# Surface profile recovery from electromagnetic field with physics–informed
neural networks

Yuxuan Chen, Ce Wang, Yuan Hui, and Mark Spivack The paper was submitted on DD
MM YYYY.Y. Chen is with School of Mathematical Sciences, Soochow University,
Suzhou, China (e-mail: chenyx@suda.edu.cn).C. Wang is with Department of
Electronic and Computer Engineering, The Hong Kong University of Science and
Technology, Hong Kong, China (e-mail: fogever@gmail.com).Y. Hui is with
Institute of Intelligent Computing Technology, Chinese Academy of Sciences in
Suzhou Branch, Suzhou, China (e-mail: huiyuan.ict@gmail.com).M. Spivack is
with Department of Applied Mathematics and Theoretical Physics, University of
Cambridge, Cambridge, UK (e-mail: ms100@cam.ac.uk).Y. Chen, C. Wang and H.
Yuan contributed equally to this work.Corresponding author: Yuxuan Chen

###### Abstract

Physics–informed neural networks (PINN) have shown their potential in solving
both direct and inverse problems of partial differential equations. In this
paper, we introduce a PINN-based deep learning approach to reconstruct one-
dimensional rough surfaces from field data illuminated by an electromagnetic
incident wave. In the proposed algorithm, the rough surface is approximated by
a neural network, with which the spatial derivatives of surface function can
be obtained via automatic differentiation and then the scattered field can be
calculated via the method of moments. The neural network is trained by
minimizing the loss between the calculated and the observed field data.
Furthermore, the proposed method is an unsupervised approach, independent of
any surface data, rather only the field data is used. Both TE field (Dirichlet
boundary condition) and TM field (Neumann boundary condition) are considered.
Two types of field data are used here: full scattered field data and phaseless
total field data. The performance of the method is verified by testing with
Gaussian-correlated random rough surfaces. Numerical results demonstrate that
the PINN-based method can recover rough surfaces with great accuracy and is
robust with respect to a wide range of problem regimes.

###### Index Terms:

Deep Learning, Electromagnetic scattering, Inverse Scattering, Method of
Moments, Physics–informed Neural network, Rough Surface Reconstruction

##  I Introduction

Wave propagation from rough surfaces is of crucial importance in the field of
electromagnetic scattering and has been extensively studied. The inverse
problem of rough surface profile recovery from scattered electromagnetic field
remains a challenging problem with a wide range of applications such as
non–destructive testing, geophysics radar, remote sensing, biomedical imaging
and so many others. Much effort has therefore been devoted to this field, for
example [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19,
20, 21]. Common methods include the Rytov–type methods [1], the Newton–type
methods [6, 7, 8, 9], reconstruction via sampling (either linear sampling or
other type) [10, 11, 12], and iterative methods with _ad hoc_ guess [19, 18,
20]. These traditional methods are generally based on physical equations of
surface scattering, and their robustness is contingent upon certain conditions
on surface and incident wave (e.g. the convergence of iterative method in [19]
is based on low grazing angle approximation).

Recently, deep learning type methods have proven to be powerful approaches
dealing with a variety of problems, e.g. image processing, natural language
processing, and scientific computing. Studies have been carried out to recover
physical scatters from the observed field data using a large number of data,
see e.g. [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32]. In these works, neural
networks are used to learn the mapping from the observed electromagnetic field
to the profile of target objects. In particular, in [31, 32], a convolutional
neural network (CNN) based deep learning method has been developed to
reconstruct rough surfaces from the scattered data. It has been shown that
deep learning methods are capable of solving the inverse scattering problem in
a more generalized way. On the other hand, these deep learning methods are
highly dependent on ‘solution’ data (ground truth). In the case of rough
surface reconstruction, it is difficult to obtain the entire profile of rough
surfaces and corresponding scattered data _a priori_ , especially when a large
number of data are required. Even if one can create data using simulation, the
generated geometric formulations may not be able to adequately capture the
features of real-world rough surfaces, resulting in a significant discrepancy
in the data’s representation, which makes the trained model hard to employ in
practice.

Aiming to inherit the efficient learning strategy of data-driven methods while
maintaining the physics of rough surface scattering, it is worthwhile to
develop a unified method. This motivates us to employ the physics–informed
neural networks (PINN) [33, 34]. PINN is a particular type of deep learning
approach developed to solve direct and inverse problems of partial
differential equations (PDEs). The main idea of PINN is to employ neural
networks as basis to approximate the ‘function’ (solution to PDE) and train
the neural network by minimizing the ‘residual’ obtained from the physical
information (PDEs), which can be realized using the automatic differentiation.
PINN is an _unsupervised_ learning method, which does not require any _a
priori_ knowledge of ‘solution’ data. PINN has now been widely applied to many
practical engineering problems; see for instance [35, 36, 37] for solving the
direct problem of electromagnetic scattering, and [38, 39, 40] for inverse
scattering problems.

In this paper, we develop a novel deep learning method that is based on PINN
to reconstruct rough surfaces in two-dimensional media given the information
of field data. The physical information (equation) used in the method is the
integral representation relating the incident field, the scattered field, and
the surface profile. Method of moments (MOM) is used to deal with integral
equations; it numerically calculates the scattered data from the surface and
incident field. A feed–forward neural network is served as basis to
approximate the surface height. The ‘predicted’ scattered field is then
calculated by inserting the predicted surface (in terms of neural network) and
its spatial derivatives (obtained by automatic differentiation) to MOM. The
neural network is trained by minimizing the loss between the predicted and
observed field data. This renders an unsupervised training scheme eliminating
the need for rough surface data, instead solving the problem using only the
field data. The problem is solved using two types of observed field data: full
scattered data (including both strength and phase) and phaseless total field
data. In both cases, knowledge of incident field is required. Both TE and TM
fields are considered. We apply the method to recover profiles of Gaussian-
correlated random rough surfaces, and test the method with a wide range of
problem settings including various noise levels, surfaces scales, surface
heights, wavenumbers and angles of incidence. A large number of numerical
results validate the effectiveness and robustness of the proposed approach.

The remainder of this paper is structured as follows. A brief review of
integral equations and method of moments is given in section II. In section
III, we introduce a PINN-based deep learning approach to reconstruct one-
dimensional rough surfaces from field data, including the detailed structure
of neural networks and loss functions. Numerical experiments are carried out
in section IV, in which the method is tested with a range of problem regimes.
Some concluding remarks and potential further directions are drawn in section
V.

##  II Background

Consider electromagnetic scattering from a perfectly electrically conducting
(PEC) 1D rough surface illuminated by an incident wave. The so–called _direct_
problem is to calculate the scattered field given the rough surface shape,
whilst, the _inverse_ problem is to recover the profile of rough surface from
a set of measured scattered data. It is well known that surface scattering can
be formulated by integral equations, which have been widely discussed [41,
42]. The method of moments (MOM) is one of the most common numerical
techniques to numerically solve the integral equations [43, 44]. In this
section, we give a brief review of integral equations and method of moments.

###  II-A Integral Equations

Let the coordinate axes be $x$ and $z$, where $x$ is the horizontal and $z$ is
the vertical. We consider the time–harmonic scalar wave field $\psi(x,z)$,
which is time independent ($e^{-i\omega t}$ time dependence is discarded). The
governing equation for wave scattering is the Helmholtz equation

$\nabla^{2}\psi(x,z)+k^{2}\psi(x,z)=0,$ (1)

where $k$ is the wavenumber, defined by $k=2\pi/\lambda$, in which $\lambda$
is the wavelength. Suppose that the 1D PEC rough surface, denoted by $z=h(x)$,
is at least second–order differentiable, i.e. $h(x)\in C^{2}(\mathbb{R})$. It
has a compact support on $\mathbb{R}$, and perturbs from zero only inside the
domain $x\in[-L,L]$. Let $V$ be the upper homogeneous medium in which the
points lie above the surface, i.e. $V=\\{(x,z)|z>h(x)\\}$, and $S$ be the set
of points lying on the surface with $S=\partial V=\\{(x,z)|z=h(x)\\}$. We
write ${\bf r},{\bf r^{\prime}}\in V$ as points in the medium, and ${\bf
r_{S}},{\bf r^{\prime}_{S}}\in S$ for the points along the surface. Denote the
incident and scattered wave fields as $\psi_{i}$ and $\psi_{s}$, respectively,
with $\psi=\psi_{i}+\psi_{s}$ being the total wave in the space.

The wave field in the upper free space $V$ can be formulated by the
Kirchoff–Helmholtz equation,

$\displaystyle\psi({\bf r})=\psi_{i}({\bf r})$ (2)
$\displaystyle+\int_{S}{\bf{n^{\prime}}}\cdot\left[\psi({\bf
r})\nabla^{\prime}G({\bf r};{\bf r^{\prime}})-G({\bf r};{\bf
r^{\prime}})\nabla^{\prime}\psi({\bf r^{\prime}})\right]ds^{\prime},$

where ${\bf{n^{\prime}}}$ is the unit normal vector to the surface, and
$G({\bf r};{\bf r^{\prime}})$ is the Green’s function to the free space
Helmholtz equation given by

$G({\bf r};{\bf r^{\prime}})=\frac{i}{4}H_{0}^{(1)}\left(k|{\bf r}-{\bf
r^{\prime}}|\right),$ (3)

where $H_{0}^{(1)}$ is the zeroth–order Hankel function of the first kind. The
Kirchoff–Helmholtz equation can be transformed into a pair of coupled
integrals by applying boundary conditions. If the Dirichlet boundary condition
is assumed, namely, $\psi=0$ on $S$, which corresponds to the transverse
electric (TE) field impinging on a PEC surface, then the coupled integrals
become:

$\psi_{i}({\bf r_{S}})=\int_{S}G({\bf r_{S}};{\bf
r^{\prime}_{S}})\frac{\partial\psi({\bf
r^{\prime}_{S}})}{\partial{\bf{n^{\prime}}}}ds^{\prime},$ (4)

and

$\psi_{s}({\bf r})=-\int_{S}G({\bf r};{\bf
r^{\prime}_{S}})\frac{\partial\psi({\bf
r^{\prime}_{S}})}{\partial{\bf{n^{\prime}}}}ds^{\prime},$ (5)

where
$\frac{\partial\psi}{\partial{\bf{n^{\prime}}}}:={\bf{n^{\prime}}}\cdot\nabla^{\prime}\psi$,
noting that the first equation is obtained via taking the limit as point ${\bf
r}$ approaches the surface. On the other hand, if we assume the Neumann
boundary condition, namely, $\bm{n}\cdot\nabla\psi=0$ on $S$, which
corresponds to the transverse magnetic (TM) field applied on the surface, then
the coupled integrals become:

$\psi_{i}({\bf r_{S}})=\frac{1}{2}\psi({\bf r_{S}})-\int_{S}\frac{\partial
G({\bf r_{S}};{\bf r^{\prime}_{S}})}{\partial{\bf{n^{\prime}}}}\psi({\bf
r^{\prime}_{S}})ds^{\prime},$ (6)

and

$\psi_{s}({\bf r})=\int_{S}\frac{\partial G({\bf r};{\bf
r^{\prime}_{S}})}{\partial{\bf{n^{\prime}}}}\psi({\bf
r^{\prime}_{S}})ds^{\prime},$ (7)

where $\frac{\partial
G}{\partial{\bf{n^{\prime}}}}:={\bf{n^{\prime}}}\cdot\nabla^{\prime}G$, noting
that the term $1/2$ in the first equation results from the singularity of the
Green’s function when limiting the point to the surface.

###  II-B Method of Moments

Method of moments (MOM) is used to solve the direct problem, namely, obtain
the scattered data. It recovers the surface currents (surface derivative
$\partial\psi/\partial\bm{n}$ for TE field and total wave along surface $\psi$
for TM field) from the first integral equation ((4) or (6)) and calculate the
scattered field from the second integral ((5) or (7)). We first discretize the
domain $[-L,L]$ uniformly into $N$ subintervals with $(N+1)$ points, and
denote them as $x_{l}$ with $l=0,1,\ldots,N$, where $x_{0}=-L$ and $x_{N}=L$.
The midpoint on each subinverval $[x_{l-1},x_{l}]$ is defined as
$X_{l}=1/2(x_{l-1}+x_{l})$ for $l=1,2,\ldots,N$. In a typical physical
situation, the scattered data in the inverse problem are usually measured
along a horizontal line in the medium $V$. We choose to measure the scattered
field along the line $z=\zeta$, where $\zeta$ is a constant with
$\zeta>h(x),\;x\in[-L,L]$. Denote the discrete points along the surface as
$\bm{r_{n}^{S}}=(X_{n},h(X_{n}))$ and points along measurement line as
$\bm{q_{n}}=(X_{n},\zeta)$.

Consider first the Dirichlet condition (TE field). To the leading order, the
surface differential on each subinterval $[x_{l-1},x_{l}]$ can be treated by a
line segment, namely,

$ds\approx dL=\sqrt{1+h^{\prime}(x)^{2}}dx.$ (8)

We denote ${\bf\Psi_{i}}$ and ${\bf\Psi_{s}}$ as the vectors of surface
incident field and target scattered field in $\mathbb{C}^{N}$ with
$\bm{\Psi_{i}}[l]=\psi_{i}\left(X_{l},h(X_{l})\right)$ and
$\bm{\Psi_{s}}[l]=\psi_{s}(X_{l},\zeta)$. Then, the discretized form of
integral equations for TE case ((4) and (5)) gives rise to a linear system
relating the surface incident field and scattered field:

$\bm{\Psi_{s}}=B_{D}A_{D}^{-1}\bm{\Psi_{i}},$ (9)

where $A_{D}\in\mathbb{C}^{N\times N}$ is given by

$A_{D}[n,l]=\int_{x_{l-1}}^{x_{l}}G({\bf{r_{n}^{S}}};{\bf
r^{\prime}_{S}})\sqrt{1+h^{\prime}(x^{\prime})^{2}}dx^{\prime},$ (10)

and $B_{D}\in\mathbb{C}^{N\times N}$ is given by

$B_{D}[n,l]=\int_{x_{l-1}}^{x_{l}}G(\bm{q_{n}};{\bf
r^{\prime}_{S}})\sqrt{1+h^{\prime}(x^{\prime})^{2}}dx^{\prime},$ (11)

for $n,l=1,\ldots,N$. The terms in $A_{D}$ and $B_{D}$ are approximated by the
trapezium rule. By inverting the matrix $A_{D}$, the surface derivative can be
obtained, and using the calculated surface derivative, the scattered field can
be measured by a matrix–vector multiplication.

Similar treatment is applied to the Neumann boundary condition (TM field).
Again, there is a linear system relating the incident wave and scattered wave
from the discretized form of the integral equations for TM case ((6) and (7)):

$\bm{\Psi_{s}}=B_{N}A_{N}^{-1}\bm{\Psi_{i}},$ (12)

where $A_{N}\in\mathbb{C}^{N\times N}$ is given by

$\displaystyle A_{N}[n,l]=$ (13)
$\displaystyle\begin{cases}\int_{x_{l-1}}^{x_{l}}\frac{\partial
G({\bf{r_{n}^{S}}};{\bf
r^{\prime}_{S}})}{\partial{\bf{n^{\prime}}}}\sqrt{1+h^{\prime}(x^{\prime})^{2}}dx^{\prime},\;\;n\neq
l\\\ \frac{1}{2}-\int_{x_{n-1}}^{x_{n}}\frac{\partial G({\bf{r_{n}^{S}}};{\bf
r^{\prime}_{S}})}{\partial{\bf{n^{\prime}}}}\sqrt{1+h^{\prime}(x^{\prime})^{2}}dx^{\prime},\;\;n=l\\\
\end{cases}$

and $B_{N}\in\mathbb{C}^{N\times N}$ is given by

$B_{N}[n,l]=\int_{x_{l-1}}^{x_{l}}\frac{\partial G(\bm{q_{n}};{\bf
r^{\prime}_{S}})}{\partial{\bf{n^{\prime}}}}\sqrt{1+h^{\prime}(x^{\prime})^{2}}dx^{\prime},$
(14)

for $n,l=1,\ldots,N$. The terms in $A_{N}$ and $B_{N}$ are calculated via the
trapezium method.

In general, the scattered field along the line $z=\zeta$ can be written in the
operator form of

$\bm{\psi_{s}}=\text{MOM}[h(x),h^{\prime}(x),h^{\prime\prime}(x)]:=B_{h}A_{h}^{-1}\bm{\psi_{i}}^{h}.$
(15)

These operators are all dependent on the surface profile $h$ and its
derivatives.

##  III Reconstruction with physics–informed neural network

In this section, a novel unsupervised learning approach is proposed to
reconstruct one–dimensional rough surfaces from the field data, which is based
on physics–informed neural networks (PINN). The neural network is utilized to
approximate the surface height and trained based on the observed field data.
The architecture of the proposed learning method is presented in fig. 1. To
illustrate the algorithm, we start with introducing some preliminary knowledge
(mesh, data, and boundary condition), followed by the neural network
structure, the physical information and loss function, and finally the
training scheme.

Figure 1: An illustrative figure for the PINN-based neural network to
reconstruct rough surfaces from field data.

###  III-A Preliminary

Sampling Points. Suppose that we want to recover the rough surface profile on
the domain $[-L,L]$. At iteration (epoch) $t$ of training, we first generate a
random integer $N_{t}$ with $N_{t}\in[N_{\rm obs},N_{\rm inv}]$ where $N_{\rm
obs}$ is the size of observation data, and $N_{\rm inv}$ is a larger number.
The domain $[-L,L]$ is discretized into $(N_{t}+1)$ equally spaced points with
$x_{j}^{t}=j\Delta x_{t}-L$ for $j=0,1,\cdots,N_{t}$ and $\Delta
x_{t}:=2L/N_{t}$. The sampling points at $t$-th iteration of training are the
midpoints $X_{j}^{t}$, for $j=1,\ldots,N_{t}$ with
$X_{j}^{t}=(x_{j-1}^{t}+x_{j}^{t})/2$.

Field Data. Suppose that the field data is collected along a horizontal line
$z=\zeta$ at observation points $(X_{j}^{\rm obs},\zeta)$ for
$j=1,2,\cdots,N_{\rm obs}$, where $N_{\rm obs}$ is the number of observation
points. Note that the observation points are not aligned with the sampling
points. Thus, a set of impulse (‘hat–like’) piecewise linear functions are
employed to interpolate the field data from the observation points
$(X_{j}^{\rm obs},\zeta)$ to the collocation points $(X_{j}^{t},\zeta)$. Two
types of field data are considered here. In the first case, we assume that
full field data can be detected, namely scattered field $\psi_{s}^{\rm
data}(X_{j}^{t},\zeta)$ is known. We also consider the case of phaseless field
data in which the amplitude of total field $\lvert\psi_{\rm tot}^{\rm
data}(X_{j}^{t},\zeta)\rvert$ is observed with $\lvert\psi_{\rm tot}^{\rm
data}\rvert=\lvert\psi_{s}^{\rm data}+\psi_{i}\rvert$.

Boundary Condition. The method also takes constraints on the boundary values.
We consider that the surface height at close-boundary points $h(X_{j}^{b})$
for $j=1,2,\ldots,N_{b}$ is known, where $X_{j}^{b}$ are close–boundary points
(they are either close to $L$ or $-L$). In the numerical examples of section
IV, a tapered rough surface is used so that surface height close to edges
decays to zero.

###  III-B Neural Network

A fully connected feed–forward deep neural network is employed as basis to
approximate the surface function. The neural network is of uniform structure
which has a fixed number of hidden layers and neurons per hidden layer. A
nonlinear activation function is applied on the outputs after each hidden
layer. The input and output are set to be one-dimensional to predict the
function of the surface profile, with the input as one spatial point and the
output being the surface height at that point. At iteration $t$ of training,
we apply the neural network to all the sampling points $X_{n}^{t}$ for
$n=1,\ldots,N_{t}$, leading to the output of surface heights on the entire
domain. It is noted that we normalize the input data to the range of $[-1,1]$,
and the output, also conceptualized within the $[-1,1]$ range, is translated
back to the target surface values through inverse normalization.

###  III-C Physical Information and Loss Function

At iteration $t$ of training, let $h^{{\rm NN}(t)}(x,\bm{\theta})$ be the
predicted surface function in terms of neural network where $\bm{\theta}$ is
the set of parameters inside the neural network including the weights and
biases. With the predicted surface height in terms of neural network, the
corresponding spatial derivatives of surface, $\partial h^{{\rm
NN}(t)}(x,\bm{\theta})/\partial x$ and $\partial^{2}h^{{\rm
NN}(t)}(x,\bm{\theta})/\partial x^{2}$, can be obtained by automatic
differentiation. Then, the scattered field along the horizontal line $z=\zeta$
is calculated from the predicted surface and its derivatives via the method of
moments (described in section II-B). This procedure can be viewed as combining
the neural network and physical equations of surface scattering. Denote the
calculated scattered field on the collocation points at $t$-th iteration of
training by $\psi_{s}^{{\rm NN}(t)}(X_{n}^{t},\zeta,\bm{\theta})$, which takes
the form of

$\psi_{s}^{{\rm NN}(t)}=\text{MOM}\left[h^{{\rm NN}(t)},\frac{\partial h^{{\rm
NN}(t)}}{\partial x},\frac{\partial^{2}h^{{\rm NN}(t)}}{\partial
x^{2}}\right].$ (16)

If full scattered data is known, the loss function at iteration $t$ is given
by a combination of two mean squared errors (MSE), MSE of scattered field
($\mathcal{L}_{\rm eqn}$) and MSE of boundary values ($\mathcal{L}_{\rm bc}$):

$\displaystyle\mathcal{L}(\bm{\theta},t):=\mathcal{L}_{\rm
eqn}+\mathcal{L}_{\rm bc}$ (17)
$\displaystyle:=\frac{1}{N_{t}}\sum\limits_{j=1}^{N_{t}}\lvert\psi_{s}^{{\rm
NN}(t)}(X_{j}^{t},\zeta,\bm{\theta})-\psi_{s}^{\rm
data}(X_{j}^{t},\zeta)\rvert^{2}$
$\displaystyle+\frac{1}{N_{b}}\sum\limits_{j=1}^{N_{b}}\lvert h^{{\rm
NN}(t)}(X_{j}^{b},\bm{\theta})-h(X_{j}^{b})\rvert^{2}.$

Note that the first MSE of scattered field is complex–valued, it can be
thought of a combination of MSE of both real and imaginary parts, namely

$\displaystyle\mathcal{L}_{\rm eqn}=$ (18)
$\displaystyle\frac{1}{N_{t}}\sum\limits_{j=1}^{N_{t}}\lvert\Re\\{\psi_{s}^{{\rm
NN}(t)}(X_{j}^{t},\zeta,\bm{\theta})\\}-\Re\\{\psi_{s}^{\rm
data}(X_{j}^{t},\zeta)\\}\rvert^{2}$ $\displaystyle+\lvert\Im\\{\psi_{s}^{{\rm
NN}(t)}(X_{j}^{t},\zeta,\bm{\theta})\\}-\Im\\{\psi_{s}^{\rm
data}(X_{j}^{t},\zeta)\\}\rvert^{2}.$

In the case of known phaseless total field data, the loss function at
iteration $t$ is composed of MSE of total field data ($\mathcal{L}_{\rm eqn}$)
and MSE of boundary values ($\mathcal{L}_{\rm bc}$):

$\displaystyle\mathcal{L}(\bm{\theta},t):=\mathcal{L}_{\rm
eqn}+\mathcal{L}_{\rm bc}$ (19)
$\displaystyle:=\frac{1}{N_{t}}\sum\limits_{j=1}^{N_{t}}\left\lvert|\psi_{\rm
tot}^{{\rm NN}(t)}(X_{j}^{t},\zeta,\bm{\theta})|-|\psi_{\rm tot}^{\rm
data}(X_{j}^{t},\zeta)|\right\rvert^{2}$
$\displaystyle+\frac{1}{N_{b}}\sum\limits_{j=1}^{N_{b}}\lvert h^{{\rm
NN}(t)}(X_{j}^{b},\bm{\theta})-h(X_{j}^{b})\rvert^{2},$

where $\psi_{\rm tot}^{{\rm NN}(t)}=\psi_{s}^{{\rm NN}(t)}+\psi_{i}$ is the
total field obtained by MOM using predicted surface at iteration $t$ of
training.

###  III-D Training

Finally, the loss function is minimized by an optimizer to achieve an
‘optimal’ set of parameters $\bm{\theta}$. In this paper, the optimizer is
taken to be the Adam (Adaptive Moment Estimation) [45]. During the
optimization, a learning rate is used to control how much the weights of
neural network are adjusted with respect to the gradient of the loss function.
After training, the neural network serves as an approximate function to the
rough surface, where inputting point $x$ yields the output $h(x)$, the
predicted surface height at that point.

##  IV Numerical examples

The performance of the proposed method is tested here. In all numerical
examples, the incident field is taken to be a plane wave with
$\psi_{i}(x,z)=\exp\left(ik(\cos\alpha x+\sin\alpha z)\right)$, where $k$ is
the wavenumber and $\alpha$ is the grazing angle (complementary to the angle
of incidence). The domain of rough surface is set as $[-L,L]$ with
$L=8\lambda$ where $\lambda$ is the wavelength of incident wave. We apply the
method to random Gaussian-correlated rough surfaces, which are generated by an
autocorrelation function (a.c.f) given by

$\rho(\eta)=\langle
h(x)h(x+\eta)\rangle=\exp\left(-\frac{\eta^{2}}{l^{2}}\right),$ (20)

where $l$ is the surface scale (autocorrelation length). The surface is
tapered to zero as $x$ approaches the edges ($x=L$ and $x=-L$) via a tanh
function. All the rough surfaces have mean zero and we scale the generated
surface vertically to vary the peak-to-trough height ($h_{\max}-h_{\min})$.
The field data is measured along the horizontal line $z=\zeta$, and the number
of observation points is set to be $N_{\rm obs}$. We consider two cases here:

  * Case A: full scattered data is known.

  * Case B: phaseless total field data (amplitude of field) is known.

If the measured field at one point is $\psi_{\rm data}$, then the noisy data
is given by

$\psi_{\rm noise}=\psi_{\rm data}(1+\epsilon\vartheta),$ (21)

where $\epsilon$ is the noise level and $\vartheta$ is a random number
uniformly generated in $[-1,1]$. Both TE field (Dirichlet boundary condition)
and TM field (Neumann boundary condition) are considered. We test the method
with respect to several characteristics of rough surface and incident field
including surface scale, peak-to-trough height, wavenumber, and angle of
grazing. Throughout sections IV-A, IV-B, IV-C and IV-D, the incident field is
taken to be the plane wave with wavenumber $k=2\pi$ (corresponding to
frequency of $300$MHz and wavelength of $1$ meter) and grazing angle
$\alpha=-\pi/4$.

As discussed in section III, uniform neural networks are employed with $N_{l}$
hidden layers and $N_{n}$ neurons per hidden layer and an activation function
applied after each hidden layer. The structures of neural networks as well as
the optimization parameters are given in table I for TE and TM fields and case
A and case B.

TABLE I: Structures of neural networks and optimization parameters Field | TE | TM
---|---|---
case A | case B | case A | case B
$N_{l}$ | 4 | 6 | 4 | 8
$N_{n}$ | 256 | 512 | 256 | 512
activation function | Sigmoid | Sigmoid
optimizer | Adam | Adam
learning rate | 0.001 | 0.001
number of iterations | 1500 | 2000 | 1500 | 2000

The number of sampling points at each iteration is randomly selected from
$[N_{\rm obs},N_{\rm inv}]$, where $N_{\rm inv}$ is a larger number. For the
boundary terms in the loss function ($\mathcal{L}_{\rm bc}$ in (17) and (19)),
we take $N_{b}=10$, and suppose that $h(X_{j}^{b})=0$ with
$x_{j}^{b}=-L+j\Delta x_{t}$ for $j=0,1,\ldots,4$, and
$X_{j}^{b}=L-(j-5)\Delta x_{t}$ for $j=5,6,\ldots,9$.

The quantitative error tested here is the $\ell^{2}$-norm error between
recovered and actual surfaces, given by

$\lVert e\rVert_{\ell^{2}}=\frac{\lVert\bm{H^{\rm
NN}}-\bm{H}\rVert_{\ell^{2}}}{\lVert\bm{H}\rVert_{\ell^{2}}}\times 100\%$ (22)

where $\bm{H^{\rm NN}}$ is the vector of reconstructed surface and $\bm{H}$ is
the vector of original surface. To analyze the error, we conduct $50$ tests
and record both the mean and the standard deviation of the $\ell^{2}$-norm
error values.

All the examples are computed using Python. The functionality of neural
networks is supported by the open–source library PyTorch [46]. The automatic
differentiation is also implemented using PyTorch [47]. The source code of
implementation is freely available in the supporting material [48].

###  IV-A Reconstruction with noiseless data

We first test the method using the field data without noise. The surface scale
is taken to be $l=2/3\lambda$, peak-to-trough height is $0.4\lambda$, and the
data observation height is $\zeta=0.5\lambda$. Here we take $N_{\rm obs}=240$,
approximately $10$ points per surface scale, and $N_{\rm inv}=480$,
approximately $20$ points per surface scale. Figure 2 shows the recovered
surface plotted against the true surface for case A (known full scattered
data) without noise for TE and TM fields.

(a) TE field

(b) TM field

Figure 2: Reconstruction of rough surfaces against the actual surfaces for
case A (known full scattered data) without noise.

(a) TE field

(b) TM field

Figure 3: Reconstruction of rough surfaces against the actual surfaces for
case B (known phaseless total field data) without noise.

The recovered surface matches well with the actual surface by capturing all
the shapes. In most parts of the surface, the recovered and actual surfaces
closely coincide. The main discrepancy occurs at peaks and troughs of the
surface. This is due to the significant change in the derivative of surface,
and has also been observed in the results obtained by other traditional
methods [2, 19].

Figure 4: Reconstruction of rough surfaces compared to actual surfaces using
data with different noise levels $\epsilon$ in the case A of known scattered
field data for TE and TM fields. Figure 5: Reconstruction of rough surfaces
compared to actual surfaces with respect to different noise levels $\epsilon$
in the case B of known phaseless total field data for TE and TM fields.

For case B where the amplitude of total field is known, the resulting solution
versus original surface with non-noisy data for TE and TM fields is shown in
fig. 3. Similar to the previous case, the solution surface closely matches the
actual surface.

###  IV-B Reconstruction with noisy data

Here we add uniformly distributed white noise to the field data, and test the
results with respect to noise level $\epsilon$. The problem setting is kept
the same as in section IV-A ($l=2/3\lambda$, $h_{\max}-h_{\min}=0.4\lambda$,
$\zeta=0.5\lambda$, $N_{\rm obs}=240$, and $N_{\rm inv}=480$). The recovered
surfaces against the original surfaces with respect to a range of noise levels
($\epsilon$) are shown in fig. 4 for TE and TM fields in case A (full
scattered field known). Clearly, with noisy data, the recovered surface still
has close agreement with the actual surface, which validates the robustness of
the method in the presence of noise. The discrepancies become more apparent
when using data with larger noise level. Surprisingly, for case A where full
scattered data is known, this method can tolerate noise with noise level up to
$20\%$.

Now we consider the performance in case B (phaseless total field known). The
recovered surface plotted with the original surface with respect to noise
level $\epsilon$ is shown in fig. 5 for TE and TM fields. It is found that the
reconstruction using phaseless data with small noise level still matches well
with the actual surface. As the noise level increases, in particular when
$\epsilon>5\%$, clear discrepancies show up, leading to much larger error.
However, all the ‘solution’ surfaces recapture the overall shape of original
ones.

We also analyze the error quantitatively. The mean $\ell^{2}$-norm error
together with the standard deviation with respect to noise level ($\epsilon$)
is presented is fig. 6 for case A and case B.

(a) case A

(b) case B

Figure 6: Mean $\ell^{2}$ norm error (points) and standard deviation (colored
band) with respect to noise level ($\epsilon$) for case A (full scattered
data) and case B (phaseless total field data).

For the case A with full scattered data, the error stays at a similar level
when noise is small, whereas it rises significantly when noise level exceeds
$15\%$. This phenomenon is more pronounced for TE field. In the case A, the
standard deviation also keeps small for small noise level, but it increases
significantly when noise level is large. In the case B with phaseless data,
there is a steady increment in the error with noise level for both TE and TM
fields. In summary, the method is able to tolerate noise level $\epsilon$ up
to around $15\%$ for case A and $5\%$ for case B.

Throughout the remainder of the numerical examples, we will add $10\%$ noise
to the data in case A (full scattered data) and $3\%$ noise to the data in
case B (phaseless total field).

###  IV-C Reconstruction with respect to surface scale

The surface scale ($l$ in eq. 20) controls the number of local extrema in the
rough surface; smaller $l$ gives rise to more oscillations. We fix
$h_{\max}-h_{\min}=0.4\lambda$ and $\zeta=0.5\lambda$. The number of
observation points is still kept as $N_{\rm inv}=240$. As the surface scale
decreases, the value of $N_{\rm inv}$ (maximum number of sampling points at
each iteration) has to be increased for two reasons: (i) it becomes more
difficult to reconstruct all the finer details for oscillatory surface and
(ii) higher resolution helps further reduce the loss during neural network
training. Some results of recovered surfaces with respect to relatively small
surface scales are shown in fig. 7 for case A and case B, TE and TM fields,
and the values of $N_{\rm inv}$ are also indicated.

(a) case A, TE, $l=\frac{1}{2}\lambda$, $N_{\rm inv}=600$

(b) case A, TM, $l=\frac{2}{5}\lambda$, $N_{\rm inv}=840$

(c) case B, TM, $l=\frac{1}{2}\lambda$, $N_{\rm inv}=720$

(d) case B, TE, $l=\frac{2}{5}\lambda$, $N_{\rm inv}=840$

Figure 7: Rough surface reconstruction with respect to different values of
surface scales ($l$) using full scattered data (case A) with $10\%$ noise and
phaseless total field data (case B) with $3\%$ noise. The maximum number of
sampling points at each iteration $N_{\rm inv}$ is also stated.

With a larger value of $N_{\rm inv}$, the recovered surface still captures the
overall shape of the original surface. As surface scale decreases,
discrepancies become more pronounced, particularly at the peaks and troughs in
the highly oscillatory region. Figure 8 presents the mean $\ell^{2}$-norm
error obtained using different values of $N_{\rm inv}$ with respect to
different surface scales ($l$).

(a) TE field, case A

(b) TM field, case B

Figure 8: The mean $\ell^{2}$-norm error obtained with different values of
$N_{\rm inv}$ (maximum number of testing points at each iteration) and
different surface scales $l$.

It is clear that the error decreases with a larger value of $N_{\rm inv}$.
Unsurprisingly, the error is small for less oscillatory surface (larger value
of $l$). The error remains steady provided that $N_{\rm inv}$ is large enough
for $l=2/3\lambda$ and $l=1/2\lambda$.

###  IV-D Reconstruction with respect to surface height

Surface height, in particular the peak-to-trough height ($h_{\max}-h_{\min}$),
is a key factor in the inverse problem. Increased height from peak to trough
typically leads to stronger surface scattering, resulting in degraded
performance of reconstruction algorithms. Here, we solve the problem with
parameters of $l=2/3\lambda$, $\zeta=2.5h_{\max}$, $N_{\rm obs}=240$ and
$N_{\rm inv}=480$. Some reconstructions of rough surfaces with large peak-to-
trough height values for TE and TM fields in case A and case B are shown in
fig. 9. The performance degrades for surfaces with larger peak-to-trough
height. Overall, despite variations, all the recovered surfaces still manage
to preserve the essential feature and shape of original surfaces. This
demonstrates the robustness of the method when dealing with surface height
variations.

(a) case A, TM,
$h_{\max}-h_{\min}=0.7\lambda$

(b) case A, TE,
$h_{\max}-h_{\min}=\lambda$

(c) case B, TE,
$h_{\max}-h_{\min}=0.7\lambda$

(d) case B, TM,
$h_{\max}-h_{\min}=\lambda$

Figure 9: Reconstruction of rough surfaces compared to actual surfaces with
respect to different peak-to-trough heights ($h_{\max}-h_{\min}$) for case A
(full scattered data) with $10\%$ noise and case B (phaseless total field
data) with $3\%$ noise.

The mean $\ell^{2}$-norm error together with the standard deviation with
respect to different values of $h_{\max}-h_{\min}$ is presented in table II
for case A data with $10\%$ noise and case B data with $3\%$ noise. There is
an obvious increase in error for larger surface height, as well as the
standard deviation. However, the error is still considerably small when the
peak-to-trough height is less than $\lambda$.

TABLE II: Mean error in $\ell^{2}$-norm (%) and standard deviation between the actual and recovered surfaces with respect to different peak-to-trough heights $h_{\max}$$-$$h_{\min}$ | case A | case B
---|---|---
TE | TM | TE | TM
$0.4\lambda$ | 6.57$\pm$0.84 | 5.87$\pm$0.67 | 8.52$\pm$1.01 | 10.72$\pm$2.24
$0.6\lambda$ | 7.13$\pm$0.95 | 6.47$\pm$0.83 | 10.26$\pm$0.96 | 11.38$\pm$2.87
$0.8\lambda$ | 7.39$\pm$1.12 | 8.91$\pm$1.28 | 11.04$\pm$1.54 | 11.57$\pm$2.79
$1.0\lambda$ | 10.55$\pm$2.43 | 10.36$\pm$2.60 | 11.37$\pm$2.77 | 12.45$\pm$3.51
$1.2\lambda$ | 15.20$\pm$3.11 | 17.49$\pm$3.52 | 28.96$\pm$5.84 | 36.08$\pm$12.39

###  IV-E Reconstruction with respect to incident wave

An important aspect of this method is its flexibility with respect to the
incident field (wavenumber and angle of incidence). We consider the problem
setting of $l=2/3\lambda$, $h_{\max}-h_{\min}=0.6\lambda$, $\zeta=0.6\lambda$,
$N_{\rm obs}=240$, and $N_{\rm inv}=480$, while the wavenumber has been
increased to $k=6.67\pi$ (corresponding to frequency of 1GHz and wavelength of
$0.3$ meters) and the angle of grazing is set to be $-\pi/9$ (corresponding to
$10^{\circ}$). The results obtained using the same structure of neural
networks (table I) for case A TM data with $10\%$ noise and case B TE data
with $3\%$ noise are shown in fig. 10.

(a) TM field, case A

(b) TE field, case B

Figure 10: Reconstruction of rough surfaces compared to actual surfaces from
the plane wave of $k=6.67\pi$ and $\alpha=-10/\pi$ for case A (full scattered
data) with $10\%$ noise and case B (phaseless total field data) with $3\%$
noise.

The reconstructed surface still closely matches the actual surface, which
validates the robustness of the method with respect to incident wave
characteristics. We make a remark here that the error increases when using
incident field with high wavenumber (frequency). However the error can be
reduced via employing deeper neural networks.

##  V Conclusions

A deep learning approach based on physics–informed neural networks (PINN) has
been developed to recover profile of rough surfaces from observed
electromagnetic field data. In this approach, the unknown surface height is
approximated by a neural network. Thy ‘physical information’ (equations) used
in the neural network are the integral equations relating the surface,
incident and scattered field. With the predicted surface (neural network) and
its spatial derivatives (automatic differentiation), we calculate the
scattered field via the method of moments (MOM). The neural network is then
trained via optimizing the loss between the observed and calculated field
data. It was shown that the method can accurately recover highly varying
random rough surfaces.

A particular aspect of this approach is that it is an unsupervised
reconstruction method, namely, it does not require any ‘ _a priori_ ’
‘solution’ data. This is particularly appealing in the rough surface
reconstruction field, where it is usually difficult and time-consuming to
generate surface data in practice, rather a more typical inverse problem is to
recover specific unknown surfaces from the measured field data. The proposed
method, therefore, opens up the possibility treating real–world applications
through a fast and robust deep learning approach, such as non–destructive
testing and microwave remote sensing.

Two types of data were considered in this paper: full scattered field data and
phaseless total field data, which results from both TE and TM incident fields.
The performance of the method is validated by a large number of test cases
with Gaussian-correlated random rough surfaces. The method has also shown a
strong robustness with respect to a range of different regimes, including
noisy data, surface scale (degree of surface oscillations), surface height
(peak-to-trough), and incident wave (wavenumber and incidence angle).

It is noted that even though this paper shows the results of rough surface
reconstruction in 2D, the approach can be easily extended to 3D problems. For
3D applications, the main difficulty becomes the time complexity of method of
moments, which involves a step of inverting a linear system of size
$N^{2}\times N^{2}$, where $N$ is the discretization size in one axis. Another
important problem to which this method can extend straightforwardly is the
design of inverse waveguide problem. In 2D version of inverse waveguide
problem, a pair of rough surfaces need to be reconstructed with respect to a
set of field data.

##  VI ACKNOWLEDGEMENT

YC was supported by the Youth Program of the Natural Science Foundation of
Jiangsu Province (No. BK20230466), the Jiangsu Funding Program for Excellent
Postdoctoral Talent (No. 2022ZB584), and Jiangsu Shuangchuang Project
(JSSCTD202209). CW was supported by the National Natural Science Foundation of
China under Grants No. 62301532, in part by the Natural Science Foundation of
Jiangsu Province under Grant No. BK20230282

## References

  * [1] A. Schatzberg and A. J. Devaney, “Rough surface inverse scattering within the Rytov approximation,” _Journal of the Optical Society of America A_ , vol. 10, no. 5, pp. 942–950, 1993.
  * [2] Y. Chen and M. Spivack, “Rough surface reconstruction at grazing angles by an iterated marching method,” _Journal of the Optical Society of America A_ , vol. 35, no. 4, pp. 504–513, 2018.
  * [3] Y. Chen, O. Rath Spivack, and M. Spivack, “Recovery of rough surface in ducting medium from grazing angle scattered wave,” _Journal of Applied Physics_ , vol. 124, no. 8, p. 084901, 08 2018.
  * [4] N. Destouches, C.-A. Guérin, M. Lequime, and H. Giovannini, “Determination of the phase of the diffracted field in the optical domain: Application to the reconstruction of surface profiles,” _Optics Communications_ , vol. 198, no. 4, pp. 233–239, 2001.
  * [5] I. Akduman, R. Kress, and A. Yapar, “Iterative reconstruction of dielectric rough surface profiles at fixed frequency,” _Inverse Problems_ , vol. 22, no. 3, p. 939, may 2006.
  * [6] R. Kress and T. Tran, “Inverse scattering for a locally perturbed half-plane,” _Inverse Problems_ , vol. 16, no. 5, p. 1541, oct 2000.
  * [7] A. Yapar, O. Ozdemir, H. Sahinturk, and I. Akduman, “A Newton method for the reconstruction of perfectly conducting slightly rough surface profiles,” _IEEE Transactions on Antennas and Propagation_ , vol. 54, no. 1, pp. 275–279, 2006.
  * [8] P. Mojabi and J. LoVetri, “Overview and classification of some regularization techniques for the Gauss-Newton inversion method applied to inverse scattering problems,” _IEEE Transactions on Antennas and Propagation_ , vol. 57, no. 9, pp. 2658–2665, 2009.
  * [9] G. Bozza and M. Pastorino, “An inexact Newton-based approach to microwave imaging within the contrast source formulation,” _IEEE Transactions on Antennas and Propagation_ , vol. 57, no. 4, pp. 1122–1132, 2009.
  * [10] J. Li, J. Yang, and B. Zhang, “A linear sampling method for inverse acoustic scattering by a locally rough interface,” _Inverse Problems and Imaging_ , vol. 15, no. 5, pp. 1247–1267, 2021.
  * [11] X. Xu, B. Zhang, and H. Zhang, “Uniqueness and direct imaging method for inverse scattering by locally rough surfaces with phaseless near-field data,” _SIAM Journal on Imaging Sciences_ , vol. 12, no. 1, pp. 119–152, 2019.
  * [12] X. Ji, X. Liu, and B. Zhang, “Inverse acoustic scattering with phaseless far field data: Uniqueness, phase retrieval, and direct sampling methods,” _SIAM Journal on Imaging Sciences_ , vol. 12, no. 2, pp. 1163–1189, 2019\.
  * [13] G. Bao and L. Zhang, “Shape reconstruction of the multi-scale rough surface from multi-frequency phaseless data,” _Inverse Problems_ , vol. 32, no. 8, p. 085002, 2016.
  * [14] G. Dolcetti, M. Alkmim, J. Cuenca, L. De Ryck, and A. Krynkin, “Robust reconstruction of scattering surfaces using a linear microphone array,” _Journal of Sound and Vibration_ , vol. 494, p. 115902, 2021.
  * [15] A. Sefer, “Locally perturbed inaccessible rough surface profile reconstruction via phaseless scattered field data,” _IEEE Transactions on Geoscience and Remote Sensing_ , vol. 60, pp. 1–8, 2022.
  * [16] A. Sefer and A. Yapar, “Inverse scattering by perfectly electric conducting (PEC) rough surfaces: An equivalent model with line sources,” _IEEE Transactions on Geoscience and Remote Sensing_ , vol. 60, pp. 1–9, 2022.
  * [17] A. Sefer, A. Yapar, and T. Yelkenci, “Imaging of rough surfaces by RTM method,” _IEEE Transactions on Geoscience and Remote Sensing_ , vol. 62, pp. 1–12, 2024.
  * [18] A. Sefer and A. Yapar, “An iterative algorithm for imaging of rough surfaces separating two dielectric media,” _IEEE Transactions on Geoscience and Remote Sensing_ , vol. 59, no. 2, pp. 1041–1051, 2021.
  * [19] Y. Chen, O. R. Spivack, and M. Spivack, “Rough surface reconstruction from phaseless single frequency data at grazing angles,” _Inverse Problems_ , vol. 34, no. 12, p. 124002, 2018.
  * [20] F. Qu, B. Zhang, and H. Zhang, “A novel integral equation for scattering by locally rough surfaces and application to the inverse problem: The Neumann case,” _SIAM Journal on Scientific Computing_ , vol. 41, no. 6, pp. A3673–A3702, 2019.
  * [21] R. J. Wombell and J. A. DeSanto, “Reconstruction of rough-surface profiles with the Kirchhoff approximation,” _Journal of the Optical Society of America A_ , vol. 8, no. 12, pp. 1892–1897, Dec 1991.
  * [22] Y. Zhou, Y. Zhong, Z. Wei, T. Yin, and X. Chen, “An improved deep learning scheme for solving 2-D and 3-D inverse scattering problems,” _IEEE Transactions on Antennas and Propagation_ , vol. 69, no. 5, pp. 2853–2863, 2020\.
  * [23] K. Xu, L. Wu, X. Ye, and X. Chen, “Deep learning-based inversion methods for solving inverse scattering problems with phaseless data,” _IEEE Transactions on Antennas and Propagation_ , vol. 68, no. 11, pp. 7457–7470, 2020\.
  * [24] K. Xu, C. Zhang, X. Ye, and R. Song, “Fast full-wave electromagnetic inverse scattering based on scalable cascaded convolutional neural networks,” _IEEE Transactions on Geoscience and Remote Sensing_ , vol. 60, pp. 1–11, 2022.
  * [25] R. Guo, X. Song, M. Li, F. Yang, S. Xu, and A. Abubakar, “Supervised descent learning technique for 2-D microwave imaging,” _IEEE Transactions on Antennas and Propagation_ , vol. 67, no. 5, pp. 3550–3554, 2019.
  * [26] M. Sabbaghi, J. Zhang, and G. W. Hanson, “Machine learning target count prediction in electromagnetics using neural networks,” _IEEE Transactions on Antennas and Propagation_ , vol. 70, no. 8, pp. 6171–6183, 2022\.
  * [27] Y. Sanghvi, Y. Kalepu, and U. K. Khankhoje, “Embedding deep learning in inverse scattering problems,” _IEEE Transactions on Computational Imaging_ , vol. 6, pp. 46–56, 2020.
  * [28] G. Chen, P. Shah, J. Stang, and M. Moghaddam, “Learning-assisted multimodality dielectric imaging,” _IEEE Transactions on Antennas and Propagation_ , vol. 68, no. 3, pp. 2356–2369, 2020.
  * [29] L. Li, L. G. Wang, F. L. Teixeira, C. Liu, A. Nehorai, and T. J. Cui, “DeepNIS: Deep neural network for nonlinear electromagnetic inverse scattering,” _IEEE Transactions on Antennas and Propagation_ , vol. 67, no. 3, pp. 1819–1825, 2019.
  * [30] Z. Zong, Y. Wang, and Z. Wei, “A wavelet-based compressive deep learning scheme for inverse scattering problems,” _IEEE Transactions on Geoscience and Remote Sensing_ , vol. 60, pp. 1–11, 2022.
  * [31] I. Aydin, G. Budak, A. Sefer, and A. Yapar, “CNN-based deep learning architecture for electromagnetic imaging of rough surface profiles,” _IEEE Transactions on Antennas and Propagation_ , vol. 70, no. 10, pp. 9752–9763, 2022.
  * [32] A. S. Izde Aydin, Guven Budak and A. Yapar, “Recovery of impenetrable rough surface profiles via CNN-based deep learning architecture,” _International Journal of Remote Sensing_ , vol. 43, no. 15-16, pp. 5658–5685, 2022.
  * [33] M. Raissi, P. Perdikaris, and G. Karniadakis, “Physics–informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations,” _Journal of Computational Physics_ , vol. 378, pp. 686–707, 2019.
  * [34] G. E. Karniadakis, I. G. Kevrekidis, L. Lu, P. Perdikaris, S. Wang, and L. Yang, “Physics–informed machine learning,” _Nature Reviews Physics_ , vol. 3, no. 6, pp. 422–440, 2021.
  * [35] M. Rasht-Behesht, C. Huber, K. Shukla, and G. E. Karniadakis, “Physics-informed neural networks (PINNs) for wave propagation and full waveform inversions,” _Journal of Geophysical Research: Solid Earth_ , vol. 127, no. 5, p. e2021JB023120, 2022.
  * [36] S. Alkhadhr, X. Liu, and M. Almekkawy, “Modeling of the forward wave propagation using physics-informed neural networks,” in _2021 IEEE International Ultrasonics Symposium (IUS)_ , 2021, pp. 1–4.
  * [37] Z. Yin, G.-Y. Li, Z. Zhang, Y. Zheng, and Y. Cao, “SWENet: a physics-informed deep neural network (PINN) for shear wave elastography,” _IEEE Transactions on Medical Imaging_ , 2023.
  * [38] Y. Chen, L. Lu, G. E. Karniadakis, and L. D. Negro, “Physics–informed neural networks for inverse problems in nano-optics and metamaterials,” _Optics Express_ , vol. 28, no. 8, pp. 11 618–11 633, Apr 2020.
  * [39] R. Guo, Z. Lin, T. Shan, X. Song, M. Li, F. Yang, S. Xu, and A. Abubakar, “Physics embedded deep neural network for solving full-wave inverse scattering problems,” _IEEE Transactions on Antennas and Propagation_ , vol. 70, no. 8, pp. 6148–6159, 2022.
  * [40] Y.-D. Hu, X.-H. Wang, H. Zhou, L. Wang, and B.-Z. Wang, “A more general electromagnetic inverse scattering method based on physics-informed neural network,” _IEEE Transactions on Geoscience and Remote Sensing_ , vol. 61, pp. 1–9, 2023.
  * [41] K. F. Warnick and W. C. Chew, “Numerical simulation methods for rough surface scattering,” _Waves in random media_ , vol. 11, no. 1, p. R1, 2001.
  * [42] A. G. Voronovich, _Wave scattering from rough surfaces_. Springer Science & Business Media, 2013, vol. 17.
  * [43] R. F. Harrington, _Field Computation by Moment Methods_. Wiley-IEEE Press, 1993.
  * [44] C. Bourlier, N. Pinel, and G. Kubické, _Method of moments for 2D scattering problems: basic concepts and applications_. John Wiley & Sons, 2013.
  * [45] D. P. Kingma and J. Ba, “Adam: A method for stochastic optimization,” _arXiv preprint arXiv:1412.6980_ , 2014.
  * [46] A. Paszke, S. Gross, F. Massa, A. Lerer, J. Bradbury, G. Chanan, T. Killeen, Z. Lin, N. Gimelshein, L. Antiga, A. Desmaison, A. Köpf, E. Yang, Z. DeVito, M. Raison, A. Tejani, S. Chilamkurthy, B. Steiner, L. Fang, J. Bai, and S. Chintala, _PyTorch: an imperative style, high-performance deep learning library_. Red Hook, NY, USA: Curran Associates Inc., 2019.
  * [47] A. Paszke, S. Gross, S. Chintala, G. Chanan, E. Yang, Z. DeVito, Z. Lin, A. Desmaison, L. Antiga, and A. Lerer, “Automatic differentiation in PyTorch,” in _NeurIPS Autodiff Workshop_ , 2017.
  * [48] Y. Chen, “Supporting material,” URL https://github.com/yc397/pinn_rough_surface, 2024.