtained from the $\chi^{2}$ minimization method. Id | mass | $Z_{i}$ | age
---|---|---|---
| ($M_{\odot}$) |  | (Myr)
HD 2280 | 1.35 ${}^{+0.1}_{-0.05}$ | 0.008 ${}^{+0.001}_{-0.002}$ | 977.7 ${}^{+202.9}_{-1.23}$ |
HD 3622 | 1.7 ${}^{+0.05}_{-0.15}$ | 0.022 ${}^{+0.002}_{-0.014}$ | 50.6 ${}^{+75.3}_{-42.04}$ |
HD 10779 | 1.6 ${}^{+0.1}_{-0.02}$ | 0.009 ${}^{+0.013}_{-0.001}$ | 9.0 ${}^{+156.43}_{-0.25}$ |
HD 17341 | 1.65 ${}^{+0.1}_{-0.01}$ | 0.012 ${}^{+0.006}_{-0.001}$ | 9.0 ${}^{+2.67}_{-0.01}$ |
HD 17693 | 1.7 ${}^{+0.05}_{-0.05}$ | 0.012 ${}^{+0.001}_{-0.003}$ | 9.7 ${}^{+0.01}_{-1.59}$ |
HD 20203 | 1.5 ${}^{+0.04}_{-0.05}$ | 0.008 ${}^{+0.001}_{-0.001}$ | 692.5 ${}^{+0.06}_{-132.78}$ |
HD 20232 | 1.68 ${}^{+0.02}_{-0.13}$ | 0.018 ${}^{+0.002}_{-0.011}$ | 229.8 ${}^{+104.08}_{-221.47}$ |
HD 24572 | 1.52 ${}^{+0.06}_{-0.02}$ | 0.013 ${}^{+0.003}_{-0.004}$ | 344.0 ${}^{+132.62}_{-331.75}$ |
HD 24975 | 1.7 ${}^{+0.05}_{-0.06}$ | 0.014 ${}^{+0.001}_{-0.003}$ | 10.4 ${}^{+0.6}_{-1.55}$ |
HD 25674 | 1.54 ${}^{+0.01}_{-0.04}$ | 0.007 ${}^{+0.001}_{-0.001}$ | 829.4 ${}^{+77.82}_{-0.08}$ |
HD 28548 | 1.7 ${}^{+0.05}_{-0.06}$ | 0.012 ${}^{+0.001}_{-0.006}$ | 163.0 ${}^{+158.0}_{-155.63}$ |
HD 30422 | 1.5 ${}^{+0.02}_{-0.05}$ | 0.006 ${}^{+0.001}_{-0.001}$ | 8.3 ${}^{+0.01}_{-0.01}$ |
HD 31322 | 1.75 ${}^{+0.05}_{-0.05}$ | 0.011 ${}^{+0.001}_{-0.002}$ | 7.7 ${}^{+0.01}_{-0.75}$ |
HD 31640 | 1.55 ${}^{+0.09}_{-0.03}$ | 0.009 ${}^{+0.004}_{-0.001}$ | 9.3 ${}^{+2.44}_{-0.23}$ |
HD 31901 | 1.58 ${}^{+0.06}_{-0.02}$ | 0.009 ${}^{+0.001}_{-0.001}$ | 9.4 ${}^{+0.01}_{-0.58}$ |
HD 32433 | 1.62 ${}^{+0.03}_{-0.1}$ | 0.018 ${}^{+0.002}_{-0.01}$ | 201.1 ${}^{+193.23}_{-191.66}$ |
HD 38597 | 1.58 ${}^{+0.06}_{-0.02}$ | 0.008 ${}^{+0.002}_{-0.001}$ | 694.8 ${}^{+0.02}_{-185.18}$ |
HD 38629 | 1.45 ${}^{+0.05}_{-0.1}$ | 0.004 ${}^{+0.001}_{-0.001}$ | 972.4 ${}^{+0.21}_{-0.86}$ |
HD 40317 | 1.7 ${}^{+0.05}_{-0.08}$ | 0.009 ${}^{+0.001}_{-0.002}$ | 8.5 ${}^{+0.16}_{-1.17}$ |
HD 42005 | 1.6 ${}^{+0.02}_{-0.02}$ | 0.008 ${}^{+0.001}_{-0.001}$ | 8.6 ${}^{+0.01}_{-0.01}$ |
HD 42608 | 1.75 ${}^{+0.05}_{-0.1}$ | 0.018 ${}^{+0.002}_{-0.01}$ | 13.8 ${}^{+96.09}_{-5.7}$ |
HD 44726 | 1.55 ${}^{+0.01}_{-0.03}$ | 0.011 ${}^{+0.001}_{-0.003}$ | 396.6 ${}^{+253.75}_{-383.59}$ |
HD 44930 | 1.64 ${}^{+0.01}_{-0.1}$ | 0.018 ${}^{+0.008}_{-0.005}$ | 13.7 ${}^{+453.78}_{-3.78}$ |
HD 44958 | 1.6 ${}^{+0.1}_{-0.02}$ | 0.009 ${}^{+0.013}_{-0.001}$ | 9.1 ${}^{+11.22}_{-0.05}$ |
HD 45424 | 1.55 ${}^{+0.01}_{-0.03}$ | 0.008 ${}^{+0.001}_{-0.001}$ | 823.0 ${}^{+54.73}_{-0.06}$ |
HD 46722 | 1.52 ${}^{+0.01}_{-0.02}$ | 0.008 ${}^{+0.001}_{-0.001}$ | 9.1 ${}^{+0.27}_{-0.01}$ |
HD 48985 | 1.64 ${}^{+0.04}_{-0.02}$ | 0.008 ${}^{+0.001}_{-0.001}$ | 8.5 ${}^{+0.01}_{-0.67}$ |
HD 50153 | 1.7 ${}^{+0.05}_{-0.02}$ | 0.02 ${}^{+0.002}_{-0.002}$ | 15.9 ${}^{+0.01}_{-0.01}$ |
HD 54711 | 1.62 ${}^{+0.06}_{-0.12}$ | 0.009 ${}^{+0.006}_{-0.003}$ | 9.8 ${}^{+378.22}_{-1.95}$ |
HD 55863 | 1.64 ${}^{+0.01}_{-0.06}$ | 0.009 ${}^{+0.001}_{-0.001}$ | 8.6 ${}^{+0.69}_{-0.09}$ |
HD 59104 | 1.56 ${}^{+0.02}_{-0.02}$ | 0.018 ${}^{+0.002}_{-0.002}$ | 19.1 ${}^{+297.27}_{-1.96}$ |
HD 59594 | 1.6 ${}^{+0.05}_{-0.1}$ | 0.015 ${}^{+0.005}_{-0.007}$ | 13.9 ${}^{+476.0}_{-4.77}$ |
HD 78198 | 1.62 ${}^{+0.06}_{-0.02}$ | 0.014 ${}^{+0.006}_{-0.001}$ | 9.9 ${}^{+4.74}_{-0.37}$ |
HD 99506 | 1.65 ${}^{+0.03}_{-0.1}$ | 0.018 ${}^{+0.002}_{-0.01}$ | 16.2 ${}^{+246.96}_{-7.1}$ |
HD 223011 | 1.62 ${}^{+0.18}_{-0.02}$ | 0.01 ${}^{+0.014}_{-0.001}$ | 8.7 ${}^{+438.49}_{-0.32}$ |
HD 290799 | 1.55 ${}^{+0.01}_{-0.01}$ | 0.004 ${}^{+0.001}_{-0.001}$ | 7.1 ${}^{+0.01}_{-0.93}$ |
TIC 349645354 | 1.75 ${}^{+0.05}_{-0.05}$ | 0.024 ${}^{+0.002}_{-0.004}$ | 12.3 ${}^{+0.01}_{-2.14}$ |
TIC 431695696 | 1.54 ${}^{+0.08}_{-0.04}$ | 0.006 ${}^{+0.002}_{-0.001}$ | 8.1 ${}^{+1.14}_{-0.01}$ |
TIC 124381332 | 1.68 ${}^{+0.07}_{-0.06}$ | 0.014 ${}^{+0.001}_{-0.007}$ | 13.3 ${}^{+145.36}_{-5.31}$ |
TIC 340358522 | 1.62 ${}^{+0.02}_{-0.02}$ | 0.008 ${}^{+0.001}_{-0.001}$ | 8.4 ${}^{+0.01}_{-0.01}$ |
HD 187547 | 1.58 ${}^{+0.02}_{-0.04}$ | 0.018 ${}^{+0.002}_{-0.002}$ | 177.9 ${}^{+0.64}_{-160.44}$ |
KIC 8415752 | 1.75 ${}^{+0.05}_{-0.05}$ | 0.024 ${}^{+0.002}_{-0.002}$ | 344.6 ${}^{+0.04}_{-0.02}$ |
KIC 9450940 | 1.62 ${}^{+0.03}_{-0.02}$ | 0.015 ${}^{+0.001}_{-0.001}$ | 702.5 ${}^{+53.63}_{-51.24}$ |
HD 37286 | 1.52 ${}^{+0.06}_{-0.02}$ | 0.006 ${}^{+0.005}_{-0.001}$ | 8.5 ${}^{+4.51}_{-0.01}$ |
HD 39060 | 1.45 ${}^{+0.05}_{-0.1}$ | 0.008 ${}^{+0.001}_{-0.001}$ | 861.4 ${}^{+0.45}_{-0.53}$ |
HD 42915 | 1.55 ${}^{+0.09}_{-0.01}$ | 0.006 ${}^{+0.001}_{-0.001}$ | 7.6 ${}^{+0.01}_{-0.87}$ |
HD 290750 | 2.1 ${}^{+0.1}_{-0.1}$ | 0.013 ${}^{+0.013}_{-0.001}$ | 5.8 ${}^{+185.38}_{-0.07}$ |
TIC 143381070 | 1.5 ${}^{+0.02}_{-0.05}$ | 0.007 ${}^{+0.001}_{-0.001}$ | 594.4 ${}^{+0.03}_{-117.6}$ |
TIC 260161111 | 1.54 ${}^{+0.06}_{-0.04}$ | 0.008 ${}^{+0.002}_{-0.003}$ | 599.3 ${}^{+51.98}_{-591.2}$ |
HD 10961 | 1.35 ${}^{+0.1}_{-0.05}$ | 0.003 ${}^{+0.001}_{-0.001}$ | 8.4 ${}^{+0.01}_{-0.01}$ |
HD 25248 | 1.5 ${}^{+0.02}_{-0.05}$ | 0.009 ${}^{+0.005}_{-0.001}$ | 10.9 ${}^{+4.14}_{-0.01}$ |
HD 67688 | 1.68 ${}^{+0.02}_{-0.03}$ | 0.008 ${}^{+0.001}_{-0.001}$ | 7.6 ${}^{+0.01}_{-0.01}$ |
HD 70510 | 1.65 ${}^{+0.03}_{-0.01}$ | 0.018 ${}^{+0.002}_{-0.002}$ | 90.0 ${}^{+0.12}_{-74.06}$ |
HD 75040 | 1.6 ${}^{+0.02}_{-0.05}$ | 0.011 ${}^{+0.001}_{-0.001}$ | 10.0 ${}^{+0.58}_{-0.09}$ |
HD 222496 | 1.8 ${}^{+0.05}_{-0.15}$ | 0.022 ${}^{+0.002}_{-0.009}$ | 12.1 ${}^{+485.56}_{-3.64}$ |
HD 34282 | 1.6 ${}^{+0.02}_{-0.1}$ | 0.009 ${}^{+0.001}_{-0.004}$ | 421.4 ${}^{+390.4}_{-0.02}$ |
HD 29783 | 1.68 ${}^{+0.02}_{-0.04}$ | 0.009 ${}^{+0.001}_{-0.001}$ | 8.5 ${}^{+0.3}_{-0.61}$ |
HD 220811 | 1.54 ${}^{+0.04}_{-0.01}$ | 0.015 ${}^{+0.003}_{-0.001}$ | 849.5 ${}^{+0.05}_{-154.81}$ |
HD 25369 | 1.58 ${}^{+0.06}_{-0.04}$ | 0.012 ${}^{+0.006}_{-0.001}$ | 9.9 ${}^{+5.47}_{-0.33}$ |
HD 89263 | 1.52 ${}^{+0.23}_{-0.07}$ | 0.011 ${}^{+0.013}_{-0.006}$ | 561.8 ${}^{+194.6}_{-555.8}$ |

##  5 Conclusion

We have deployed a least-squares minimization technique to obtain the
structure parameters of 60 $\delta$ Sct stars that show regular p-mode
pulsation patterns. This method allows us to carry out spectrum fitting
without prior mode identification. It is semi-automated in the sense that we
fit very few modes in the trial-and-error effort, and mode identification
comes as a byproduct of the fitting routine. We provided the first inferences
of $M$, $Z$ ([Fe/H]) and age ($\tau$) for these stars.

We found the masses of most of these stars to be distributed around
$1.6M_{\odot}$, with the exception of two stars of lower mass (HD 2280, HD
10961: $\sim 1.4M_{\odot}$) and one star with the highest mass (HD 290750:
$2.1M_{\odot}$). Metallicity of a significant fraction of the stars are found
to hover around $Z=0.010$ (or [Fe/H] = -0.23). In this sample, we also found a
few stars having very low as well as very high values of metallicity. TIC
349645354 and KIC 8415752 have the highest metallicity, at $Z=0.024$. Finally,
more than half of the stars turned out to be younger than 30 Myr, with the
rest several hundreds of Myrs old.

We had originally developed three machine learning based neural networks to
carry out similar parameter inferences. Although we were unable to obtain
reliable results, they still carry the potential of simpler as well faster
parameteric fitting. On the synthetic data, our neural networks were able to
infer $M$ and $Z$ much more accurately using
{$L,T_{\mathrm{eff}},\Delta\nu,\epsilon$} as inputs. However, age inferences
were in general not as accurate as $M$ or $Z$. This is because, over the
course of their MS evolution, stars cross their pre-MS trajectory, which make
all their structure parameters degenerate in these two stages of evolution. A
similar behaviour was demonstrated in Murphy et al. 2021a. Therefore, age
inference is not expected to be confident.

With the exception of the parameter inference, neural networks allowed us to
carry out an additional analysis - we were able to determine the relative
importance of different input quantities . We determined that
$T_{\mathrm{eff}}$ plays a critical role in the inference of all parameters.
$\epsilon$ was found to have significance as important as $\log Z$ and other
parameters like $M$, $\Delta\nu$, and $L$, while constraining the age.

We observed that a longer pattern of radial modes (starting from $n=1$)
contains critical information about $\delta$ Sct structure. This set of
frequencies may be treated as an essential substitute to
{$L,T_{\mathrm{eff}}$} as we found that they can constrain stellar parameters
even without $L$ and $T_{\mathrm{eff}}$. Doing the feature importance
experiment, we found that $\nu_{6}$, $\nu_{2}$ and $\nu_{3}$ are the most
significant radial modes.

Finally, we noticed that inclusion of dipole modes led to more precise
determine $\delta$ Sct age. We used $L$, $T_{\mathrm{eff}}$ and frequencies of
both radial ($\ell=0$) and dipole ($\ell=1$) modes with radial order
$n\in[4-7]$ as inputs to infer different parameters. The degeneracy problem in
age was reduced while using these inputs. We therefore conclude that dipole
modes act as independent quantities with which to constrain stellar parameters
and they add supplementary information to radial modes and
{$\Delta\nu,\epsilon$}.

Our models lack the implementation of rotation and gravity darkening - which
have significant impacts over stellar evolution and pulsation. While 5 km s-1
fluctuation in rotation velocity can perturb the pulsation frequencies by
$2-3\%$ (Deupree 2011), gravity darkening can cause as high as $\sim 50\%$ and
$2.5\%$ departures in $L$ and $T_{\mathrm{eff}}$ (Paxton et al. 2019)
respectively. Depending on these facts, our parameter inferences can be
subject to modification once we introduce such phenomena. However, this is
beyond our current scope and we look forward to study their impacts in a
future project. Such a detailed analysis will help us in putting more
realistic constrains over the ages of multiple $\delta$ Sct stars observed in
missions like TESS and Kepler.

## acknowledgments

S.D. acknowledges SERB, DST, Government of India, CII and Intel Technology
India Pvt. Ltd. for the Prime minister’s fellowship and facilitating research.
We have performed all computations in the Intel Lab Academic Compute
Environment using Intel® Xeon® Platinum 8280 CPU. MESA (Paxton et al. 2010)
and GYRE (Townsend et al. 2013) codes have been very useful in simulating the
data, without which this project would not have been possible. We have used
SciPy (Oliphant 2007), Matplotlib (Hunter 2007), NumPy (van der Walt et al.
2011), Keras (Chollet et al. 2015), and Tensorflow (Abadi et al. 2015) like
Python packages. This research made use of Lightkurve (Lightkurve
Collaboration et al. 2018), a Python package for Kepler and TESS data
analysis. GNU parallel (Tange 2011) has been used to do parallel computation.
SJM was supported by the Australian Research Council (ARC) through Future
Fellowship FT210100485. TRB was also supported by the ARC, through DP210103119
and FL220100117, and by the Danish National Research Foundation (Grant
DNRF106) through its funding for the Stellar Astrophysics Centre (SAC). We
thank Antonio García Hernández for his suggestions regarding the impacts of
rotation and gravity darkening. Finally we are thankful to the referee for his
very useful comments.

##  Appendix A Inference for each star

We show below in Figures 14 \- 16, $M$, $Z$, and $\tau$ inferences from
method-4 for 60 $\delta$ Sct stars taken from Bedding et al. 2020. The x-axis
contains the IDs of the stars and the y-axis displays the inferred quantities.

Figure 14: $M$ inference for 60 $\delta$ Scuti stars. Figure 15: $Z$ inference
for 60 $\delta$ Scuti stars. Figure 16: $\log_{10}(\tau/{\rm Myr})$ inference
for 60 $\delta$ Scuti stars.

##  Appendix B Group-wise inference accuracy

For most of the stars, measured values of $L$ and $T_{\mathrm{eff}}$ are
available, although these quantities are not sufficient to constrain different
stellar parameters. Additional independent quantities are expected to assist
in constraining these structure parameters.

Asteroseismic quantities $\Delta\nu$ and $\epsilon$ are suitable for this
purpose. But neither ($L,T_{\mathrm{eff}}$) nor ($\Delta\nu,\epsilon$) alone
can accurately infer the stellar parameters. However, when combined, these
parameters can efficiently constrain those stellar parameters.

We considered {$L,T_{\mathrm{eff}},R$} as inputs, trained the networks with
these, and measured the maximum absolute errors between true values and
network predictions (over the validation samples). Similarly, we took
asteroseismic quantities {$\Delta\nu,\epsilon$} and repeated this validation
process. Finally, we combined all of them and again measured the maximum
absolute error associated with the predictions. During age inference, we
supplemented $M$, $Z$ like quantities and also assembled the values of
validation errors. We visualize all of these errors in Figure 17, which
emphasizes that validation error decreases drastically when observables and
asteroseismic quantities are simultaneously considered as inputs.

Figure 17: Maximum absolute errors between network prediction and true values,
for a different group of input quantities shown in the x-axis. Figures for
prediction of (a) $M$ in $M_{\odot}$, (b) $Z$ and (c) $\log\tau$, where $\tau$
is in Myr.

In Table 5, we show the Pearson-R correlation coefficients (Rodgers et al.
1988) between network predictions (over validation samples) and corresponding
true values. It is a measure of inference accuracy, i.e., correlation between
true and inferred values. The Pearson-R coefficient can have values between -1
and +1. The higher the R coefficient, the higher the correlation between true
values and network prediction. This Pearson-R correlation coefficient between
the two sets of measurements {p} and {q} is calculated by equation B1.

$R=\dfrac{\sum_{i}(p_{i}-\overline{p})(q_{i}-\overline{q})}{\sqrt{\sum_{i}(p_{i}-\overline{p})^{2}\sum_{i}(q_{i}-\overline{q})^{2}}}$
(B1)

where, $\overline{p},\overline{q}$ are the means of the measurements.

Adding ($M,Z$) as additional inputs increases the Pearson-R coefficient by
only 0.01. However, from Figure 17 (c), it is evident that ($M,Z$) inputs
actually assist in reducing the absolute error associated with $\log\tau$
prediction.

| {$L,T_{\mathrm{eff}}$} | {$\Delta\nu,\epsilon$} | {$L,T_{\mathrm{eff}}$, | {$L,T_{\mathrm{eff}}$,
---|---|---|---|---
|  |  |  $\Delta\nu,\epsilon$} |  $\Delta\nu,\epsilon$,
|  |  |  |  $M,Z$}
$M$ | 0.93 | 0.86 | 0.99 |
$\log Z$ | 0.77 | 0.85 | 0.99 |
$\log\tau$ | 0.67 | 0.78 | 0.98 | 0.99
Table 5: Pearson-R correlation coefficients between network predictions and
true values, for different groups of input quantities.

##  Appendix C supplementary resources

This article provides a supplementary file demonstrating the stellar model
fittings to observations of all stars in the sample.

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