# Physics-informed MTA-UNet: Prediction of Thermal Stress and Thermal
Deformation of Satellites

Zeyu Cao<EMAIL_ADDRESS>Wen Yao  Wei Peng  Xiaoya Zhang  Kairui Bao
Defense Innovation Institute, Chinese Academy of Military Science, No. 53,
Fengtai East Street, Beijing 100071, China  The No.92941st Troop of PLA,
Huludao 125001, China

###### Abstract

The rapid analysis of thermal stress and deformation plays a pivotal role in
the thermal control measures and optimization of the structural design of
satellites. For achieving real-time thermal stress and thermal deformation
analysis of satellite motherboards, this paper proposes a novel Multi-Task
Attention UNet (MTA-UNet) neural network which combines the advantages of both
Multi-Task Learning (MTL) and U-Net with attention mechanism. Besides, a
physics-informed strategy is used in the training process, where partial
differential equations (PDEs) are integrated into the loss functions as
residual terms. Finally, an uncertainty-based loss balancing approach is
applied to weight different loss functions of multiple training tasks.
Experimental results show that the proposed MTA-UNet effectively improves the
prediction accuracy of multiple physics tasks compared with Single-Task
Learning (STL) models. In addition, the physics-informed method brings less
error in the prediction of each task, especially on small data sets. The code
can be downloaded at: https://github.com/KomorebiTso/MTA-UNet.

###### keywords:

Physics-informed surrogate model , Multi-Task Learning , U-Net , Thermal
stress and deformation

##  1 Introduction

Satellites are important performers of space missions and play an
indispensable role in fields such as communication, remote sensing,
navigation, and military reconnaissance [1]. Increasingly complicated
aerospace missions pose greater challenges to the stability and reliability of
satellites [2]. The components installed on satellite motherboards inevitably
generate much heat during operation due to their high power density [3]. The
heat then leads to a raft of thermal responses such as thermal deformation,
thermal buckling, and thermally induced vibration, affecting the accuracy of
the satellite mission [4, 5]. Ground-based simulation test and numerical
analysis are two traditional methods commonly adopted in engineering to
analyze thermo-mechanical coupling problems such as thermal stress and thermal
deformation of satellites [6, 7, 8, 9]. However, these two methods are usually
costly and require a long computational time. As a result, in terms of
optimization of the design of thermal control measures and structural layout
for satellites, the iterative design by the above two methods is likely to
face a significant cost and computational burden or even fail to be completed
in a given time [10, 11, 12]. It is urgent and necessary to conduct a real-
time analysis of the thermo-mechanical coupling problems of the satellite.
This paper aims to achieve fast and accurate predictions of thermal stress and
thermal deformation of satellite motherboards when given a temperature field.

A common method for the aforementioned engineering requirements is to build a
surrogate model, which implements a high-precision prediction quickly. Once
iterated in optimization procedure, surrogate model could achieve a compromise
between computational accuracy and computational cost, improving the
efficiency of prediction tasks [13, 12, 14]. At present, the mainstream
surrogate model methods include polynomial-based response surface [15],
support vector machine regression [16], radial basis function [17], and
Kriging interpolation [18]. However, when dealing with ultra-high dimensional
regression problems, these traditional surrogate modeling methods all face the
challenge of “curse of dimensionality” (i.e. the difficulty in constructing
surrogate models between high-dimensional variables). As artificial
intelligence advances, many deep learning-based surrogate model methods have
been applied to the prediction of physics-related problems, and neural
networks such as fully connected neural networks [19], fully convolutional
neural networks [20], and conditional generative adversarial networks [21]
have been widely used. In particular, the U-Net, as an advanced neural
network, is coming into vogue. Rishi Sharma et al. [22] take different
boundary conditions as input and applies the U-Net to predict the temperature
field in a two-dimensional plane. Cheng et al. [23] analyze the two-
dimensional velocity and pressure fields around arbitrary shapes in laminar
flow through the U-Net. In addition, Saurabh Deshpande et al. [24] use the
U-Net to predict the large deformation response of hyperelastic bodies under
loading. These related works prove the potential of U-Net in ultra-high
dimensional regression problems.

The focus of the aforementioned deep learning-based surrogate models is to dig
a huge number of training data built by finite element or experiments.
Nevertheless, these models are driven by data loss only, and they often suffer
from high costs of training data or a small number of training data. Physics-
informed neural network (PINN) [25] is an effective way to alleviate this
problem. In the face of the physics prediction tasks where some existing
experience has been accumulated in the real world, the PINN method enables the
integration of prior physical information with neural networks, so as to
reduce the dependence of neural networks on large amounts of training data.
Raissi et al. [25] combine the residual error of PDEs with the loss function
of FCN and predict the velocity and pressure fields around a cylinder, while
Liu et al. [26] use PINN to solve the forward and inverse problem of the
temperature field. In addition, motivated by PINN, PDE-based loss function can
also be explored for surrogate modeling. Bao et al. [27] and Zhao et al. [28]
discrete the heat conduction equation by finite difference, making it a loss
function.

It is noticed that all the physics prediction models mentioned above are only
trained in a Single-Task Learning (STL) manner (i.e., each task is trained
separately). When faced with multiple physics prediction tasks, the STL has
two drawbacks. On the one hand, the separate training for each sub task takes
up much memory space and computational time. On the other hand, if each sub
task is modeled separately, the STL tends to ignore the relationship between
tasks, such as correlation, conflict, and constraint, making it hard to
achieve high precision for the prediction of multiple tasks. The Multi-Task
Learning (MTL) shows more obvious advantages in dealing with such multiple
tasks. In the MTL, multiple tasks share one model, which reduces memory usage
and improves inference speed. MTL combines related tasks together and enables
the tasks to complement each other by sharing feature information between
tasks, which can effectively reduce over-fitting and improve prediction
performance [29, 30].

Existing MTL is divided into soft parameter sharing and hard parameter
sharing. In recent years, there have been a proliferation of research and
successful practices in the industrial community on soft parameter sharing,
such as the Cross-Stitch [31], PLE [32], ESMM [33] and SNR [34]. In
particular, the MMOE [35] proposed by Google has gained great success. It
consists of expert networks shared by multiple tasks and task-specific gate
networks which can decide what to share. Such a flexible design allows the
model to automatically learn how to assign expert parameters based on the
relationship of the low-level tasks. The gate networks thus effectively
improve the generalization performance of the model and the prediction
accuracy across tasks. The MTL has made remarkable achievements in the field
of computer vision and has been widely applied in engineering problems such as
recommendation systems. However, such MTL models have rarely been used in
high-dimensional regression problems with multiple physics tasks.

In this paper, the regression tasks of temperature fields to thermal
deformation and thermal stress of satellite motherboards are regarded as
mappings between images. To achieve accurate and fast prediction of these
multiple physics tasks, three strategies are adopted.

  1. (1)

This paper integrates the advantages of the U-Net and MTL, and proposes the
Multi-Task Attention UNet network (MTA-UNet). This network not only shares
feature information between high-level layers and low-level layers, but also
shares feature information between different tasks. Specially, it shares the
parameters in a targeted way through the attention mechanism. The MTA-UNet
effectively reduces the training time of the model and improves the accuracy
of prediction compared with STL U-Net.

  2. (2)

A Physics-informed approach is applied in training the deep learning-based
surrogate model, where the finite difference is applied to discrete
thermoelastic and thermal equilibrium equations. The equations are encoded
into a loss function to fully exploit the existing physics knowledge.

  3. (3)

Faced with multiple physics tasks, an uncertainty-based loss balancing
strategy is adopted to weigh the loss functions of different tasks during the
training process. This strategy solves the problem that the training speed and
accuracy between different tasks are difficult to balance if trained together,
effectively reducing the phenomenon of competition between tasks.

The rest of the paper is structured as follows. Section 2 presents the
specific definition of the mathematical model. Section 3 introduces the
strategies applied in the construction of deep learning-based MTA-UNet in
detail. Section 4 shows the training steps and the experimental results.
Section 5 is the conclusion.

##  2 Mathematical Modeling of Thermal Stress and Deformation Prediction

In this paper, as previously defined in [3] and [11], a two-dimensional
satellite motherboard with partial openings is used as a study case. The
objective is to realize a rapid prediction of thermal stress and thermal
deformation.

Figure 1: Schematic diagram of a two-dimensional satellite motherboard.

The satellite motherboard with four circular holes is shown in Fig.1, and the
holes represent screw holes in engineering assembly. There are $n$ electronic
components installed on the motherboard, which generate a large amount of heat
during operation and can be regarded as internal heat sources. It is assumed
that the temperature of the satellite motherboard changes $T$ under the joint
action of the internal heat sources and external environment. Due to the
thermal expansion and contraction, the elastomer tends to undergo thermal
deformation, and the deformation trend is restricted to a certain extent
because of the external constraints and mutual constraints between various
parts of the body. With thermal stress generated inside the elastomer, the
thermal stress in turn leads to new additional tension and affects the thermal
deformation.

In this case, non-displacement boundary conditions are adopted at the edges of
the holes. The thermoelastic properties on the motherboard tend to be
isotropic, with the parameters consistent with the thermoelastic properties of
aluminum. The linear elasticity coefficient of the motherboard is considered
to be around the reference temperature $T_{0}=273$ K and is assumed not to
change with temperature. Non-stress and adiabatic boundary conditions are
adopted on the outer boundary of the motherboard. The side length of the
rectangular computational domain is set to $L=H=20$ cm. The radius of the
holes is $r=0.5$ cm, and the thermal conductivity coefficient within the
domain is $k=1$ W/(m $\times$ k). The coefficient of linear expansion is
$\alpha=1e-5$, Young’s modulus is $E=50e3$, and Poisson’s ratio is $\mu=0.2$.

The thermoelasticity motherboard is a two-dimensional planar elastomer and its
thermal stress components and temperature satisfy the following equations:

$\left\\{\begin{array}[]{l}\sigma_{xx}=\frac{E}{1-\mu^{2}}\left(\frac{\partial
u_{x}}{\partial x}+\mu\frac{\partial u_{y}}{\partial y}\right)-\frac{E\alpha
T}{1-\mu}\\\ \sigma_{yy}=\frac{E}{1-\mu^{2}}\left(\frac{\partial
u_{y}}{\partial y}+\mu\frac{\partial u_{x}}{\partial x}\right)-\frac{E\alpha
T}{1-\mu}\\\ \sigma_{xy}=\frac{E}{2(1+\mu)}\left(\frac{\partial
u_{y}}{\partial x}+\frac{\partial u_{x}}{\partial y}\right)\end{array}\right.$
(1)

where $\sigma_{xx}$ means the x-directional thermal stress, $\sigma_{yy}$
means the y-directional thermal stress, and $\sigma_{xy}$ denotes the
tangential directional thermal stress. In addition, $u_{x}$ and $u_{y}$
represent x-directional thermal displacement and y-directional thermal
displacement respectively. According to the differential equation of thermal
equilibrium, the $u_{x}$ and $u_{y}$ satisfy the following set of equations:

$\left\\{\begin{array}[]{l}\frac{\partial^{2}u_{x}}{\partial
x^{2}}+\frac{1-\mu}{2}\frac{\partial^{2}u_{x}}{\partial
y^{2}}+\frac{1+\mu}{2}\frac{\partial^{2}u_{y}}{\partial x\partial
y}-(1+\mu)\alpha\frac{\partial T}{\partial x}=0\\\
\frac{\partial^{2}u_{y}}{\partial
y^{2}}+\frac{1-\mu}{2}\frac{\partial^{2}u_{y}}{\partial
x^{2}}+\frac{1+\mu}{2}\frac{\partial^{2}u_{x}}{\partial x\partial
y}-(1+\mu)\alpha\frac{\partial T}{\partial y}=0\end{array}\right.$ (2)

and the boundary conditions further satisfy:

$\left\\{\begin{array}[]{l}l\left(\frac{\partial u_{x}}{\partial
x}+\mu\frac{\partial u_{y}}{\partial
y}\right)_{s}+m\frac{1-\mu}{2}\left(\frac{\partial u_{x}}{\partial
y}+\frac{\partial u_{y}}{\partial x}\right)_{s}=l(1+\mu)\alpha(T)_{s}\\\
m\left(\frac{\partial u_{y}}{\partial y}+\mu\frac{\partial u_{x}}{\partial
x}\right)_{s}+l\frac{1-\mu}{2}\left(\frac{\partial u_{y}}{\partial
x}+\frac{\partial u_{x}}{\partial
y}\right)_{s}=m(1+\mu)\alpha(T)_{s}\end{array}\right.$ (3)

Assuming that the direction of the outer normal to the elastic plane is N,
then $l$ and $m$ can be expressed as the directional cosine of the $x$ axis
and $y$ axis, respectively.

Based on the above set of equations and the corresponding parameters, the
thermal stress components and the thermal displacement components can be
calculated when given an arbitrary temperature field $T$. However, the
numerical methods are usually costly and time-consuming. In this paper, a deep
learning-based surrogate model is constructed to quickly predict five physics
components of satellite motherboard, $u_{x}$, $u_{y}$, $\sigma_{xx}$,
$\sigma_{yy}$, and $\sigma_{xy}$. The five multiple physics prediction tasks
are divided into three multiple sub tasks, $u_{x}$ and $\sigma_{xx}$ in task
1, $u_{y}$ and $\sigma_{yy}$ in task 2, and $\sigma_{xy}$ in task 3. Three sub
tasks are trained together and the temperature field $T$ is the input of
model.

##  3 Method

This section details the technical strategies proposed during the construction
of the deep learning surrogate model. Firstly, on account of multiple physics
tasks, to reduce the training cost and improve the prediction performance of
the model, we design a neural network structure MTA-UNet in section 3.1.
Subsequently, in section 3.2, a physics-informed approach is applied to reduce
the sample size of training data. Lastly, section 3.3 adopts a uncertainty-
based loss balancing strategy to weight the loss function of multiple tasks,
adjusting the training speed and accuracy of multiple tasks. Fig.2 shows the
frame diagram of the main technical strategies.

Figure 2: The frame diagram of the main technical strategy. This paper applies
three strategies when build the surrogate model, the MTA-UNet network,
physics-informed model, and uncertainty-based loss balancing method
respectively.

###  3.1 MTA-UNet Network Structure

This section describes the MTA-UNet in detail, and Fig.3 shows the actual
structure. This model takes the U-Net as the basic architecture, and the
improvement is the sharing method between different tasks. In MTA-UNet,
multiple sub tasks are trained together and can share feature parameters with
other tasks. Different tasks share the coding layer and each of them has their
specific decoding layer. In addition, when the features of different layers
are concatenated, feature selection is carried out through the Attention Gate
(AG). The AG distinguishes which shared features are highly correlated with
specific tasks and extracts the relevant shared features from the shared
coding layer. The MTA-UNet combines the merits of U-Net and MTL, having the
advantage of a “purposeful” combination of feature parameters from different
layers and physics tasks. This design fully learns information from the
training sets, reducing the training time of the model and memory space of the
system and also improving the prediction performance of the model. This
section is divided into the architecture of the MTA-UNet (3.1.1), the sharing
mechanism of the MTA-UNet (3.1.2), and the Attention Gate of the MTA-UNet
(3.1.3).

Figure 3: Structure of MTA-UNet.

####  3.1.1 Architecture of the MTA-UNet Network

To fully share feature information of different layers, MTA-UNet adopts the
U-Net [36] as its basic architecture. In recent years, the U-Net has exhibited
good performance when dealing with various ultra-high dimensional regression
problems. According to Fig.4, U-Net has a U-shaped symmetric structure which
combines feature parameters of different layers. Owing to this characteristic,
the U-Net network is suitable for image-to-image regression tasks as it takes
multi-scale feature fusion into account and increases the amount of
information.

Figure 4: Structure of U-Net.

The MTA-UNet network built in this paper fully retains the advantages of the
U-Net network. The first half of MTA-UNet is a classical downsampling process,
which is the feature extraction part similar to the coding process, capable of
learning multi-scale features of an image through pooling and convolution
operations. The second half is an upsampling process similar to the decoding
part, capable of restoring the size of the image layer by layer through
deconvolution operations. These feature maps are effectively used in
subsequent calculations through skip connections.

In MTA-UNet, the downsampling layers are shared layers and the upsampling
layers are exclusive to multiple tasks, respectively. The MTA-UNet consists of
two repetitive applications of 3x3 convolutions (padding=1), and each
convolution is followed by a BatchNorm2d normalized activation function layer
and a 2x2 max-pooling operation with a stride of 2. We double the number of
feature channels in each step of the shared downsampling and, in contrast,
halve the number of feature channels in each step of the exclusive upsampling.
After the feature selection through the AG, feature parameters of upsampling
are skip connected to downsampling with the corresponding scale of the feature
map. In the last layer, each component feature vector is mapped to its
corresponding class by a 1x1 convolution.

####  3.1.2 Feature Sharing Mechanism of MTA-UNet

To enable the sharing of feature parameters across multiple physics tasks, the
MTA-UNet uses a partially shared coding layer to enable simultaneous learning
of multiple tasks. During the downsampling process, multiple tasks share the
coding layer, each having an independent decoding layer. This information
concatenation operation of shared and specific feature parameters realizes the
deep sharing of multi-task feature parameters.

For the task $k$ in the given $K$ tasks, $k=1,2,\ldots,K$. The model consists
of the shared downsampling function $f$, the exclusive upsampling $g_{k}$ ,
and $k$ task networks $h_{k}$. The shared layer follows the input layer, and
the task networks are built upon the output of the shared-bottom. The $y_{k}$
is the output of the task $k$, follows the corresponding task-specific tower.
Assuming that for the task $k$, the $m^{th}$ downsampling layer of the model
is $f^{m}$, and its corresponding upsampling layer is $g_{k}^{m}$, the output
of the task $k$ is:

$\displaystyle
y_{k}=h_{k}\left(\text{SkipConnect}\left(AG_{k}^{m}\left(f^{m},g_{k}^{m}\right),g_{k}^{m}\right)\right)$
(4)

where $AG_{k}^{m}$ denotes the AG operation for the $m^{th}$ layer of the task
$k$.

####  3.1.3 Attention Gate of MTA-UNet

Inspired by CNN-based MTL models such as MMOE [35] and PLE [32], the AG
operation is added to the MTA-UNet model to realize the selection of shared
features. In this way, shared features that are highly relevant to specific
tasks are extracted from the shared network layer.

Figure 5: Attention Gate operation.

Fig.5 illustrates the AG operation of task $k$, where a series of operations
such as Relu, Sigmoid, and resampler are performed to evaluate the similarity
between $f^{m}$ and $g_{k}^{m}$. The $m^{th}$ upsampling layer $f^{m}$ and its
corresponding downsampling layer $g_{k}^{m}$ are input of $AG_{k}^{m}$. Since
the underlay network $g_{k}^{m}$ is generally more accurate, when the
similarity between $f^{m}$ and $g_{k}^{m}$ is calculated, the features in
$f^{m}$ with higher similarity with $g_{k}^{m}$ are given higher weights,
equals chosen of feature parameters of $f^{m}$. The selection is reflected in
the calculation of the weight coefficient, and the operation is presented as:

$\displaystyle
AG_{k}^{m}\left(f^{m},g_{k}^{m}\right)=\sum_{i=1}^{L_{x}}\operatorname{Similarity}\left(f^{m},g_{k}^{m}\right)\times
f^{m}$ (5)

###  3.2 A Physics-informed Training Strategy

To make full use of the prior physical laws among the physical quantities
being predicted, reducing the size of the training sample, a physics-informed
strategy is used in the training process. The existing physics knowledge
(Eq.1-Eq.3) is discretized by the Finite Difference Method (FDM) and
integrated into a loss function to construct a physics-informed surrogate
model. The FDM is a numerical solution method that expresses basic equations
and boundary conditions in the form of function approximation.

It was assumed that an arbitrary two-dimensional elastomer is divided into a
uniform grid as shown in Fig.6, in which the intersections of the lines are
the nodes.

Figure 6: Grid diagram of finite differences.

Let $f=f(x,y)$ be a continuous physical quantity, which can be expressed as
stress component, displacement component, temperature, etc. Assuming that the
grid spacing is sufficiently small, then $\left|x-x_{0}\right|=h$. The central
difference formulas of the first and second derivatives of $x$ at node 0 can
be obtained :

$\displaystyle\left(\frac{\partial f}{\partial
x}\right)_{0}\approx\frac{f_{1}-f_{3}}{2h},\left(\frac{\partial^{2}f}{\partial
x^{2}}\right)_{0}\approx\frac{f_{1}+f_{3}-2f_{0}}{h^{2}}$ (6)

Similarly, the central difference formulas for the first and second
derivatives at node 0 in the y direction are:

$\displaystyle\left(\frac{\partial f}{\partial
y}\right)_{0}\approx\frac{f_{2}-f_{4}}{2h},\left(\frac{\partial^{2}f}{\partial
y^{2}}\right)_{0}\approx\frac{f_{2}+f_{4}-2f_{0}}{h^{2}}$ (7)

According to the above equations, the central difference formulas for mixed
second derivatives are:

$\displaystyle\left(\frac{\partial^{2}f}{\partial x\partial
y}\right)_{0}=\left[\frac{\partial}{\partial x}\left(\frac{\partial
f}{\partial y}\right)\right]_{0}=\frac{\left(\frac{\partial f}{\partial
y}\right)_{1}-\left(\frac{\partial f}{\partial
y}\right)_{3}}{2h}\approx\frac{1}{4h^{2}}\left[\left(f_{6}+f_{8}\right)-\left(f_{5}+f_{7}\right)\right]$
(8)

For boundary points where central difference cannot be used, forward
difference or backward difference are used for discretization:

$\displaystyle\left(\frac{\partial f}{\partial
x}\right)_{0}\approx\frac{-3f_{0}+4f_{1}-f_{9}}{2h}\approx\frac{3f_{0}-4f_{3}+f_{11}}{2h},$
(9)

$\displaystyle\left(\frac{\partial f}{\partial
y}\right)_{0}\approx\frac{-3f_{0}+4f_{2}-f_{10}}{2h}\approx\frac{3f_{0}-4f_{4}+f_{12}}{2h}$
(10)

To deal with the prediction tasks of thermal stress and thermal displacement,
a neural network $\hat{Y}(x,\theta)$ is constructed, where
$\theta=\left\\{W^{\ell},b^{\ell}\right\\}_{1\leq\ell\leq D}$, and $W^{\ell}$
is the weight matrix, $b^{\ell}$ is the deviation vector in the neural
network, and the physics-informed loss function of the neural network is
defined as:

$\displaystyle\mathcal{L}_{\text{PINN}}(\theta)=\mathcal{L}_{\text{data
}}+\mathcal{L}_{\text{pde}}$ (11)

where $\mathcal{L}_{\text{data}}$ is the data loss obtained from labeled data
with the loss function of Mean Square Error (MSE), and the data loss of the
task $k$ is defined as:

$\displaystyle\mathcal{L}_{\text{data}_{k}}=\frac{1}{\left|\mathcal{P}_{f}\right|}\sum_{(\mathbf{x}_{k},\mathbf{y}_{k})\in\mathcal{P}_{f}}\left(Y_{{k}}-\hat{Y}_{{k}}\right)^{2}$
(12)

where $\mathcal{P}_{f}$ denotes the size of data set, $\hat{Y}_{k}$ represents
the prediction of the model, $Y_{k}$ is the ground truth.

The aforementioned finite difference equations (Eq.6-Eq.10) are used to
discretize the thermoelasticity and the thermal equilibrium equations, with
the central difference method for interior points and a forward/backward
difference method for boundary points. The discretized PDEs are encoded into a
loss function $\mathcal{L}_{\text{pde}}$ ($\mathcal{L}_{\text{pde}}$ denotes
the physical losses of $K$ tasks), which is defined as:

$\displaystyle\mathcal{L}_{pde}=\frac{1}{\left|\mathcal{P}_{f}\right|}\sum_{(x,y)\in\mathcal{P}_{f}}\left(\mathcal{L}_{\sigma_{xx_{-}}pde}+\mathcal{L}_{\sigma_{yy_{-}}pde}+\mathcal{L}_{\sigma_{xy_{-}}pde}+\mathcal{L}_{u_{x_{-}}pde}+\mathcal{L}_{u_{y_{-}}pde}\right)$
(13)

Where $\mathcal{L}_{u_{x_{-}}pde}$, $\mathcal{L}_{u_{y_{-}}pde}$,
$\mathcal{L}_{\sigma_{xx_{-}}pde}$, $\mathcal{L}_{\sigma_{xy_{-}}pde}$,
$\mathcal{L}_{\sigma_{yy_{-}}pde}$ mean the PDE loss of Eq.1 and Eq.2
respectively.The boundary loss $\mathcal{L}_{bc}$ is added to the physical
loss function matrix $\mathcal{L}_{pde}$ in the form of the matrix mask. A
convolution operation can also realize the physics-informed strategy.

By training the neural network and minimizing the loss function until the best
parameter $\theta^{*}$ is found, we construct a physics-informed surrogate
model of the thermal deformation and thermal stress of satellite motherboards.

###  3.3 Uncertainty-based Multi-Task Loss Balancing Strategy

The training goal of the MTA-UNet model is to minimize the difference between
the prediction results and the labels. For MTL, however, errors between
multiple tasks features cause competition between multiple tasks. To overcome
this seesaw phenomenon, it is an integral part of MTL to balance the training
speed and accuracy of multiple tasks through some loss balancing strategies.

Since the loss functions of different tasks tend to have different magnitudes,
this paper first normalized their inputs. Subsequently, to solve the problem
of competition between different tasks, the most direct approach is to use the
weight coefficient $w_{i}$ to adjust the proportions of the loss functions of
different tasks, as shown in the following equation.

$\displaystyle\operatorname{Loss}=\sum_{k}^{K}w_{k}\times\operatorname{Loss}_{k}$
(14)

However, this loss balancing method of fixed weights may not be applicable in
some cases as different tasks are very sensitive to the setting of $w_{i}$,
and different settings of $w_{i}$ vary greatly in terms of performance.

We would like to use a dynamic weight method to balance the loss functions of
different tasks. The mainstream methods include GradNorm [37], DWA [38], DTP
[39], and Uncertainty [40, 41], among which GradNorm consumes a longer
computational time as it requires gradient computation, while DWA and DTP need
additional weighting operations. This paper adopts the uncertainty-based
method to dynamically adjust the weight coefficients for the loss functions of
different tasks. The model in this study is constructed based on task-
dependent and homoscedastic uncertainty in aleatoric uncertainty [40].

For a certain task $k$, the model predicts both a prediction $y_{k}$ and the
homoscedastic uncertainty $\sigma_{k}$ of the model. Thus the loss function
for MTL is defined as:

$\displaystyle\mathcal{L}\left(W,\sigma_{1},\sigma_{2},\ldots,\sigma_{k}\right)=\sum_{k}\frac{1}{2\sigma_{k}^{2}}\mathcal{L}_{k}(W)+\log\sigma_{k}^{2}$
(15)

where $W$ means the weight matrix, $\mathcal{L}_{k}(W)$ is the loss function
of task k. $\sigma_{k}$ means noise of the model of task $k$,
$\frac{1}{2\sigma_{k}^{2}}$ is the weight coefficient of the corresponding
task, and $\text{log}\sigma_{k}^{2}$ is the regularization item to prevent
$\sigma_{k}$ from learning too large. In this way, higher weights can be given
to simple tasks, and the learning of other tasks are driven by them.

For the total of $K$ regression tasks driven by physics, the overall loss
function is defined as:

$\displaystyle\mathcal{L}=\sum_{k=1}^{K}\left(\frac{1}{2\sigma_{k}^{2}}\mathcal{L}_{\text{data}_{k}}(W)+\log\sigma_{k}^{2}\right)+\frac{1}{2\sigma_{pde}^{2}}\mathcal{L}_{pde}(W)+\log\sigma_{pde}^{2}$
(16)

##  4 Experiments

In this section, the training details and the results of experiments are
shown. Section 4.1 introduces the data set and training metrics. Section 4.2
discusses the effect of dynamic loss balancing strategy, MTA-UNet, and the
physics-informed method in detail, respectively.

###  4.1 Training Steps

Data set: In MTA-UNet, the input is a two-dimensional planar temperature field
on a $200\times 200$ uniform grid and the output is five corresponding thermal
stress matrices and thermal deformation matrices, including x-directional
displacement $u_{x}$, y-directional displacement $u_{y}$, x-directional
thermal stress $\sigma_{xx}$, y-directional thermal stress $\sigma_{yy}$, and
tangential thermal stress $\sigma_{xy}$. The OpenFOAM software is applied to
build the data sets, and the computational domain is a uniform grid. Fig.7
shows one of the data set samples.

Figure 7: A sample of data set. This figure shows a sample of the data set.
The temperature field is the input of surrogate model, and $u_{x}$, $u_{y}$,
$\sigma_{xx}$, $\sigma_{yy}$, $\sigma_{xy}$ are outputs.

The input temperature field is a Gaussian Random Field (GRF) [42], and the
roughness of the temperature field matrix changes when the mean and covariance
of GRF are adjusted. This paper generates training sets with sample sizes of
100, 200, 500,1000, 2000, 5000, and 10,000. Firstly, the prediction of the STL
model U-Net and the MTL model MTA-UNet are compared in the training set with a
sufficient number of 10,000 samples to confirm the superiority of our model.
Subsequently, the MTA-UNet model is trained in training sets with sample sizes
of 100, 200, 500, 1000, 2000, and 5000 respectively to verify the
effectiveness of the physics-informed approach. The trained models are tested
in a general test set with a sample size of 500.

Metrics: AdamDelda [43] is selected as the optimizer and the MTA-UNet model is
implemented by PyTorch 1.8. The training batch size is set to 16. The
prediction performance of different models for task $k$ is measured by metrics
MAE and MRE, defined as:

$\displaystyle
MAE_{k}=\frac{1}{\left|\mathcal{P}_{f}\right|}\sum_{(\mathbf{x}_{k},\mathbf{y}_{k})\in\mathcal{P}_{f}}\left|Y_{k}-\hat{Y}_{k}\right|$
(17)

$\displaystyle
MRE_{k}=\frac{1}{\left|\mathcal{P}_{f}\right|}\sum_{(\mathbf{x}_{k},\mathbf{y}_{k})\in\mathcal{P}_{f}}\frac{\left|Y_{k}-\hat{Y}_{k}\right|}{\hat{Y}_{k}}$
(18)

###  4.2 Experiment Results

####  4.2.1 Effectiveness of Dynamic Loss Balancing Strategies

To verify the necessity and effectiveness of the dynamic loss balancing
strategy in our MTL regression model, training is carried out with the fixed
weighted loss balancing strategy and uncertainty-based dynamic loss balancing
strategy, respectively. Fig.8 shows the training curves, indicating that if
the loss function of MTL is balanced by fixed weight, multiple tasks will
deviate in different directions due to different task objectives. In this way,
the better learning of one task decreases the accuracy of another. Unlike
fixed weight, the loss functions of multiple tasks simultaneously decrease at
a stable speed with the uncertainty-based dynamic balancing strategy. The
dynamic strategy avoids the phenomenon of competition and enables multiple
tasks to achieve good performance simultaneously.

(a) Fixed weighted loss balancing strategy

(b) Uncertainty-based loss balancing strategy

Figure 8: Training curves of different loss balancing strategies. The training
curves of fixed weighted strategy shock in a certain range, making it
difficult for training among multiple tasks. While the training curves of
dynamic uncertainty-based loss balancing strategy decrease in a stable speed,
enable good prediction for multiple tasks simultaneously.

####  4.2.2 Performances of Models

We train the MTA-UNet model and the U-Net model in the training set of 10,000
samples and test them in the general test set, respectively. The MAE and MRE
errors obtained from the training are shown in Table.1 and Table.2. The
$\sigma_{MAE}$ and $\sigma_{MRE}$ mean the standard deviations of MAE and MRE
respectively, and the data in bold represent the result of MTA-UNet.

Table 1: The MAE of different networks Task | $MAE$ | $\sigma_{MAE}$
---|---|---
U-Net | MTA-UNet | U-Net | MTA-UNet
$u_{x}$ | 1.9754$\times 10^{-6}$ | 1.1556$\times 10^{-6}$ |  $\pm$8.2727$\times 10^{-7}$ |  $\pm$1.9903$\times 10^{-7}$
$u_{y}$ | 2.0947$\times 10^{-6}$ | 1.0721$\times 10^{-6}$ |  $\pm$8.3263$\times 10^{-7}$ |  $\pm$2.2980$\times 10^{-7}$
$\sigma_{xx}$ | 0.0945 | 0.0713 |  $\pm$0.01 |  $\pm$9.5297$\times 10^{-5}$
$\sigma_{yy}$ | 0.0924 | 0.0728 |  $\pm$0.01 |  $\pm$0.0098
$\sigma_{xy}$ | 0.0474 | 0.0402 |  $\pm$0.01 |  $\pm$0.0087
Table 2: The MRE of different networks Task |  $MRE/\%$ |  $\sigma_{MRE}$
---|---|---
U-Net |  MTA-UNet |  U-Net |  MTA-UNet
$u_{x}$ |  2.23 |  1.30 |  $\pm$0.0056 |  $\pm$0.0044
$u_{y}$ |  2.39 |  1.21 |  $\pm$0.0058 |  $\pm$0.0039
$\sigma_{xx}$ |  1.27 |  0.96 |  $\pm$0.0036 |  $\pm$0.0027
$\sigma_{yy}$ |  1.29 |  0.98 |  $\pm$0.0039 |  $\pm$0.0027
$\sigma_{xy}$ |  1.36 |  1.03 |  $\pm$0.0038 |  $\pm$0.0030

We first investigate the prediction performance. According to Table.2, the
minimum per-pixel MRE of the proposed MTA-UNet model is only 0.96%, and the
model shows good performance in predicting thermal stress and thermal
deformation of satellite motherboards. Fig.9 shows the prediction result of
MTA-UNet. The mean per-pixel error is calculated with high precision according
to the ground truth, demonstrating the feasibility of using the MTA-UNet model
for regression between ultra-high dimensional variables. In addition, as for
the comparison between the STL U-Net and the MTL MTA-UNet model proposed, it
can be seen that the MTA-UNet exhibits better prediction performance on each
physical prediction task by fully sharing feature information between
different tasks. Especially in $u_{x}$, the proposed MTL MTA-UNet model
reduces the $MAE$ by at least 41% lower than the STL U-Net model in our task.
Since different tasks have different noise, the sharing of feature parameters
between tasks by MTA-UNet cancels part of the noise in different directions to
a certain extent, improving the prediction accuracy. According to variances of
MAE and MRE for different models, the MTA-UNet model is more robust than
U-Net. The sharing mechanism in MTA-UNet plays the role of data enhancement.
Moreover, some parameters required in a single task are better trained by
other tasks, and the sharing of feature between different tasks make sense in
this respect.

Then, the training cost has been discussed. This deep learning surrogate model
can reduce the prediction time of thermal stress and thermal deformation from
2 minutes to 0.23 seconds, effectively saving the time cost, which also
verifies the effectiveness of the deep learning-based thermal stress and
thermal deformation surrogate model.

(a) Sample 1

(b) Sample 2

Figure 9: Prediction of the MTA-UNet model. This figure shows two samples of
results by MTA-UNet. The up row of each sample is the label of $u_{x}$,
$u_{y}$, $\sigma_{xx}$, $\sigma_{yy}$ and $\sigma_{xx}$, the next line are
corresponding predictions of MTA-UNet.

####  4.2.3 Effects of the Physics-informed Strategy

To verify the contributions of the physics-informed strategy to the accuracy
of the model and sample size, the MTA-UNet surrogate model is trained in data
sets with different sample sizes and tested in a general test set, and the
comparison of MRE between data-driven only and the physics-informed is shown
in Table.3. The $\sigma_{MRE}$ means the standard deviation of MRE, and the
data in bold represent the result of the physics-informed strategy.

Table 3: MRE of training strategies on dataset with different scales Scale | Method | $MRE/\%$ | $\sigma_{MRE}$
---|---|---|---
$u_{x}$ | $u_{y}$ | $\sigma_{xx}$ | $\sigma_{yy}$ | $\sigma_{xy}$ | $u_{x}$ | $u_{y}$ | $\sigma_{xx}$ | $\sigma_{yy}$ | $\sigma_{xy}$
200 | Data | 6.46 | 6.42 | 4.43 | 4.55 | 5.52 |  $\pm$0.0421 |  $\pm$0.0396 |  $\pm$0.0145 |  $\pm$0.0143 |  $\pm$0.0200
PDE | 6.07 | 6.02 | 4.02 | 4.17 | 5.11 |  $\pm$0.0305 |  $\pm$0.0297 |  $\pm$0.0103 |  $\pm$0.0102 |  $\pm$0.0142
500 | Data | 3.72 | 3.76 | 2.77 | 2.78 | 2.93 |  $\pm$0.0185 |  $\pm$0.0193 |  $\pm$0.0083 |  $\pm$0.0088 |  $\pm$0.0104
PDE | 3.48 | 3.50 | 2.37 | 2.45 | 2.66 |  $\pm$0.0095 |  $\pm$0.0096 |  $\pm$0.0065 |  $\pm$0.0056 |  $\pm$0.0093
1000 | Data | 3.11 | 3.19 | 1.99 | 2.01 | 2.29 |  $\pm$0.0114 |  $\pm$0.0115 |  $\pm$0.0056 |  $\pm$0.0057 |  $\pm$0.0086
PDE | 2.96 | 2.98 | 1.82 | 1.78 | 2.09 |  $\pm$0.0073 |  $\pm$0.0073 |  $\pm$0.0046 |  $\pm$0.0044 |  $\pm$0.0071
2000 | Data | 2.62 | 2.65 | 1.55 | 1.61 | 1.72 |  $\pm$0.0089 |  $\pm$0.0092 |  $\pm$0.0049 |  $\pm$0.0052 |  $\pm$0.0065
PDE | 2.49 | 2.51 | 1.41 | 1.40 | 1.53 |  $\pm$0.0063 |  $\pm$0.0062 |  $\pm$0.0044 |  $\pm$0.0042 |  $\pm$0.0044
5000 | Data | 2.28 | 2.41 | 1.19 | 1.25 | 1.41 |  $\pm$0.0057 |  $\pm$0.0069 |  $\pm$0.0039 |  $\pm$0.0040 |  $\pm$0.0046
PDE | 2.17 | 2.30 | 1.11 | 1.12 | 1.22 |  $\pm$0.0048 |  $\pm$0.0052 |  $\pm$0.0029 |  $\pm$0.0030 |  $\pm$0.0031

According to the experimental results, the physics-informed surrogate model
exhibits better performance than data-driven only in data set with different
sample sizes. Relatively speaking, the physics-informed strategy improves more
significantly in the case of small samples, and the gap between the data-
driven strategy and the physics-informed strategy gradually narrows as the
number of samples increases, which demonstrates the potential of the physics-
informed strategy in cases with small sample size.

In addition, we conduct training on smaller size of samples. Fig.10 shows the
prediction obtained by the data-driven strategy and the physics-informed
strategy respectively for 100 samples. The comparison shows that physics-
informed strategy obtains predictions that are more in line with the practical
meaning of physics and realistic working conditions.

Figure 10: Predictions of the data-driven strategy and the physics-informed
strategy.

##  5 Conclusion

To solve the regression problems of multiple physical quantities in thermal-
mechanical coupled fields, this paper proposes a novel MTL network MTA-UNet.
The MTA-UNet integrates the strengths of U-Net and MTL, having the advantage
of selective sharing parameters of different tasks and layers. With shared
encoding layers, task-specific decoding layers, and AG feature filtering
operations of MTA-UNet, the correlation between different tasks is fully
learned. The performance of the deep learning-based surrogate model is greatly
enhanced, and the training time and system memory are also effectively
decreased. Subsequently, an uncertainty-based dynamic loss weighted strategy
is used among loss functions of different tasks. With this strategy, the
training speed and accuracy of each sub task are well balanced. In addition, a
physics-informed method is applied during training to rapidly predict the
thermal stress and thermal deformation of satellite motherboards. In the
process, a set of PDEs are encoded into a loss function, making full use of
the prior physics knowledge. Compared with the data-driven only strategy, the
physics-informed method exhibits better performance in training sets with
different sample sizes and obtains solutions more consistent with the laws of
physics.

Despite the MTA-UNet neural network we constructed for the regression problems
of multiple physical quantities, further research is still required. As our
model structure is strongly general, it is worth exploring strategies to
enhance the balance between multiple tasks further. Extension to more complex
boundaries and more complicated nonlinear problems is also valuable. Besides,
embedding physics knowledge into the network structure remains a challenging
work.

## Conflict of interest statement

On behalf of all authors, the corresponding author states that there is no
conflict of interest.

## Acknowledgments

This work was supported by the National Natural Science Foundation of China
(Nos.11725211 and 52005505).

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# Strong nonlinear optical response and transient symmetry switching in Type-
II Weyl semimetal $\beta$-WP2

T. C. Hu  International Center for Quantum Materials, School of Physics,
Peking University, Beijing 100871, China  B. Su  Beijing National Laboratory
for Condensed Matter Physics, Institute of Physics, Chinese Academy of
Sciences, 100190 Beijing, China  L. Y. Shi  International Center for Quantum
Materials, School of Physics, Peking University, Beijing 100871, China  Z. X.
Wang  International Center for Quantum Materials, School of Physics, Peking
University, Beijing 100871, China  L. Yue  International Center for Quantum
Materials, School of Physics, Peking University, Beijing 100871, China  S. X.
Xu  International Center for Quantum Materials, School of Physics, Peking
University, Beijing 100871, China  S. J. Zhang  International Center for
Quantum Materials, School of Physics, Peking University, Beijing 100871, China
Q. M. Liu  International Center for Quantum Materials, School of Physics,
Peking University, Beijing 100871, China  Q. Wu  International Center for
Quantum Materials, School of Physics, Peking University, Beijing 100871, China
R. S. Li  International Center for Quantum Materials, School of Physics,
Peking University, Beijing 100871, China  X. Y. Zhou  International Center for
Quantum Materials, School of Physics, Peking University, Beijing 100871, China
J. Y. Yuan  International Center for Quantum Materials, School of Physics,
Peking University, Beijing 100871, China  D. Wu  Beijing Academy of Quantum
Information Sciences, Beijing 100913, China  Z. G. Chen  Beijing National
Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy
of Sciences, 100190 Beijing, China  Songshan Lake Materials Laboratory,
Dongguan, Guangdong 523808, China  T. Dong  International Center for Quantum
Materials, School of Physics, Peking University, Beijing 100871, China  N. L.
Wang<EMAIL_ADDRESS>International Center for Quantum Materials, School of
Physics, Peking University, Beijing 100871, China  Beijing Academy of Quantum
Information Sciences, Beijing 100913, China

###### Abstract

The topological Weyl semimetals with peculiar band structure exhibit novel
nonlinear optical enhancement phenomena even for light at optical wavelengths.
While many intriguing nonlinear optical effects were constantly uncovered in
type-I semimetals, few experimental works focused on basic nonlinear optical
properties in type-II Weyl semimetals. Here we perform a fundamental static
and time-resolved second harmonic generation (SHG) on the three dimensional
Type-II Weyl semimetal candidate $\beta$-WP2. Although $\beta$-WP2 exhibits
extremely high conductivity and an extraordinarily large mean free path, the
second harmonic generation is unscreened by conduction electrons, we observed
rather strong SHG response compared to non-topological polar metals and
archetypal ferroelectric insulators. Additionally, our time-resolved SHG
experiment traces ultrafast symmetry switching and reveals that polar metal
$\beta$-WP2 tends to form inversion symmetric metastable state after photo-
excitation. Intense femtosecond laser pulse could optically drive symmetry
switching and tune nonlinear optical response on ultrafast timescales although
the interlayer coupling of $\beta$-WP2 is very strong. Our work is
illuminating for the polar metal nonlinear optics and potential ultrafast
topological optoelectronic applications.

###### pacs:

Valid PACS appear here

##  I INTRODUCTION

As topological quantum materials, Weyl semimetals (WSMs) exhibit a unique
linear band dispersion in close vicinity of the Fermi energy described by the
Weyl equationWeyl (1929). Pairing Weyl nodes with left and right chirality
occur in the crossing of conduction and valence bandsXu  _et al._ (2016a); Yao
_et al._ (2019). These nodes act as monopoles of Berry curvature in momentum
space. The emergent low-energy quasiparticle excitations in WSMs are regarded
as massless fermionsXu  _et al._ (2015); Lv  _et al._ (2015). WSMs with the
quasiparticles require the breaking of at least space inversion or time-
reversal symmetryWeng  _et al._ (2015); Xu  _et al._ (2020). Compare to type-I
WSMs, the Lorentz invariance is violated and the Weyl cone is tilted in the
momentum space in type-II WSMsSoluyanov  _et al._ (2015); Armitage  _et al._
(2018). The novel topological band structure can be manifested in intriguing
electronic and optical properties in condensed-matter physicsAli  _et al._
(2014); Kumar  _et al._ (2017); Ma  _et al._ (2019).

Recently, WSMs have aroused tremendous interests for their spectacular
nonlinear optical properties and future potential technological
applicationsChi  _et al._ (2018); Liu  _et al._ (2020); Zhang  _et al._
(2020). Some nonlinear optical phenomena such as the giant second harmonic
generation, colossal mid-infrared bulk photovoltaic effect and chiral
Terahertz wave emission has been observed in type-I WSM TaAsWu  _et al._
(2017); Osterhoudt  _et al._ (2019); Sirica  _et al._ (2019); Gao  _et al._
(2020). While it was proposed that type-II Weyl fermions could emerge from
nonmagnetic layered compound Td phase WTe2/MoTe2Soluyanov  _et al._ (2015);
Sun  _et al._ (2015) and three dimensional compound $\beta$-WP2/MoP2Autès  _et
al._ (2016). Minority works mainly focus on their primary nonlinear optical
propertiesDrueke  _et al._ (2021); Lv  _et al._ (2021). SHG is a fundamental
nonlinear optical process and a sensitive probe for tracing crystal symmetry
switchingHsieh  _et al._ (2011); Zhao  _et al._ (2016); Harter  _et al._
(2017). Electric-dipole (ED) allowed SHG response is expected to be dominant
in the bulk crystal with lacking inversion symmetry structure and the SHG
intensity reflects the magnitude of second-order susceptibility $\chi^{(2)}$,
defined by the relation
P${}^{(2)}_{i}$=$\epsilon_{0}\chi_{ijk}E{{}_{j}}E{{}_{k}}$ Fiebig  _et al._
(2005). The magnitude of $\chi^{(2)}$ is significant in nonlinear optics and
close connected to some other nonlinear optical processes such as optical
rectification and photocurrents generating.

Figure 1: Sample structure and characterizations for $\beta$-WP2. (a) Crystal
structure of $\beta$-WP2 (b) Schematic of the reflection static/time-resolved
SHG spectroscopy geometry, focus lens are omitted. Polarization SHG scans (c)
for the parallel-polarized (P-Out) and (d) for the perpendicular-polarized SH
photons (S-Out) relative to the scattering plane while the incidence wired-
polarizer is rotated. The solid curves are theory fit results to C2v point
group symmetry. The polarization of incidence is kept 45° and the polarization
of outgoing is kept parallel as depicted in (c) with 45° incidence for fluence
dependent TR-SHG measurement.

In this work, we report an experimental study of static and time-resolved SHG
in the three dimensional type-II WSM candidate bulk $\beta$-WP2 single
crystal. In static SHG measurement, we verify the C2v point group of
$\beta$-WP2 and estimate the size of the second-order susceptibility
$\chi^{(2)}$. Although $\beta$-WP2 exhibits extremely good metallic properties
with large mean free path, we observe rather strong SHG response at near-
infrared 800 nm wavelength excitation and the largest nonlinear susceptibility
$|\chi_{xxz}^{(2)}|$ of WP2 is as the same level of the widely-used electro-
optics crystal GaAsBergfeld and Daum (2003) and ZnTeWagner  _et al._ (1998).
Additionally, we find that light can non-thermally drive novel transient
symmetry switching on hundreds of picoseconds timescales. The maximum SHG
response decreases at about 70$\%$ with 4 mJ/cm2 laser pump fluence. It
indicates that $\beta$-WP2 has tendency to switch to the inversion symmetric
(topological trivial) metastable state and the topological nonlinear optical
properties could be tuned by light. The transient large SHG response comes
from the lattice distortion which is tied to topology. The ultrafast switching
is accompanied by a distinct softening of the A1 phonon. This behaviour is
generally related to the structural instability. In previous work, light-
driven interlayer shear displacements plays an essential role in two
dimensional layered type-II WSMs Td-XTe2 (X=W,Mo) ultrafast symmetry
switching. They could be induced a structural phase transition from Td to 1T'
phase and completely to the topological trivial metastable state by lightSie
_et al._ (2019); Zhang  _et al._ (2019). For three dimensional compound
$\beta$-WP2, since the strength of interlayer coupling is very strong and
robust, ultrafast bond softening mechanism driving by light or other mechanism
should be considered. The results have implications for optical control of the
symmetry and topological properties of the transition metal phosphides and
related materials.

Figure 2: Benchmark static SHG measurements. Benchmark 11° incident 800 nm SHG
experiments on (a) (110) ZnTe, (b) (111) intrinsic insulating GaAs wafer
($\rho$ $>$1$\times$107 $\Omega\cdot$cm) and (c) (010) $\beta$-WP2, the signal
is normalized to the maximum SHG intensity of ZnTe.

##  II RESULTS AND DISCUSSION

$\beta$-WP2 is a three dimensional compound and typically needle-shaped with
shiny luster (Optical micrograph of $\beta$-WP2 is shown in Fig.S1.). Previous
Raman experimental results indirectly indicate that $\beta$-WP2 crystallizes
into a non-centrosymmetric orthorhombic Cmc21 (No.36) space groupSu  _et al._
(2019); Wulferding  _et al._ (2020); Osterhoudt  _et al._ (2021). A simple
illustration of the crystal structure is shown in Figure 1(a). It contains a
twofold screw axis along the crystalline z-axis, a glide plane perpendicular
to the y-axis, and a mirror plane perpendicular to the x-axis. Compare to the
first predicted type-II WSMs candidate layered Td-WTe2/MoTe2, the interlayer
coupling of $\beta$-WP2 are much stronger while crystal point group is the
same as C2v. No structural or other phase transition has been observed in
$\beta$-WP2 at 4-300 K range yetWang  _et al._ (2017); Wulferding  _et al._
(2020); Su  _et al._ (2019).

Since $\beta$-WP2 is a nonmagnetic compound, inversion symmetry breaking is
prerequisite for topological nontrivial band structure. It is expected to
produce fundamental SHG response from ED contribution. Our model reflecting
staic/time-resolved SHG optical measurement setup is depicted in Figure 1(b).
In our setup, the WP2 sample is kept static while the polarization of
incidence or outgoing beam is rotated at least. The long x-axis of WP2 xz
plane is attached on the sample holder along the parallel direction. The
electric field polarizations of the obliquely incident fundamental beam (in)
and outgoing SHG beam (out) can be independently selected to lie either
parallel or perpendicular to the crystal high-symmetry axis. The SHG response
data are acquired by measuring the parallel(P-out) and perpendicular (S-out)
component of SHG intensity reflected from the (010) surface of WP2 as a
function of the polarization at 45° incidence beam. In crystals with C2v point
group symmetry, five independent non-vanishing elements of $\chi^{(2)}$, there
are $\chi^{(2)}_{xxz}$=$\chi^{(2)}_{xzx}$,
$\chi^{(2)}_{yyz}$=$\chi^{(2)}_{yzy}$, $\chi^{(2)}_{zxx}$, $\chi^{(2)}_{zyy}$,
$\chi^{(2)}_{zzz}$. Hence rotating oblique 45° xz plane fundamental incidence
polarization could simultaneously detect ED-SHG responses from all the
symmetry-allowed tensor elements. The fit lobes for P-out and S-out
configurations are
$I_{P-out}(\theta)$$\propto$$\frac{1}{4}$${|\chi_{xxz}+\chi_{yyz}|}^{2}$$sin^{2}$$(2\theta)$,
$I_{S-out}(\theta)$
$\propto$${|\frac{1}{2}(\chi_{zxx}+\chi_{zyy})cos^{2}(\theta)+\chi_{zzz}sin^{2}(\theta)|}^{2}$,
respectively, where the polarization of incidence rotates at an angle
$\theta$. The results of the two scan patterns are shown in Figure 1(c) and
1(d). All the solid fit curves for SHG responses derived from the C2v point
group. Detailed formula derivation procedure for SHG patterns can be found in
supplementary information. Our static rotation SHG results offer a very direct
and explicit evidence that the point group symmetry of the $\beta$-WP2 is C2v.

The intensity of ED-SHG response reflect the magnitude of second-order
nonlinear susceptibility. And the order of magnitude of $\chi^{(2)}$ is
significative and generally related to some other nonlinear optical processes
such as optical rectification, difference frequency generation and generating
photocurrents. Our optical setup and measuring scheme for the estimation of
$\chi^{(2)}$ refers to previous work in TaAsWu  _et al._ (2017). Since it is
liable to distinguish each tensor component $\chi^{(2)}$ contribution, the
angle of incidence is set as near-normalized (11°). We used insulating (111)
GaAs wafer ($\rho>$1$\times$107 $\Omega\cdot$cm) and 1mm-thick (110) ZnTe
archetypal electro-optic crystals as benchmarks. Both benchmark crystals have
large and well-characterized SHG responses, only one non-vanishing tensor
$\chi_{xyz}^{(2)}$ at fundamental 800 nm wavelength ($\hbar\omega$=1.55 eV)
for their high symmetry point group (Td). The SHG probe fluence is kept with
0.30 mJ/cm2 for the measurements. Surprisingly, the SHG response of
$\beta$-WP2 is rather strong that unscreened by a number of conduction
electrons. As seen in the Fig.2(a), 2(b) and 2(c), the signal intensity is
normalized to the peak SHG intensity in ZnTe. The peak intensity formulas are
derived in Supporting information. In previous literature, the experimental
refractive index of bulk GaAs, ZnTe and $\beta$-WP2 can be found. Taking
$|\chi_{xyz}^{(2)}|$ value of ZnTe as reference, by Bloembergan and Pershan
expressionBloembergen and Pershan (1962), we could calculate the maximum of
$\beta$-WP2 is about 150 pV/m. The order of the estimated $|\chi_{xxz}^{(2)}|$
is approximate to the theoretical calculation peak value of $|\chi^{(2)}|$ for
representative ferroelectric insulators BiFeO3 ($|\chi_{zzz}^{(2)}|$= 260
pV/m, fundamental wavelength 500 nm) and BaTiO3 ($|\chi_{zzz}^{(2)}|$= 200
pV/m, fundamental wavelength 170 nm)Ju  _et al._ (2009); Young and Rappe
(2012), even one or two order larger than LiNbO3, LiTaO3 and topological
trivial polar metal LiOsO3 at 800 nm fundamental wavelength Padmanabhan  _et
al._ (2018).

Figure 3: Time-resolved SHG responses of $\beta$-WP2. (a) Pump-induced SHG
time traces of WP2 at various pump fluences. (b) SHG intensity changes at
increasing fluences of 1.55 eV photon energy. (c) SHG time traces of WP2 with
2.5 mJ/cm2 fluence at different temperature points. Inset: The SHG intensity
change with 1.5mJ/cm2 for much longer timescales. (d) The nearly isotropic SHG
reduction response in pump polarization with 2 mJ/cm2 fluence.

The magnitude of ED-SHG response depends on several possible factors.
Generally, it is recognized that the intensity of SHG is close-linked to the
degree of the crystal polarization. So insulating ferroelectric is more ideal
for strong ED-SHG response in a sense. Appearance of conduction electrons in a
crystal can screen and weaken the polarization and SHG responsesAnderson and
Blount (1965). Since $\beta$-WP2 exhibits extremely high conductivity and an
extraordinarily large mean free path, it is unexpected to observe strong SHG
signal. Meanwhile, strong SHG response was hardly observed in topological
trivial polar metal such as LiOsO3Padmanabhan  _et al._ (2018). Obviously, the
conventional displacive ferroelectric polarization perspective inadequately
illuminates the strong SHG response from topological polar metal. The strong
SHG response may arise from the peculiar band topology contributions in
theoretical model proposed by Morimoto and Nagaosa. This theory deduced
concise expression with explicit geometrical meaning for $\chi^{(2)}$ and its
possible relation to the existence of Weyl nodes. However, the energy of
fundamental excitation at optical wavelength is away from the low energies
Weyl physics. Recently, resonance-enhanced effect from other bands was
proposed and consist with their first-principle calculationPatankar  _et al._
(2018); Lu  _et al._ (2022). Previous infrared spectroscopy studies indicate
much higher carrier densities in $\beta$-WP2Xu  _et al._ (2016b); Su  _et al._
(2019), thus it is reasonable to expect that the SHG response is weakened in
view of carriers screening effect in ferroelectric polarization perspective
when compared with type-I WSM TaAs. The origin of strong SHG response in polar
WSMs is still controversial, topological band contributions, band resonance-
enhanced and carriers screening effect are all possible influences for the SHG
responseWu  _et al._ (2017); Patankar  _et al._ (2018); Li  _et al._ (2018);
Lu  _et al._ (2022). In $\beta$-WP2, the weyl nodes are located hundreds of
meV below fermi energyAutès  _et al._ (2016); Yao  _et al._ (2019). The
relative large energy scale provides a better platform for studying the strong
topological nonlinear optical response. Since the value of $\chi^{(2)}$ is
dependent on the frequency, more theoretical and calculation work should be
stimulated for the attractive strong topological nonlinear optical responses
in type-I and type-II polar WSMs based on band structure and wavefunctions.

Figure 4: Snapshots of transient SHG patterns. With a linearly polarized pump
excitation, the SHG patterns for S/P-Out configurations with 45° probe
incidence for various delay time and pump fluence. Fits of the pattern
assuming the crystal point symmetries of C2v are shown as solid.

Symmetry plays a central role in conventional and topological phases of
matter, intense femtosecond laser pulse can optically drive transient symmetry
switching and make novel metastable or hidden states in some quantum
materialsStojchevska  _et al._ (2014); Wang  _et al._ (2021); Bao  _et al._
(2022); Sirica  _et al._ (2022). Electric dipole allowed SHG arises from a
non-zero second-order susceptibility $\chi^{(2)}$, as shown in non-
centrosymmetric topological systemsMorimoto and Nagaosa (2016); Wu  _et al._
(2017); Drueke  _et al._ (2021). Thus, time-resolved SHG measurement is a very
sensitive technique for monitoring ultrafast transient symmetry switching
processes in these topological materials. Our optical setup for tr-SHG
measurements is shown in 1(b). The pump pulse is normal to the xz plane, the
polarization of incidence and reflecting probe pulse is set to make the static
SHG response at maximum value, with 45° incidence P-out configuration. (The
polarization direction of incidence is depicted in Fig.1(c).)

After the femtosecond pump pulse arrives ($\Delta$t$\approx$1.5 ps), the SHG
intensity decreases saliently. Figure.3(a) shows the measured SHG time traces
from WP2 at various pump fluences. At rather low fluence excitation (about 0.5
mJ/cm2, red curve), the time dependent SHG intensity shows remarkable
reduction. With 4.0 mJ/cm2 pump fluence, the SHG intensity plummets
drastically, about 70$\%$ reduction. Continue to increase pump fluence until
the damage threshold, the value of $\Delta$Imax/I keeps almost unchanged. The
much longer timescales intensity variation is shown in inset of Fig.3(c). The
metastable state persists with more than one hundred picoseconds and slowly
relax to the initial state before the next pump pulse arrives. Figure.3(c)
shows the temperature-dependent relative change of the SHG intensity at fixed
pump fluence about 2.5 mJ/cm2 as a function of the delay time. Rather slight
maximum SHG intensity reduction difference of temperature variations in
Fig.3(c) and no high temperature structural transition was observed,
manifesting non-thermal symmetry switching process. Varying the linear
polarization of the excitation pulse in the xz plane relative to the x-axis as
shown in Fig.3(d), no distinct amplitude change could be found in
$\Delta$Imax/I. The results suggest the tr-SHG responses are almost isotropic
in the pump polarization. Compare to the previous work in layered Td-WTe2,
light with sufficient electric field strength can drive it to a completely
topological trivial state, indicating by almost no SHG responseSie  _et al._
(2019). Firstly, one possible reason is that the Gaussian-like pump beam
profile is nonuniform, evidently much larger intensity in the center of the
beam than at the edge region. Secondly, the bonding state in WP2 is very
strong rather than weak van del Waals coupling. For this reason, it is very
tough to drive the atoms move to the totally centrosymmetric positions with
SHG response decreasing to almost undetected. In general, the lattice symmetry
is dominant in ED-SHG response that it is often taken as a sensitive tool to
detect the structural symmetry change of a compound such as GaAsSaeta  _et
al._ (1991). It is appropriate to take the structural change into account for
the transient symmetry switching.

Figure 5: The fluence-dependent coherent vibrational dynamics. (a) The
transient pump-induced reflectivity at 800 nm with 0.1-4 mJ/cm2 fluence. (b)
Fourier transformation of the oscillation after subtracting the background.
(c) The pump fluence dependence of A1 phonon mode frequency.

Following near-infrared (1.55 eV) light excitation, figure.4(a) and 4(b) show
pronounced SHG signal changes in the two polarization scan lobes before/after
pump pulse arriving. In the P/S-out SHG scan lobes, the SHG response is still
consistent with the equilibrium C2v point group symmetry fit at 10 ps, but
with much less SHG responses. The maxmium SHG tensor element $|\chi_{xxz}|$
shows a reduction of nearly 80$\%$. Considering the pump electric field
polarization and perturbations are in the xz-plane, the results suggest a
connection between the polar axis and the strong optical nonlinearity of the
$\beta$-WP2. The snapshots of SHG polarization patterns indicate the specific
reversible ultrafast symmetry switching and a strong tendency to the inversion
symmetric state. Thus, intense laser pulse could manipulate the nonlinear
optical properties drastically in WP2 in an ultrafast fashion.

Due to rather weak van der Waals bonding, interlayer shear displacements could
be driven by intense light in Td-XTe2(W,Mo). The intensity of low-frequency
shear mode phonon would recede in time-resolved reflectivity experiment
accompanied by the light-induced structural transition. However, previous
Raman results of $\beta$-WP2 point out that no such a low-frequency shear
motion phonon existed. In order to get further insight into the latent
switching mechanism, we also carry out the fluence-dependent ultrafast optical
pump-probe reflectivity experiment to investigate coherent phonon
oscillations. The data under different pump fluences are shown in Fig.5(a),
all the decay processes can be described by a double-exponential function to
fit the decay process: $\Delta
R/R=A_{1}\exp\left(-t/\tau_{1}\right)+A_{2}\exp\left(-t/\tau_{2}\right)+C$,
Where $A_{1}$, $A_{2}$ in the formula represents the amplitude of the photo-
induced reflectivity change, $\tau_{1}$, $\tau_{2}$ stands for the two fast
relaxation time and $C$ is constant for long lived thermal diffusion process.
After subtracting the background decay processes, only one A1 phonon mode
could be extracted from time domain coherent oscillation by Fourier transform.
This A1 mode phonon frequency is about 5.4 THz ($\sim$180 cm-1) in Fig.5(b)
under low 130 $\mu$J/cm2 pump excitation, which is in accord with the previous
static Raman scattering measurementsWulferding  _et al._ (2020). The
dependence of the phonon frequency on the pump fluence is depicted in
Fig.5(c). With the pump fluence further increasing, the A1 phonon frequency in
Fig.5(c) shows distinct continuous red-shifted from 5.4 THz to 5.1 THz than
equilibrium state ($\sim$5$\%$ frequency red-shifted), indicating a phonon
mode softened behavior. Similar phonon softening behavior has been observed in
previous fluence-dependent transient optical reflectivity experiments of
tellurium(Te), bismuth(Bi) and tin selenide(SnSe) crystalsHunsche  _et al._
(1995); Hase  _et al._ (2002); Huang  _et al._ (2022). This behavior could be
regarded as an extra evidence for the structural distortionFritz  _et al._
(2007).

In general, valence electrons and ionic cores can be regarded as distinct
components that couple through the electron-phonon interaction. Previous
hydrodynamic transport and Raman spectroscopy results indicated that the
electron-phonon interaction was strong in $\beta$-WP2Coulter  _et al._ (2018);
Osterhoudt  _et al._ (2021), similar to tellurium (Te) and antimony (Sb).
Electronic ultrafast bond softening mechanism is usually taken into
consideration for lattice instability in these materials. The lattice
structure is very sensitive to the population distribution of electrons within
the conduction bands. The A1 coherent optical phonon mode at symmetric zone
center could be driven by light-induced charge carriers and the vibrational
excitation is generally believed to be displaciveZeiger  _et al._ (1992). The
population of valence electrons are redistributed after photo-excitation so
that makes the bond softening through electron-phonon coupling. With high
excitation fluence, the interatomic forces that bind solids and the atoms
positions can be altered compare to equilibrium state. The observed phonon
softening is consistent with behavior related to the carriers-induced
ultrafast bond softening Fritz  _et al._ (2007). Perturbation by light makes
structural instability in the xz plane connected to the polar axis, resulting
SHG and dominant $\chi_{zxx}$ reduction as the previous high-pressure strain
manipulation effect in TaAs SHG response and dominant $\chi_{zzz}$Li  _et al._
(2022). However, since time-dependent optical reflectivity monitors the A1
vibration mode indirectly and can not provide messages about atomic positions,
detailed femtosecond x-ray diffraction is required to elucidate the dynamics
of the high-amplitude phonon and to proof this mechanism. For topological
semimetals, where structural symmetry is intimately tied to topology, other
possibility is that the band structure transformation is also induced by
light, resulting SHG response change. To determine the origin of the specific
ultrafast symmetry switching, further time-resolved angle-resolved
photoemission spectroscopy (tr-ARPES) is also indispensable. Since very strong
strength of interlayer coupling in WP2, it is certain that the essence of
driving mechanism is entirely different with the shear displacements in
layered Td-XTe2 (X= W,Mo).

##  III CONCLUSION

In summary, we performed combined static and time-resolved SHG measurements in
Type-II Weyl semimetal $\beta$-WP2. As a topological polar metal with
extremely high conductivity, WP2 exhibits rather strong SHG responses that
maximum $|\chi_{xxz}|$ is about 150 pV/m, which is one or two order larger
than ferroelectric LiNbO3, LiTaO3 and polar metal LiOsO3 with 1.55 eV photon
excitation. The static SHG results enrich the visions of intriguing polar
metal nonlinear optics and enable potentially practical applications for
topological optoelectronic devices. After photo-excitation, intense laser
pulse could optically drive novel symmetry switching and tune nonlinear
optical responses on ultrafast timescales. The results indicate that
$\beta$-WP2 have strong tendency to the topological trivial state. The
ultrafast switching is accompanied by a distinct softening of the lowest-
frequency A1 phonon. The essence of driving mechanism is entirely different
with the Td phase WTe2/MoTe2 light-induced shear displacements. Our study
provides explicit evidence of ultrafast symmetry switching, which opens up
experimental possibilities for ultrafast manipulation of the topological
nonlinear optics properties of solids. This ultrafast symmetry switching will
have wide ranging applications, particularly for topological semimetals, where
symmetry is intimately tied to topology. Further time-resolved x-ray
diffraction and time-resolved angle-resolved photoelectron spectroscopy is
pursued for the transient lattice and topological electronic structure
switching in future.

Methods

Crystal growth and structure characterization.

Single crystals of $\beta$-WP2 were grown via chemical vapor transport with
iodine as the transport agent. All capsulation processes were operated in a
sealed glove box. Detailed growth method could be found in referenceSu  _et
al._ (2019). The as-grown three-dimensional crystals with needle-shaped
exhibit multiple shiny surface facets. $\beta$-WP2 is air stable. Crystal
structure and crystal orientation were confirmed using single crystal X-ray
diffractometer (SXRD) Bruker D8 Venture with Mo-Kα radiation ($\lambda$= 0.71
$\AA$) at room temperature. The determined lattice parameters were consist
with the previous report in the literature.

Optics set-up for reflecting static/time-resolved second-harmonic generation
and pump-probed reflectively measurement

The optics set-up for measuring reflecting static/time-resolved SHG is
depicted in the Fig.1b. The measurements were performed in a Ti:sapphire
amplified laser (Spitfire Ace) system of 800-nm centre-wavelength pulses for a
duration of 35 fs (1 kHz repetition rate). Generator pulses pass through a
mechanical chopper that provides amplitude modulation at 377 Hz. And the
pulses are focused at 11°/45° incidence onto the sample. The spot sizes of
pump and probe are measured about 150,120 $\mu$m in 45° incidence
configuration by a 100-$\mu$m diameter pinhole, respectively. The pump is in
normal incidence relative to xz plane with a half waveplate to vary linear
polarization. The probe fluence of the time-resolved SHG is set as 0.30
mJ/cm2. A Quarter waveplate and a rotatable wired-polarizer in the incidence
beam path are used to generate circular polarization and to vary the direction
of linear polarization. Both the outgoing fundamental and the second-harmonic
beams are directed to a short-pass, band-pass filter group that ensures only
the second-harmonic light to reach the photomultiplier tube photodetector
(PMM01, Thorlabs, Bias voltage set as 0.65V). Another rotatable wire-grid
polarizer placed before the PMT allows for analysis of the polarization of the
second-harmonic beam. Temperature-dependence measurements were performed by
mounting the WP2 sample on a holder of helium-free closed-cycle cryostat
(Montana Instruments). Benchmark measurements on $\beta$-WP2, ZnTe and GaAs
were performed in ambient conditions and the samples mounted flat on a xyz-
micrometer stage to maximize the signal at room temperature.

For fundamental time-dependent reflectivity measurement, the standard pump-
probe scheme can be found elsewhereHu  _et al._ (2022).

ACKNOWLEDGMENTS

This work was supported by the National Natural Science Foundation of China
(No. 11888101), the National Key Research and Development Program of China
(No. 2017YFA0302904).

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