# Inter-SubNet: Speech Enhancement with Subband Interaction

###### Abstract

Subband-based approaches process subbands in parallel through the model with
shared parameters to learn the commonality of local spectrums for noise
reduction. In this way, they have achieved remarkable results with fewer
parameters. However, in some complex environments, the lack of global spectral
information has a negative impact on the performance of these subband-based
approaches. To this end, this paper introduces the subband interaction as a
new way to complement the subband model with the global spectral information
such as cross-band dependencies and global spectral patterns, and proposes a
new lightweight single-channel speech enhancement framework called Interactive
Subband Network (Inter-SubNet). Experimental results on DNS Challenge -
Interspeech 2021 dataset show that the proposed Inter-SubNet yields a
significant improvement over the subband model and outperforms other state-of-
the-art speech enhancement approaches, which demonstrate the effectiveness of
subband interaction.

Index Terms—  subband interaction, global spectral information, Inter-SubNet,
speech enhancement

##  1 Introduction

The interference from environmental noise is one of the main factors that
hinder hearing aids, audio communication and automatic speech recognition. In
these regards, single-channel speech enhancement approaches, which eliminate
interference noise in the noisy speech and aim to improve the clarity and
comprehensibility of voice, are highly desired. Traditional speech enhancement
methods use statistical signal theory to effectively suppress stationary
noise, but they do not perform well under conditions of low signal-to-noise
ratio (SNR). Recently, with the aid of the powerful capabilities of deep
neural networks, deep learning-based speech enhancement approaches have
yielded exceptional results, particularly in tackling challenging situations,
such as low SNR, reverberation, etc. Noisy audio can be denoised through deep
neural networks in the time or frequency domain. The time-domain methods [1,
2, 3, 4, 5] generate the clean voice waveform straight from the noisy audio
waveform. Alternatively, the frequency-domain methods [6, 7, 8, 9] typically
adopt the noisy spectrogram feature as their model’s input feature, and the
predicting target of them is the clean spectrogram feature or the spectrogram
mask including Ideal Binary Mask (IBM) [10], Ideal Ratio Mask (IRM) [11] and
complex Ideal Ratio Mask (cIRM) [12], just to name a few. Overall, considering
the robustness of system and computational complexity, the frequency-domain
approaches are more preferable [13].

The subband-based approaches [14, 15] are effective for single-channel
frequency-domain speech enhancement. This type of approaches divides the input
spectrogram into multiple subbands and then processes them in parallel through
the model with shared parameters, which helps the model learn the commonality
of local spectrums and the frequency-wise signal stationarity to suppress
noise[14]. Thus, these subband-based approaches reduce the number of
parameters and computational burden while maintaining the model performance.

However, since they are only sensitive to the local information, the subband-
based models can hardly recover the clean speech when encountering a complex
acoustic environment with unclear local spectral patterns. This challenge can
be tackled with the help of the global spectral information, such as long-
distance cross-band dependencies and global spectral pattern [16]: the model
is capable of deducing the local pattern of the current subband from the
pattern of the global spectrum as well as from other relatively clean
subbands. FullSubNet [16] and FullSubNet+ [17] provide global spectral
information to the subband model by serially connecting the fullband model
with the subband model. They work exceptionally well on the noise reduction
task, which demonstrates the value of replenishing the global spectral
information for the subband model. Nevertheless, due to the introduction of
the fullband model, these methods lead to a large number of overall model
parameters as well as increase the complexity of the model.

In this paper, to extract global spectral information like cross-band
dependencies and global spectral patterns while maintaining a low model
complexity, a novel subband interaction (SubInter) module is proposed for
speech enhancement. We incorporate the newly designed SubInter module into the
subband model [14] to complement the model with the global spectral
information. This new lightweight speech enhancement framework is called
Inter-SubNet, which consists of alternating SubInter module and Long Short-
Term Memory (LSTM) layer to capture the global spectral information while
retaining the ability to focus on the local spectral patterns. Experimental
results on DNS Challenge - Interspeech 2021 dataset show that our Inter-SubNet
yields a dramatic improvement compared to the subband model. Moreover, the
Inter-SubNet outperforms FullSubNet+ and FullSubNet in terms of both number of
parameters and performance, and also exceeds other state-of-the-art speech
enhancement methods. These results demonstrate that the subband interaction is
a lightweight and efficient method for modeling the global spectral
information.

Fig. 1: The general schematic of the proposed Inter-SubNet, where “SIL block
1” and “SIL block 2” refer to the first and second SubInter-LSTM block
respectively. The “G-norm” denotes group normalization [18]. The Inter-SubNet
$G_{is}$ mainly consists of two stacked SIL blocks and one fully-connected
layer. Taking the subband units $\\{\mathbf{b}_{i}\\}_{i=1}^{F}$ as input, the
model $G_{is}$ generates the final output cIRM $\mathbf{M}^{r}$ and
$\mathbf{M}^{i}$.

##  2 Methodology

We consider single-channel signal with 16 kHz sampling rate after short-time
fourier transform (STFT):

$X_{t,f}=S_{t,f}+N_{t,f}.$ (1)

where $t=1,...,T$ and $f=0,...,F-1$ denote the frame and frequency indices,
respectively. $X_{t,f}$, $S_{t,f}$ and $N_{t,f}$ are the complex-domain STFT
factors of the noisy speech, noisy-free speech (the reverberant signal picked
up by the microphone) and interference noise. This paper only concerned with
the denoising mission in the frequency-domain (STFT), and the goal is to
filter the noise and recover the noisy-free speech signal with 16 kHz sampling
rate.

Towards this mission, we propose a framework called Inter-SubNet, which
extracts the global spectral information such as cross-band dependencies and
global spectral patterns through SubInter module, while preserving the
capacity of the subband-based model to concentrate on the local spectral
patterns. The overall diagram of Inter-SubNet $G_{is}$ is shown in Fig.1. The
input magnitude spectrogram $\mathbf{X}\in\mathbb{R}^{F\times T}$, where $F$
and $T$ refer to the overall quantity of frequency bins and frames
respectively, is first transformed to subband units
$\\{\mathbf{b}_{i}\\}_{i=1}^{F}$ by the “unfold” procedure. Taking the subband
units as input, the model $G_{is}$ generates the final output cIRM
$\mathbf{M}^{r}$ and $\mathbf{M}^{i}$. The $G_{is}$ is mainly composed of two
stacked SubInter-LSTM (SIL) blocks followed by one fully-connected layer. We
will elaborate each part in the following sub-sections.

###  2.1 Subband units

Subband units are obtained from magnitude spectrogram $\mathbf{X}$ through the
“unfold” operation and serve as the input to the Inter-SubNet. Specifically,
for each frequency $i$ from the magnitude spectrogram $\mathbf{X}$, we take a
frequency bin vector $\mathbf{X}_{i}\in\mathbb{R}^{T}$ and the $2\times n$
frequency bin vectors neighboring it as a subband unit $\mathbf{b}_{i}$:

$\mathbf{b}_{i}=[\mathbf{X}_{i-n},\cdots,\mathbf{X}_{i},\cdots,\mathbf{X}_{i+n}]\in\mathbb{R}^{F_{s}\times
T}.$ (2)

where “$[,]$” denotes the concatenation operation and $F_{s}=2n+1$. Besides,
the boundary frequencies with $i+n>F-1$ or $i-n<0$ are assigned circular
fourier frequencies. With the aforesaid process, we can ultimately acquire
totally $F$ such subband units having a dimension of $(F_{s},T)$, which are
noted as $\\{\mathbf{b}_{i}\\}_{i=1}^{F}\in\mathbb{R}^{F\times F_{s}\times
T}$.

###  2.2 SubInter module

In multi-channel speech separation, [19] supplements global information with
each channel through the interaction among the audio of distinct channels.
Inspired by this, we introduce SubInter module, which exploits global spectral
information including cross-band dependencies and global spectral patterns by
the interaction among parallel subband units, and incorporates the global
spectral information into these subband units. In detail, following section
2.1, we have a set of $F$ subband units $\\{\mathbf{b}_{i}\\}_{i=1}^{F}$ with
$\mathbf{b}_{i}\in\mathbb{R}^{F_{s}\times T}$, where $F_{s}$ represents the
number of frequency bins in a subband units. The SubInter module is applied on
these subband units for complementary global spectral information:

$\\{\mathbf{\hat{b}}_{i}\\}_{i=1}^{F}=\mathcal{H}(\\{\mathbf{b}_{i}\\}_{i=1}^{F}).$
(3)

where $\mathbf{\hat{b}}_{i}\in\mathbb{R}^{F_{s}\times T}$ is the subband
feature that contains global spectral information as well as local spectral
pattern, and $\mathcal{H}(\cdot)$ is the mapping function that is intended to
describe the SubInter module.

Fig. 2: The details of the SubInter Module. The “Hidden” indicates the hidden
representations. The “Global” denotes the output of the second linear layer
containing global spectral information.

With regard to the design of specific SubInter module, as shown in Fig.2, for
the set of subband units $\\{\mathbf{b}_{i}\\}_{i=1}^{F}$, we transform the
dimensions and then apply a linear layer with the mapping function
$\mathcal{F}(\cdot)$ to obtain the hidden representation
$\mathbf{h}_{i}\in\mathbb{R}^{T\times H}$:

$\\{\mathbf{h}_{i}\\}_{i=1}^{F}=\mathcal{F}(transform(\\{\mathbf{b}_{i}\\}_{i=1}^{F})).$
(4)

The $\\{\mathbf{h}_{i}\\}_{i=1}^{F}$ internally represent the information of
parallel subband units. Next, all of the $\mathbf{h}_{i}$s are averaged and
sent to the second linear layer:

$\mathbf{\tilde{h}}=\mathcal{R}(\frac{1}{F}\sum_{i=1}^{F}\mathbf{h}_{i}).$ (5)

where $\mathcal{R}(\cdot)$ represents the mapping function defined by the
second linear layer. This manipulation enables the otherwise parallel subband
units to have an interaction, and hence the module extracts the
$\mathbf{\tilde{h}}\in\mathbb{R}^{T\times H}$ containing the global spectral
information as in cross-band dependencies and global spectral patterns.
Eventually, each $\mathbf{h}_{i}$ is concatenated with $\mathbf{\tilde{h}}$ to
complement the global spectral information. And then they are fed into the
third linear layer with the mapping function $\mathcal{P}(\cdot)$ for feature
fusion. After the transformation of the dimensions and adding a residual
connection from the module input $\mathbf{b}_{i}$, we get the output
$\mathbf{\hat{b}}_{i}\in\mathbb{R}^{F_{s}\times T}$:

$\\{\mathbf{\hat{b}}_{i}\\}_{i=1}^{F}=\\{\mathbf{b}_{i}\\}_{i=1}^{F}+transform(\mathcal{P}(\\{[\mathbf{h}_{i},\mathbf{\tilde{h}}]\\}_{i=1}^{F})).$
(6)

where “$[,]$” is the concatenation operation.

Through the above-mentioned module, we extract the global spectral information
by interacting the subband units that are originally processed in parallel,
and then fuse the global information into the former subband units.
Consequently, the output subband features
$\\{\mathbf{\hat{b}}_{i}\\}_{i=1}^{F}$ contain both global spectral
information and local spectral pattern.

###  2.3 Model architecture and workflow

The original subband model is composed of two stacked LSTM layers, which
process the subband units in parallel to model the signal stationarity and
local spectral pattern. To extract and integrate global spectral information,
we propose the Inter-SubNet, which supplements the original subband model with
SubInter module. In detail, SIL block is applied to replace the LSTM layer in
original subband model. As shown in Fig.1, the Inter-SubNet $G_{is}$ mainly
consists of two stacked SIL blocks and one fully-connected layer. SubInter
module, LSTM network and group normalization (G-norm) [18] layer are stacked
in each SIL block.

According to section 2.1, a total of $F$ subband units
$\\{\mathbf{b}_{i}\\}_{i=1}^{F}$ are fed as the input to the $G_{is}$. Inside
SIL block, $\\{\mathbf{b}_{i}\\}_{i=1}^{F}$ are first interacted by the
SubInter module to extract and fuse global spectral information to output the
subband features $\\{\mathbf{\hat{b}}_{i}\\}_{i=1}^{F}$. Then, the LSTM learns
the local spectral patterns and the complementary global spectral information
contained in these subband features. After that, the G-norm layer normalizes
the outputs of LSTM. Finally, the fully-connected layer outputs the cIRM as
the learning target.

Table 1: The performance in terms of WB-PESQ [MOS], NB-PESQ [MOS], STOI [%],
and SI-SDR [dB] on the DNS Challenge test dataset.

Model | Feat. |  # Para (M) | With Reverb | Without Reverb
---|---|---|---|---
WB-PESQ | NB-PESQ | STOI | SI-SDR | WB-PESQ | NB-PESQ | STOI | SI-SDR
Noisy | - | - | 1.822 | 2.753 | 86.62 | 9.033 | 1.582 | 2.454 | 91.52 | 9.07
DCCRN-E[20] | RI | 3.70 | - | 3.077 | - | - | - | 3.266 | - | -
Conv-TasNet[2] | Waveform | 5.08 | 2.750 | - | - | - | 2.730 | - | - | -
PoCoNet[21] | RI | 50.0 | 2.832 | - | - | - | 2.748 | - | - | -
DCCRN+[15] | RI | 3.30 | - | 3.300 | - | - | - | 3.330 | - | -
TRU-Net[22] |  Mag∗ | 0.38 | 2.740 | 3.350 | 91.29 | 14.87 | 2.860 | 3.360 | 96.32 | 17.55
CTS-Net[23] | Mag+RI | 4.99 | 3.020 | 3.470 | 92.70 | 15.58 | 2.940 | 3.420 | 96.66 | 17.99
FullSubNet[16] | Mag | 5.64 | 3.057 | 3.584 | 92.11 | 16.04 | 2.882 | 3.428 | 96.32 | 17.30
FullSubNet+[17] | Mag+RI | 8.67 | 3.177 | 3.648 | 93.64 | 16.44 | 3.002 | 3.503 | 96.67 | 18.00
Subband model[14] | Mag | 1.82 | 2.885 | 3.475 | 92.10 | 15.54 | 2.552 | 3.242 | 95.10 | 16.46
Subband modelL | Mag | 3.00 | 2.953 | 3.504 | 92.44 | 16.02 | 2.593 | 3.265 | 95.36 | 16.94
Inter-SubNet | Mag | 2.29 | 3.207 | 3.659 | 93.98 | 16.76 | 2.997 | 3.504 | 96.61 | 18.05

  * 1

$*$ More precisely, TRU-Net takes the per-channel energy normalization (PCEN)
[24] feature as input.

##  3 Experimental Setup

###  3.1 Datasets

The Inter-SubNet was trained on a sub-set of the DNS Challenge - Interspeech
2021 dataset with 16 kHz sampling rate, which will be called as DNS Challenge
dataset in the following sections. The clean speech dataset consists of 563
hours of clips from 2,150 speakers, while there are 181 hours of over 60,000
clips from 150 classes of noise in the noise dataset. To make the most of the
dataset, dynamic mixing was utilized to mimic the noisy speech during model
training. In particular, room impulse responses (RIRs), which are from the
openSLR26 and openSLR28 [25], were randomly chosen and added to 75% of the
clean speech. Afterwards, the noisy speeches were dynamically produced through
mixing the clean or reverberant speeches and noise at a randomly selected SNR
varying from -5 to 20 dB. The DNS Challenge also delivers a publically
accessible test set containing synthesized clips of two classes [26], namely
with and without reverberations. There are 150 noisy clips with a SNR ranging
from 0 to 20 dB in each class. The effectiveness of the different models was
verified on this test set.

###  3.2 Training setup and baselines

The Hanning window, which had 32 ms window length and 16 ms frame shift, was
applied to convert the signal into the frequence-domain. We employed Adam
optimizer having a 1e-3 learning rate. With respect to the subband units,
$n=15$ was set by according to [14], which means 15 neighbor frequencies are
taken on each side of each input frequency bin. During the training procedure,
the input-target sequence pairs were generated into a constant length sequence
of $T=192$ frames (approximately 3 s).

To testify the effectiveness of the subband interaction method, the following
models were compared. Aiming at a fair comparison, the identical experimental
setup together with predictive objectives (cIRM) were implemented for all
models.

Subband model: Following [14], the subband model contained 2 LSTM layers and
each layer consisted of 384 hidden cells. The overall number of parameters of
the subband model was 1.82 M.

Subband modelL: In order to verify that the performance improvement of Inter-
SubNet is caused by the SubInter module rather than the increase in the number
of parameters, we designed a larger model named subband modelL. By adding more
LSTM layers, we increase the amount of parameters in the model. The subband
modelL, which comprised 3 LSTM layers with 384 hidden units for each layer,
had a total of 3.00 M parameters.

Inter-SubNet: In the first SIL block, the SubInter module had 102 hidden
units, while the LSTM had 384 hidden units. In the second block, there were
SubInter module containing 307 hidden units and the LSTM with 384 hidden
units. The number of parameters for Inter-SubNet was 2.29 M.

##  4 Results and Discussions

###  4.1 Comparison with baseline and state-of-the-art methods

Table 1 shows the performance of different speech enhancement models on the
DNS Challenge dataset. Over the table, “With Reverb” and “Without Reverb” mean
the test sets with and without reverberation, respectively. “# Para” denotes
the overall number of parameters of the model, which is measured in millions.
“Feat.” stands for the input features of the model, where “Waveform” refers to
the time-domain waveform, “Mag” indicates the frequency-domain magnitude
spectrogram, and “RI” indicates the frequency-domain real and imaginary
spectrograms. It should be noted that “Mag+RI” means that all the magnitude,
real and imaginary spectrograms are treated as input features by the model.

In the last three rows of Table 1, the performances of subband model, subband
modelL, and the proposed Inter-SubNet are compared and the results show that
Inter-SubNet outperforms the subband model in all evaluation metrics. It also
shows that subband modelL has improved performance compared to subband model
after increasing the number of parameters. However, when compared to subband
modelL, the Inter-SubNet achieves superior results with fewer parameters,
demonstrating that the performance of Inter-SubNet is improved by the SubInter
module rather than the increase in the number of parameters.

Additionally, from the table we can have a comparison of the Inter-SubNet with
FullSubNet[16] and FullSubNet+[17]. It can be noticed that with much smaller
number of parameters, our model outperforms FullSubNet in all metrics.
Moreover, compared with the more complex FullSubNet+, which takes all of
magnitude, real and imaginary spectrograms as its input features, Inter-SubNet
outperforms or equals the performance of FullSubNet+ using only the magnitude
spectrogram as input. These illustrate that, compared with serially connecting
the fullband model with the subband model, our proposed SubInter module is a
more efficient and lightweight way for supplementing global spectral
information.

We further compare the proposed Inter-SubNet to some other state-of-the-art
time-domain and frequency-domain methods [20, 2, 21, 15, 22, 23] on the DNS
Challenge dataset throughout Table 1. The conclusion can be drawn that, the
Inter-SubNet shows superb performance in the denoising mission without
reverberation compared to the latest methods, whereas the improvement in
performance is even more prominent in the presence of reverberation. These
indicate that the proposed Inter-SubNet succeeds the remarkable capability of
subband model in terms of reverberation effects that is reported in [14], and
drastically boosts the denoising capacity with SubInter module for modeling
global spectral information.

###  4.2 Ablation study

In this section, ablation study was conducted to further explore the role of
the SubInter module. Table 2 presents the results of the ablation experiment,
where “1st SubInter” and “2nd SubInter” refer to the SubInter module in the
first and the second SIL block, respectively. “$\Delta$” represents the
difference between the value after removing the module and the value before
removing the module. We can see that removing both the first SubInter module
and the second SubInter module lead to a decline in the performance of the
model in terms of signal (SI-SDR) and perceptual quality (WB-PESQ). This shows
that subband interaction before each layer of subband processing LSTM network
brings gains, and these gains can be superimposed without conflict. In
addition, the performance degradation caused by removing the first SubInter
module is greater than removing the second SubInter module. This is probably
because that, without the second SubInter module, the model still has the
first SubInter module to provide global spectral information; whereas when the
first SubInter module is removed, the model degrades to the subband model
lacking global spectral information, which does not perform well in complex
acoustic environments. In general, the usage of the SubInter module does
effectively improve the noise reduction capability of Inter-SubNet.

Table 2: WB-PESQ, $\Delta$WB-PESQ, SI-SDR and $\Delta$SI-SDR performance in
the ablation study 4.2 on the without reverberation test set.

Models |  WB- PESQ |  $\Delta$WB- PESQ |  SI- SDR |  $\Delta$SI- SDR
---|---|---|---|---
Inter-SubNet | 2.997 | - | 18.05 | -
$-$ 2nd SubInter | 2.867 | -0.130 | 17.37 | -0.68
$-$ 1st SubInter | 2.552 | -0.315 | 16.46 | -0.91

##  5 Conclusions

This paper introduces a novel subband interaction (SubInter) module as a new
way to model global spectral information for speech enhancement and proposes a
new framework named Inter-SubNet111Code and examples:
https://github.com/RookieJunChen/Inter-SubNet. This new lightweight speech
enhancement framework is designed to exploit the global spectral information,
meanwhile retaining the ability to focus on the local spectral patterns.
Experimental results demonstrate the effectiveness of the proposed SubInter
module. We also compare Inter-SubNet with other top-ranked methods on the DNS
Challenge dataset, which shows that the superior performance of the proposed
Inter-SubNet.

In the future, we will explore whether the subband interaction is applicable
to other subband-based speech enhancement approaches. The implementation of
causal subband interaction will also be the focus of our future work.

Acknowledgement: This work is supported by National Natural Science Foundation
of China (62076144), Tencent AI Lab Rhino-Bird Focused Research Program
(RBFR2022005) and Tsinghua University - Tencent Joint Laboratory.

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∎

11institutetext:  ✉Wen Yao
11email<EMAIL_ADDRESS>
22institutetext: ✉Xiaoya Zhang
22email<EMAIL_ADDRESS>
33institutetext: 1 College of Aerospace Science and Engineering, National
University of Defense Technology, No. 109, Deya Road, Changsha 410073, China
44institutetext: 2Defense Innovation Institute, Chinese Academy of Military
Science, Beijing 100071, China

# A physics and data co-driven surrogate modeling approach for temperature
field prediction on irregular geometric domain

Kairui Bao 1 Wen Yao 2,* Xiaoya Zhang 2,* Wei Peng 2 Yu Li 2

(Received: date / Accepted: date)

###### Abstract

In the whole aircraft structural optimization loop, thermal analysis plays a
very important role. But it faces a severe computational burden when directly
applying traditional numerical analysis tools, especially when each
optimization involves repetitive parameter modification and thermal analysis
followed. Recently, with the fast development of deep learning, several
Convolutional Neural Network (CNN) surrogate models have been introduced to
overcome this obstacle. However, for temperature field prediction on irregular
geometric domains (TFP-IGD), CNN can hardly be competent since most of them
stem from processing for regular images. To alleviate this difficulty, we
propose a novel physics and data co-driven surrogate modeling method. First,
after adapting the Bezier curve in geometric parameterization, a body-fitted
coordinate mapping is introduced to generate coordinate transforms between the
irregular physical plane and regular computational plane. Second, a physics-
driven CNN surrogate with partial differential equation (PDE) residuals as a
loss function is utilized for fast meshing (meshing surrogate); then, we
present a data-driven surrogate model based on the multi-level reduced-order
method, aiming to learn solutions of temperature field in the above regular
computational plane (thermal surrogate). Finally, combining the grid position
information provided by the meshing surrogate with the scalar temperature
field information provided by the thermal surrogate (combined model), we reach
an end-to-end surrogate model from geometric parameters to temperature field
prediction on an irregular geometric domain. Numerical results demonstrate
that our method can significantly improve accuracy prediction on a smaller
dataset while reducing the training time when compared with other CNN methods.

###### Keywords:

Co-driven surrogate model Mesh learning Reduced-order model Irregular
geometric domain

##  1 Introduction

Thermal analysis plays a vital role in the aircraft structure design and
optimization Bodie et al. (2010); Sanchez and Liscouet-Hanke (2020); Munk et
al. (2017). In the design process, each shape modification involves repetitive
thermal analysis. Thus, a fast and accurate temperature field prediction could
greatly help to improve the efficiency of aircraft design Yao et al. (2011);
Zheng et al. (2020, 2019). An ideal circumstance is to reach real-time
prediction while guaranteeing accuracy simultaneously. However, for most cases
setting on the irregular geometric domain, it is hard to realize this due to
the lack of analytical solutions, and many efforts have been made towards this
end.

Since temperature field prediction heavily relies on solving partial
differential equations (PDEs), traditional numerical methods, such as finite
difference (FD), finite element (FE), and finite volume (FV) methods, are
always employed. These numerical methods usually follow the same formula:
first grid generation and then temperature fields solving. Most of the present
commercial simulation software take this recipe, like ANSYS, ABAQUS, COMSOL.
For the cases on the irregular geometric domain, the mesh should be fine
enough to depict the corner accurately. Therefore, both these two stages are
time-consuming due to massive iterations Majumdar et al. (2001); Chen and Taha
(2014). Overall, such numerical methods could be applied to temperature field
prediction on the irregular geometric domain but with the time-consuming
drawback.

Another natural intuition is constructing a surrogate model to learn the
mapping from design parameters to temperature fields. Many machine learning
(ML) methods have been proven promising for surrogate modeling of PDE-based
systems Capuano and Rimoli (2019); Yao et al. (2020); Zhao et al. (2021a).
Popularly, ML methods are divided into two classes: data-driven and physics-
informed methods. Data-driven methods are widely used by training models with
labeled data, and many data-driven models are proposed for temperature field
prediction. Back in the early 21st century, classic methods, such as Kriging
Hannat et al. (2014), Radial Basis Function Li et al. (2001), Random Forest
Dasari et al. (2019), were introduced as mainstream surrogate modeling in
aircraft design. However, these methods have limitations on solving high-
dimensional problems. Recently, by taking full advantage of deep learning
methods in dealing with high-dimensional problems, many interdisciplinary
integration milestone works emerged. Similarly, deep neural networks (DNNs)
have been introduced to serve as regression models by enforcing temperature
prediction into an image-to-image regression problem. For instance, the fully
connected neural network(FCNN) models Zakeri et al. (2019), the fully
convolutional neural networks Edalatifar et al. (2021), the conditional
generative adversarial networks Farimani et al. (2017) and others Chen et al.
(2021). Especially, Chen et al. Chen et al. (2020) proposed a new deep
learning surrogate-assisted heat source layout optimization method, using the
feature pyramid network Lin et al. (2017) as the surrogate model to evaluate
thermal performance accurately under different input layouts. Compared with
the FPN model, the U-Net model Ronneberger et al. (2015) also has a fairly
good performance. Ma et al. Ma et al. (2020) used a U-Net model and proposed a
combined data-driven and physics-driven method to improve prediction accuracy.
Li et al. Li et al. (2020) raised a new neural operator by parameterizing the
integral kernel directly in Fourier space (FNO), allowing for an expressive
and efficient architecture. However, since all of the above models predict the
entire temperature field, these models are needed to be trained on vast
labeled training data. Furthermore, Heiß et al. Heiß et al. (2021) and Lye et
al. Lye et al. (2021) proposed a novel neural method for high-dimensional
parametric PDEs. They adopt the idea of combining DNNs and Proper Orthogonal
Decomposition (POD), which allows an efficient approximation of the solution.
They also present a multi-level scheme to improve the performance of the
algorithm and reduce the solution error.

Another branch give birth to physics-informed methods. The perspective is
using neural networks as solvers by constraining PDEs into the loss function.
When governing PDEs are known, their solutions can be learned in a physics-
informed way with little or even no data. A milestone work was developed by
Raissi et al. Raissi et al. (2019), named physics-informed neural network
(PINN), which combined the PDE residual with the loss function of FCNN as a
regulative function. Furthermore, Gao et al. Gao et al. (2021) proposed an
unsupervised solver based on CNN for solving the PDE on irregular geometric
domains. They used a coordinate mapping method to map irregular domains to
regular domains and then proposed a physics-constrained CNN learning
architecture to learn solutions of parametric PDEs on irregular domains
without any labeled data. However, every time the shape changes, the method
requires remeshing and retraining to get the physical field. Besides,
motivated by PINN, PDE-based loss function can also be explored for surrogate
modeling. Zhao et al. Zhao et al. (2021b) and Sharma et al. Sharma et al.
(2018) developed a physics-informed CNN for the thermal surrogate modeling by
using the U-Net model, which can learn the mapping from the boundary
conditions of heat area to the steady temperature field without label data.
The work Zhao et al. (2021b) complied with a method that the heat conduction
equation was discretely constructed as a loss function using the finite
difference method; the Dirichlet and Neumann boundary conditions were
represented as image filling. As a surrogate model, it is challenging to deal
with irregular geometric domains because regular geometries with the image-
like format are needed. Whether data-driven or physics-informed surrogate
modeling methods are primarily used to deal with rectangular domains, few
works have been explored for irregular geometric.

In this paper, for the TFP-IGD problem, motivated by the classic numerical
formula, we propose a physics and data co-driven surrogate modeling method.
Firstly, to describe irregular geometric regions more efficiently and in a
regular image-like format (i.e., rectangular domains with uniform grids). We
parameterize the irregular boundary with a Bezier curve and then use body-
fitted coordinate technology to map irregular physical planes to regular
computational planes. Secondly, for the sake of overcoming the time-consuming
shortcoming of traditional simulation software: we construct the meshing
surrogate with nonlinear elliptic equation residuals as the loss function in
the mesh generating stage; then, the thermal surrogate is adopted for
temperature field prediction in an image format. It is worth noting that, in
order to improve the prediction accuracy and reduce the dependence on the
dataset, the thermal surrogate adopts a data-driven-based multi-level POD
model instead of a graph-to-graph regression model. Finally, the results of
the two surrogates are combined to form an irregular temperature field.

In conclusion, this work develops a novel physics and data co-driven surrogate
model, which could reach a fast prediction from geometric shape parameters to
temperature field. On the one hand, our method can overcome the large
computational burden of traditional numerical methods; on the other hand, and
the proposed method can deal with the problem of temperature field prediction
in irregular geometric regions that is difficult for CNN to handle. The main
contributions could be concluded as follows.

  1. (1)

In order to deal with irregular geometric domains, we use the Bezier curve to
approximate the irregular area; and apply the body-fitted coordinate technique
to complete the mapping from the irregular physical plane to the regular
computational plane.

  2. (2)

We develop a physics-informed network for mesh generation based on nonlinear
elliptic equation and finite difference method. The network can be trained by
the physics-informed loss with few labeled data and can be applied to
construct a structural mesh in various irregular domains.

  3. (3)

A data-driven surrogate model based on the multi-level POD method is proposed
to map from geometric parameters to temperature fields. The experiment
verifies that the proposed method can accurately predict even with few
training samples.

  4. (4)

We propose a physics and data co-driven surrogate model framework for
temperature field prediction on irregular domains. The experiments demonstrate
that the temperature field can be accurately and quickly obtained with given
geometric parameters.

The remainder of this paper is structured as follows. Mathematical modeling
for irregular temperature field prediction is introduced in Section 2. The
physics and data co-driven surrogate model framework is presented in Section
3. Numerical results on heat equations and the comparison of our approach with
other methods are presented in Section 4. Finally, Section 5 concludes the
paper.

##  2 Mathematical modeling for temperature field prediction on irregular
geometric domain

In the aircraft thermal structure design, computing the temperature field with
a large variety of geometric shapes is required. Each shape modification based
on the surrogate modeling means a new input and corresponding prediction
followed. In this section, we first define a temperature field prediction
problem on an irregular geometric domain. In a two-dimensional plane, the
steady-state temperature field without internal heat source can be computed by
solving the Laplace equation,

$\displaystyle\frac{\partial^{2}T}{\partial
x^{2}}+\frac{\partial^{2}T}{\partial y^{2}}=0,$ (1)
$\displaystyle\text{Boundary}:\quad T=T_{b}.$

The Dirichlet boundary condition is given for the convenience of explaining
the proposed approach, and $T_{b}$ represents temperature values on different
curves. Figure 1 shows one irregular quadrangle, where geometry shape of the
irregular domain is controlled by curves $P_{1}P_{4}$ and $P_{5}P_{6}$.

Figure 1: The illustration of the irregular geometric domain.

Due to the wide variation range of curve $P_{1}P_{4}$ and $P_{5}P_{6}$, it is
challenging to predict the temperature field under different irregular
geometric domains accurately. Intuitively, the domain could be more complex.

Motivated by curve fitting, an irregular geometric boundary can be represented
by geometric parameters, which can simplify the representation of the boundary
curve. In this paper, the Bezier curve is chosen as the boundary curve of
irregular geometric domains, which is extensively utilized in the geometry
design of aircraft and automobile fields. Once the control vertex
$P_{i}(x_{i},y_{i})\left(i=0,1,\cdots,n\right)$ is fixed, the parametric
equation of the n-order Bezier curve segment can be expressed as follows:

$B(t)=\sum_{i=0}^{n}B_{n,i}(t)P_{i}(x_{i},y_{i}),$ (2)

where $B_{n,i}(t)$ is the Bernstein base function, has the following form:

$B_{n,i}=C_{n}^{i}t^{i}(1-t)^{n-i}.$ (3)

Specifically, Figure 1 also shows a schematic depiction of the irregular
geometric domain fitted by a cubic Bezier curve with four control vertices. In
this case, we make an assumption that this domain is symmetric as the x-axis.
We set $P_{1},P_{2},P_{3},P_{4}$ are the four control vertices of the cubic
Bezier curve $P_{1}P_{4}$, which is similar for curve $P_{5}P_{6}$.

For solving such a problem, the coordinates of these control vertices are
considered as input parameters. Consequently, our goal is translated into
constructing an end-to-end surrogate model, which can efficiently learn the
mapping from the given geometric parameters to the temperature field.

##  3 A physics and data co-driven surrogate model for TFP-IGD

In this section, the physics and data co-driven surrogate modeling framework
is presented for TFP-IGD. Motivated by the classic numerical analysis formula,
the framework has been translated into a two-stage model: the meshing
surrogate for structural mesh generating and the thermal surrogate for
predicting the regular temperature field mapping from the irregular domain.
The meshing surrogate is implemented by constructing a PDE-based loss
function. In contrast to the end-to-end approach, by adopting the proper
orthogonal decomposition (POD) technology, the thermal surrogate is used to
predict the POD coefficients of fixed temperature field basis rather than the
complete temperature field. Figure 2 illustrates the proposed physics and data
co-driven surrogate model framework.

Figure 2: The framework of the physics and data co-driven surrogate.

As shown in Figure 2, with each afferent irregular geometric boundary, we
first input the boundary into the meshing surrogate for meshing and obtain the
coordinates of all mesh points. Then parameterize the boundary and put it into
the thermal surrogate to obtain the mapped regular temperature field. Finally,
the irregular temperature field is obtained by combining the grid position
information in the meshing surrogate and the scalar temperature field
information in the thermal surrogate.

It is organized as follows. In Section 3.1, the body-fitted coordinate mapping
is introduced. Section 3.2 presents a physics-driven surrogate model for the
meshing surrogate, and Section 3.3 describes a multi-level data-driven method
for the thermal surrogate.

###  3.1 Body-fitted coordinate mapping

As shown in Figure 3, irregular grid points generally appear on the physical
plane, making it difficult to predict the temperature field with the image
format. In order to overcome this tricky issue, body-fitted coordinate systems
Thompson et al. (1982) were proposed. The key idea of the body-fitted
coordinate systems is utilizing coordinate transformation techniques to map
solution fields from the irregular physical plane to the rectangular
calculation plane. (as shown in Figure 3).

Figure 3: A schematic diagram of coordinate mapping between irregular physical
and computational plane.

The forward/inverse mapping between coordinates of the irregular physical
plane ($\Omega_{p}$) and regular computational plane ($\Omega_{c}$) can be
defined as,

$\mathbf{x}=\mathbf{f}(\bm{\xi}),\quad\bm{\xi}=\mathbf{f}^{-1}(\mathbf{x}),$
(4)

where $\mathbf{f}:\Omega_{c}\mapsto\Omega_{p}$ denotes the forward map, and
$\mathbf{f}^{-1}:\Omega_{p}\mapsto\Omega_{c}$ denotes the inverse map;
$\mathbf{x}\doteq[x,y]\in\Omega_{p}$ and
$\bm{\xi}\doteq[\xi,\eta]\in\Omega_{c}$ represent mesh coordinates of the
physical plane and computational plane, respectively. In our work, elliptic
coordinate mapping is introduced to complete the coordinate transformation in
Figure 3. The generation of body-fitted coordinate can be treated as solving
the differential equation with the Dirichlet boundary condition. Motivated by
the work of Thompson et al. (1974), elliptic equations with Dirichlet boundary
condition can be applied to approximate the mapping. For example, the mapping
can be obtained by solving a diffusion equation as follows:

$\displaystyle\nabla^{2}\bm{\xi}(\mathbf{x})=\xi_{xx}+\xi_{yy}=0,$ (5)
$\displaystyle\nabla^{2}\bm{\eta}(\mathbf{x})=\eta_{xx}+\eta_{yy}=0,$

with the boundary conditions:

$\displaystyle\bm{\xi}(\mathbf{x})=\bm{\xi}_{b},\quad\text{for}~{}~{}\forall~{}\mathbf{x}\in\partial\Omega_{p}^{i},\quad
i=1,\cdots,4.$ (6)

The coordinates of the physical plane and computational plane are denoted by
$\mathbf{x}\doteq[x,y]$ and $\bm{\xi}\doteq[\xi,\eta]$. However, considering
that the area on the physical plane is invariably irregular, it is challenging
to solve (5). By interchanging the independent and dependent variables in (5),
the following diffusion equations corresponding to the calculation plane in
terms of physical coordinate $x,y$ is derived,

$\displaystyle\alpha\frac{\partial^{2}x}{\partial\xi^{2}}-2\beta\frac{\partial^{2}x}{\partial\xi\partial\eta}+\gamma\frac{\partial^{2}x}{\partial\eta^{2}}=0,$
(7)
$\displaystyle\alpha\frac{\partial^{2}y}{\partial\xi^{2}}-2\beta\frac{\partial^{2}y}{\partial\xi\partial\eta}+\gamma\frac{\partial^{2}y}{\partial\eta^{2}}=0,$

with the boundary conditions:

$\displaystyle\mathbf{x}(\bm{\xi})=\mathbf{x}_{b},$
$\displaystyle\text{for}~{}~{}\forall~{}\bm{\xi}\in\partial\Omega_{c}^{i},\quad
i=1,\cdots,4,$ (8)

and $\alpha,\beta,\gamma$ are given by

$\displaystyle\alpha=\left(\frac{\partial
x}{\partial\eta}\right)^{2}+\left(\frac{\partial y}{\partial\eta}\right)^{2},$
(9) $\displaystyle\gamma=\left(\frac{\partial
x}{\partial\xi}\right)^{2}+\left(\frac{\partial y}{\partial\xi}\right)^{2},$
$\displaystyle\beta=\frac{\partial x}{\partial\xi}\frac{\partial
x}{\partial\eta}+\frac{\partial y}{\partial\xi}\frac{\partial
y}{\partial\eta}.$

By solving (7) with boundary condition defined in (8) numerically (i.e.,
iterative method), the discrete values of the forward map $f$ are obtained.

###  3.2 A physics-driven meshing surrogate

It is very time-consuming to use numerical methods to complete the structural
meshing for involving multiple iterative calculations. Therefore, a physics-
driven approach for the meshing surrogate is introduced in this part. We first
introduce the finite difference discretization of the governing equations,
then we use it to build a physics-informed loss for the meshing surrogate.

####  3.2.1 The finite difference discretization of the governing equations

In order to transform the differential equation residuals into loss function
in the deep learning model, we employ the finite difference method to obtain
the discrete values of (7).

Figure 4: Illustration of mesh point in the regular domain.

More specifically, the discretization of this nonlinear system of equations is
accomplished by using the central difference method. As shown in Figure 4,
taking the $x$ direction as an example, the first-order derivative of the
coordinate system is discrete as follows,

$\displaystyle\frac{\partial
x}{\partial\xi}\approx\frac{x_{i+1,j}-x_{i-1,j}}{2},$ (10)
$\displaystyle\frac{\partial
x}{\partial\eta}\approx\frac{x_{i,j+1}-x_{i,j-1}}{2}.$

Then, the second-order derivative can be approximated by

$\displaystyle\frac{\partial^{2}x}{\partial\xi^{2}}\approx
x_{i+1,j}+x_{i-1,j}-2x_{i,j},$ (11)
$\displaystyle\frac{\partial^{2}x}{\partial\eta^{2}}\approx
x_{i,j+1}+x_{i,j-1}-2x_{i,j},$
$\displaystyle\frac{\partial^{2}x}{\partial\xi\partial\eta}\approx\frac{x_{i+1,j+1}+x_{i-1,j-1}-x_{i+1,j-1}-x_{i-1,j+1}}{4}.$

Similarly, second-order derivative in the $y$ direction is available as

$\displaystyle\frac{\partial^{2}y}{\partial\xi^{2}}\approx
y_{i+1,j}+y_{i-1,j}-2y_{i,j},$ (12)
$\displaystyle\frac{\partial^{2}y}{\partial\eta^{2}}\approx
y_{i,j+1}+y_{i,j-1}-2y_{i,j},$
$\displaystyle\frac{\partial^{2}y}{\partial\xi\partial\eta}\approx\frac{y_{i+1,j+1}+y_{i-1,j-1}-y_{i+1,j-1}-y_{i-1,j+1}}{4}.$

In addition, $\alpha,\beta$ and $\gamma$ can also be gained,

$\displaystyle\alpha=\left(\frac{\partial
x}{\partial\eta}\right)^{2}+\left(\frac{\partial
y}{\partial\eta}\right)^{2}\approx\left(\frac{x_{i,j+1}-x_{i,j-1}}{2}\right)^{2}+\left(\frac{y_{i,j+1}-y_{i,j-1}}{2}\right)^{2},$
(13) $\displaystyle\beta=\frac{\partial x}{\partial\xi}\frac{\partial
x}{\partial\eta}+\frac{\partial y}{\partial\xi}\frac{\partial
y}{\partial\eta}\approx\left(\frac{x_{i+1,j}-x_{i-1,j}}{2}\right)\left(\frac{x_{i,j+1}-x_{i,j-1}}{2}\right)+\left(\frac{y_{i+1,j}-y_{i-1,j}}{2}\right)\left(\frac{y_{i,j+1}-y_{i,j-1}}{2}\right),$
$\displaystyle\gamma=\left(\frac{\partial
x}{\partial\xi}\right)^{2}+\left(\frac{\partial
y}{\partial\xi}\right)^{2}\approx\left(\frac{x_{i+1,j}-x_{i-1,j}}{2}\right)^{2}+\left(\frac{y_{i+1,j}-y_{i-1,j}}{2}\right)^{2}.$

By combining (11), (12), and (LABEL:14), the discretization of (7) is
completed. Further, (7) can be integrated into the loss function of the deep
learning model to form a physics-driven framework in Section 3.2.2.

####  3.2.2 Physics-driven surrogate model for fast meshing

Based on (11), (12), and (LABEL:14), the structured mesh can be generated
through iterative methods. However, the disadvantage is that generating
structured mesh would be time-consuming, especially when the number of grid
points is burgeoning. To improve the efficiency of mesh generation, a novel
physics-constrained CNN learning architecture has been proposed without any
labeled data. The loss function of this framework is set as

$\displaystyle\operatorname{loss}_{x}=\alpha\frac{\partial^{2}x}{\partial\xi^{2}}-2\beta\frac{\partial^{2}x}{\partial\xi\partial\eta}+\gamma\frac{\partial^{2}x}{\partial\eta^{2}},$
(14)
$\displaystyle\operatorname{loss}_{y}=\alpha\frac{\partial^{2}y}{\partial\xi^{2}}-2\beta\frac{\partial^{2}y}{\partial\xi\partial\eta}+\gamma\frac{\partial^{2}y}{\partial\eta^{2}},$
$\displaystyle\operatorname{loss}=\left|\operatorname{loss}_{x}\right|+\left|\operatorname{loss}_{y}\right|.$

Combining formula (11),(12), and (LABEL:14), we can get the discrete values of
formula (LABEL:16). In this way, we can get the loss function of the network.
Then we can train the CNN based upon backpropagation. Figure 5 states the
framework of physics-constrained CNN learning.

Figure 5: The illustration of fast meshing method based on physics-driven
method.

We encode the coordinates of grid points $(x,y)$ on irregular domains into a
two-channel image format. The input of the network is a two-channel image
containing only boundary values, where the rest of the image is filled with
value $0$. The U-net model acts as a mesh generator, which generates
coordinates corresponding to grid points on different irregular geometric
domains. Similarly, the output of the network is also a two-channel image,
which represents coordinates of boundary and internal grid points $(x,y)$
respectively.

###  3.3 A multi-level data-driven thermal surrogate

Using numerical methods to solve the temperature field contains multiple
iteration calculations, making the solution time-consuming. Therefore, a
multi-level data-driven approach for temperature field prediction is
introduced in this part. In order to facilitate the subsequent optimization
process, we use the geometric parameters as the model input and adopt the
multi-level POD model with the POD coefficients as output. Then the
temperature field can be obtained. It is organized as follows. We first
introduce two key techniques for the multi-level data-driven model: The
numerical method under the multi-level framework and the POD technique; then,
we describe the multi-level data-driven model for the regular temperature
field after mapping.

####  3.3.1 Numerical method under the multi-level framework

Motivated by the multi-grid technique, the numerical method under the multi-
level framework is proposed Heiß et al. (2021). For the sake of making the
prediction more precise, integrating into the idea of multi-grid, the
complicated prediction process has been divided into numerous simpler sub-
processes. We define $\phi_{1}\subset\cdots\subset\phi_{L}\subset M(D)$ as a
nested grid space obtained by successive refinement on the basis of coarse
grid over the spatial domain $D$, as shown in Figure 6, with
$v_{i},i=1,\cdots,l$ being obtained by the numerical solution, such as FDM,
FEM, and FVM. The $v_{L}$ is the finest solution.

Figure 6: A schematic for a sequence of nested grids in multi-stage finite
difference method.

Thus, we can bring the modified form of the solution $v_{L}$ on the most
refined mesh $\phi_{L}$ as

$v_{L}^{{}^{\prime}}=v_{1}+\sum_{l=2}^{L}(v_{l}-v_{l-1})=v_{1}+\sum_{l=2}^{L}v_{cl}=\sum_{l=1}^{L}\widetilde{v_{l}}.$
(15)

As can be seen from (15), the computation of the approximate solution for the
finest mesh consists of two operations: the first is a subtraction operation
to get the residuals as in Figure 7(a), and the second is an addition
operation between residuals like in Figure 7(b). For the subtraction
operation, we define $I_{l-1}^{l}$ as the transfer operator from sparse mesh
($\phi_{l-1}$) to the precise mesh ($\phi_{l}$), then the residual can be
expressed as

$v_{cl}=v_{l}-v_{l-1}=v_{l}-I_{l-1}^{l}\left(v_{l-1}\right)\quad
l=2,\cdots,L.$ (16)

In fact, $I_{l-1}^{l}$ can be realized in many ways, such as interpolation.
Equally, the addition operation can be approached similarly. In this way, we
transform the finest mesh solution $v_{L}^{{}^{\prime}}$ into a combination of
sub-process solutions by using the multi-stage finite difference method.

(a) The subtraction operation of getting the residuals

(b) The addition operation between residuals

Figure 7: The schematic diagram of two operations in the multi-stage finite
difference method. The dots represent the temperature values on the grid
points, the red represents the results obtained by interpolation, and the
other colors represent the previously calculated temperature values of the
grid points with different precisions.

####  3.3.2 POD technology

Then, the POD technology will be adopted to reduce the order of each sub-
process solution. From Figure 6, we can apparently see that dimension
$\phi_{l}$ grows exponentially with the level $l$. Aiming to solve such high-
dimensional problem, the POD technology is adopted to generate a basis space
$V_{l}=\operatorname{span}\left\\{v_{l}^{1},\ldots,v_{l}^{\operatorname{dim}V_{l}}\right\\}$
to represent approximations of $\tilde{v_{l}}$ in (15) in an efficient way.
The final solution $v_{L}$ consists of many subprocesses $\tilde{v_{l}}$.

$\tilde{v_{l}}=\sum_{i=1}^{\operatorname{dim}V_{l}}c_{l}^{i}v_{l}^{i},$ (17)

where $c_{l}^{i}$ are the POD coefficients.

In this way, the solution $\tilde{v_{l}}$ could be decomposed as a linear
combination of $c_{l}^{i}$ and $v_{l}^{i}$. Then given the POD basis $V_{l}$,
only POD coefficients $c_{l}^{i}$ need to be predicted. Due to the mapping
between PDE parameters and POD coefficients, the model training process is
simplified and time-saving.

####  3.3.3  A multi-level data-driven method for the thermal surrogate

##### Data preparation

Different from other end-to-end data-driven methods, our method constructs a
mapping between geometric parameters (described in Section 2) and POD
coefficients (described in Section 3.3.2). To get the multi-level POD
coefficients, two essential procedures need to be completed during the data
preparation. Figure 8 illustrates the whole process.

Figure 8: The procedure of data generation.

First, we use the multi-stage finite difference method to generate finite-
difference decomposition of different mesh precision and compute the residuals
between them as in Section 3.3.1. Then, the POD technology has been used to
reduce the order of residuals at different grid precision in Section 3.3.2. In
this way, we can get the multi-level POD coefficients and complete the data
preparation.

##### Model training and prediction

As mentioned earlier, the basis and coefficients are obtained by the POD
technique. In this way, a data-driven surrogate model can be constructed to
map from geometric parameters to the POD coefficients. Then the temperature
field can be obtained by the inner product between the POD coefficients and
the POD basis. Since both geometric parameters and coefficients are low
dimensions, many machine learning models could be capable, such as DNNs,
Random forest regression, Gaussian process regression, etc.

In our multi-level data-driven surrogate model, we use the Gaussian process
(GP) regression model:
$\mathbb{R}^{p}\rightarrow\mathbb{R}^{\operatorname{dim}V_{l}}$ to learn the
basis coefficients $c_{l}^{i}$ ($p$ is the dimension of the input geometry
parameters),

$v_{L}^{{}^{\prime}}=\sum_{l=1}^{L}\tilde{v_{l}}\approx\sum_{l=1}^{L}\sum_{i=1}^{\operatorname{dim}V_{l}}c_{l}^{i}v_{l}^{i}\approx\sum_{l=1}^{L}\sum_{i=1}^{\operatorname{dim}V_{l}}[GP]_{l}v_{l}^{i}.$
(18)

In addition, since it is a coefficient-to-coefficient mapping rather than
predicting the entire temperature field, the whole training process will be
easily and fast predictable.

In general, this is a quite concise data-driven framework. As long as the
geometric parameters representing the irregular geometric domain are input,
the corresponding temperature field under given boundary conditions can be
obtained.

##  4 Numerical results

In this section, Firstly, we evaluate the performance of the meshing surrogate
on the prediction accuracy. And to prove the validity of the meshing
surrogate, we compare the mesh quality of the physics-driven and data-driven
method on U-net with the same structure. Secondly, the experiments for TFP-IGD
based on the thermal surrogate are explored. We evaluate the irregular
temperature field results, which are obtained by the combination of the
meshing surrogate and thermal surrogate. In order to validate the efficacy of
the thermal surrogate, we compare the result with some new combine models with
the thermal surrogate that have been changed to other models, such as the
single-level model, U-net, and FNO model (see Section 4.2.2).

In the experiments, mean absolute error (MAE) and mean relative error (MRE)
are adopted as the metrics to evaluate the performance of the proposed
combined model, where the results solved by finite difference method (FD) are
set as the benchmark. In terms of dataset generation (describe in Section
3.3.3), 2000 samples of different irregular geometry are generated by the
Latin Hypercube sampling method. Among them, $70\%$ of the samples are used
for training, and the remaining $30\%$ for testing. We set the mesh series as
$L=4$, and the finest mesh size is $64\times 256$ while the coarsest mesh size
is $8\times 32$.

###  4.1 Performance of mesh generation by the meshing surrogate

In our study, the U-net model has been adopted in the meshing surrogate model,
and Adam is chosen as the optimizer. For the training process, the training
epoch has been set to 1000 with batch size 16, and the adaptive learning rate
strategy has been applied. A data-driven model will be used for comparison,
and the result shows that the meshing surrogate can generate high-quality
meshes like numerical methods.

####  4.1.1 Prediction performance

We first train a U-Net model on the training set, denoted by $Unet_{phy}$.
Then, we evaluate $Unet_{phy}$ on the test set. Taking the $x$ and $y$
coordinates of the curve boundary as input and through $Unet_{phy}$, we get
the $x$ and $y$ coordinates of all grid points as the output of $Unet_{phy}$.
After getting the prediction results, the overall grid is obtained by
combining the $x$ and $y$ coordinates of each grid point as in Figure 10.

Figure 9: An illustration of the prediction performance of $Unet_{phy}$ on one
test sample.

To better illustrate the prediction accuracy of the meshing surrogate, the MRE
in $x$ and $y$ directions are also used as the evaluation metric apart from
the MAE, which is called $MRE_{x}$ and $MRE_{y}$ separately. $Unet_{phy}$
maintains the MAE of 0.023$K$, while $MRE_{x}$ and $MRE_{y}$ are 0.11$\%$ and
0.25$\%$ respectively. As shown in Figure 9, we take one sample in the test
set to deliver the performance. We display the input $x$ and $y$ coordinates,
the predicted $x$ and $y$ coordinates, the ground-truth $x$ and $y$
coordinates, and prediction error. It can be seen that $Unet_{phy}$ has a very
high prediction accuracy on both $x$ and $y$ coordinates. Based on the three
evaluation metrics and Figure 9, the physics-driven model performs well.
However, it is worth noting that the overall mesh result depends not only on
the error of the two coordinate axes alone but also depends on the overall
mesh quality because it is entirely possible to generate a chaotic mesh.
Therefore, we also analyzed the quality of grid generation from the
perspective of a structured grid. Figure 10 shows the general mesh predicted
by the above sample on the test dataset and the ground-truth mesh.

Figure 10: An illustration of the overall mesh on two test samples.”Mesh
Reference” represents the mesh calculated by the numerical method, and ”Mesh
$Unet_{phy}$” represents the mesh obtained by the meshing surrogate.

As previously mentioned, the physics-driven model has promising results on the
test dataset. Besides, we also found that our method can also obtain high-
quality meshes when selecting geometric parameters outside the test set range
for mesh generation. The model performs on geometric parameters outside the
test set are shown in Figure 11. Since the physics-driven model involves the
discretized mesh control equation into the loss function and can learn the
relevant information of the mesh control equation, which has strong
generalization performance and will have a good performance on geometric
parameters outside the test set range either. Whether on the test set or
outside the test set, the $Unet_{phy}$ performs admirably.

Figure 11: An illustration of the overall mesh on unseen data that geometric
parameters outside the test set range. ”Mesh Reference” represents the mesh
calculated by the numerical method, and ”Mesh $Unet_{phy}$” represents the
mesh obtained by the meshing surrogate.

####  4.1.2 Comparison with data-driven method

For structural mesh generating, a data-driven approach is used in this
experiment as a comparison denoted by $Unet_{data}$, which takes boundary
points as input and mesh points as output the same as the physics-driven
method. Using the U-net model equally while the loss function turns into L2
loss, the label data of actual mesh points are obtained by the iterative
approach.

The MAE of $Unet_{data}$ on the test set after training is 0.029$K$, which has
little difference compared to $Unet_{phy}$, indicating that the data-driven
method can also learn the actual mesh points. However, the data-driven method
is more like a black-box model, which can only predict the coordinates of each
grid point independently. In other words, it is unable to add the connections
between grid points into the model, which will lead to some grid points
appearing misaligned (Figure 12).

Figure 12: The comparison between the data-driven and physics-driven method.

###  4.2 Performance of the combined model for temperature field prediction

In this work, the Gaussian process model has been adopted in the data-driven
model for mapping geometric parameters to POD coefficients. Here, the kernel
of the Gaussian process is the rational quadratic kernel. To describe the
geometric parameters in more detail, as shown in Figure 1, curve $P_{1}P_{4}$
and $P_{5}P_{6}$ are symmetrical about the $x$-axis. Further, make
$x_{1}=2x_{2}=-2x_{3}=-x_{4}$, so only five coordinates
$\left(x_{1},y_{1},y_{2},y_{3},y_{4}\right)$ are required to represent the
irregular geometric boundary. The five coordinates are considered as the input
parameters, and the parameter variation range is shown in Table 1,

Table 1: The value range of the five design parameters Parameter | Lower Bound/$mm$ | Upper Bound/$mm$
---|---|---
$x_{1}$ | 100 | 150
$y_{1}$ | 10 | 16
$y_{2}$ | 0 | 30
$y_{3}$ | 20 | 50
$y_{4}$ | 25 | 75

To demonstrate the effectiveness of the data-driven framework, firstly, the
experimental results could explain why using multi-level method rather than
single-level, and then compare our model with CNN models. It is worth noting
that the temperature field predicted by the data-driven model is a regular
area temperature field. Still, to better show the results, the temperature
maps we present in this part are all irregular temperatures field, which was
created by merging the irregular grid with the regular temperature field.

####  4.2.1 On the prediction performance

We first train the multi-level data-driven model on the training set, denoted
by $ML$, and then evaluate the model on the test set. $ML$ maintains the MAE
of 0.0122K with the MRE of $0.0299\%$ on the test set, and Figure 13 shows the
results of two examples in the test set.

Figure 13: The prediction performance of two samples in the test set. The ”FD”
represents the ground-truth temperature field obtained by the finite
difference method, the temperature field obtained by our approach is expressed
by ”ML”, and ”error” means the error of our method.

It can be seen from Figure 13, although the geometric boundary changes, our
method can still make accurate predictions. We make predictions on geometric
parameters outside the test set range either. The results can be seen in
Figure 14. According to the findings, $ML$ performed well in the test dataset
and has a strong generalization performance.

Figure 14: The prediction performance of two samples on geometric parameters outside the test set range. The ”FD” represents the ground-truth temperature field obtained by the finite difference method, the temperature field obtained by our approach is expressed by ”ML”, and ”error” means the error of our method. Table 2: Performances on the test set for different models Dataset size | 500 | 1000 | 1500 | 2000 | 2500
---|---|---|---|---|---
MAE/K | 0.0157 | 0.0131 | 0.0127 | 0.0123 | 0.0125
MRE/$\%$ | 0.0378 | 0.0320 | 0.0309 | 0.0299 | 0.0305

To illustrate the effect of the training data scale on prediction accuracy, we
take 500, 1000, 1500, 2500 samples as dataset to train separate Gaussian
process models. The performance of the above surrogates is concluded in Table
2. According to the results, the scale of the dataset has little effect on
prediction accuracy, especially when the number of samples is greater than
1000. In other words, our model can still have high prediction accuracy under
a small-scale dataset.

####  4.2.2 Comparison with other methods

As a novel data-driven model for temperature field prediction, there are two
major improvements. One is the utilization of multi-level grids, which
enhances the prediction accuracy. The other uses the POD model, translating
the high-dimensional temperature field prediction problem into a low-
dimensional POD coefficient prediction, which lowers the training complexity.
To prove the above two points, we first compare the results with POD usage
only on the finest mesh and then compare our model with the CNN models, such
as U-net and FNO.

Table 3: Performances on the test set for the single-level and multi-level model Surrogate | MAE/K | MRE/$\%$
---|---|---
$ML$ | 0.0123 | 0.0299
$SL$ | 0.0185 | 0.0448
Figure 15: The prediction performance of single-level and multi-level model on
two test samples. The ”SL” represents the result predicted by the single-level
model, and the temperature field obtained by our approach is expressed by
”ML”.

##### Comparison with the single finest mesh

We investigate the prediction performance of the two models on the same data
set and the single-level model denoted by $SL$. The comparisons of MAE and MRE
are shown in Table 3 and Figure 15 show the two samples of predicting results
on the test set.

Figure 16: The prediction performance of three models on two test samples. The
temperature field obtained by our approach is expressed by ”ML”, the ”FNO”
means the temperature field predicted by the FNO model and the temperature
field predicted by the U-net model expressed as ”U-net”.

In Figure 15, both the single-level and multi-level models can capture the
temperature field information. However, notable discrepancies are observed in
the single-level predicted temperature field, especially in the region that is
near the boundary.

##### Comparison with the CNN models

The U-net Zhao et al. (2021b) and FNO Li et al. (2020) models are employed as
a comparison. Specifically, the boundary of the curve is set as model input,
while the output of the model is the temperature field.

We first investigate the prediction performance. As can be seen from the error
in Figure 16, three models all can accurately predict the temperature field.
However, significant differences are observed in the temperature field
predicted by U-net and FNO model.

(a) MAE

(b) MRE

Figure 17: Performances with different dataset scale for different models.

Then, the relationship between the scale of the dataset and the prediction
performance is discussed. Without loss of generality, due to the common
phenomenon that more training data makes the model less prone to overfitting,
we assume that more training data leads to better performance. Figure 17 shows
the MAE and MRE of three models with different dataset sizes. It can be
observed that the MAE and MRE of the multi-level model are very small equally,
and both two indicators have little changes with the increasing of dataset
size. In contrast to the U-net and FNO model, the prediction accuracy varies
greatly with the scale of the data set, which clearly proves the effectiveness
of the proposed multi-level model.

Besides, the training cost has been investigated, observing that the U-net or
FNO model needs about 10 times more than the multi-level model does to reach
convergence. It takes about 20 minutes to train the U-net or FNO model, but
only around 2 minutes to train the multi-level model with higher predictive
accuracy.

##  5 Conclusions

In this paper, a novel physics and data co-driven surrogate model is proposed
to deal with the TFP-IGD problem. First, the irregular region is parameterized
with the Bezier curve and mapped to the regular computational plane. Then,
using a physics-driven meshing surrogate for structure meshing. And a multi-
level data-driven thermal surrogate is adopted to predict the temperature
field in the mapped regular area. Finally, we combine the result of the
meshing surrogate and thermal surrogate to get the irregular temperature
field. Experimental results show that the meshing surrogate can produce meshes
of high quality and low error. In addition, the thermal surrogate can make an
accurate prediction on smaller datasets. These advantages highlight the
potential of the proposed method to solve more complex scalability problems.

Although we explore a novel model for the TFP-IGD problem, many topics still
exist for further investigations. Due to the strong generality of our model,
it is promising to apply the proposed model to more complex boundary
conditions and more intricate problems, even to handle the three-dimensional
temperature field prediction problem. Furthermore, it is worth predicting more
complex meshes using deep learning models.

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