# Generating extreme quantum scattering in graphene with machine learning

Chen-Di Han  School of Electrical, Computer and Energy Engineering, Arizona
State University, Tempe, Arizona 85287, USA  Ying-Cheng Lai  Ying-
<EMAIL_ADDRESS>School of Electrical, Computer and Energy Engineering,
Arizona State University, Tempe, Arizona 85287, USA  Department of Physics,
Arizona State University, Tempe, Arizona 85287, USA

###### Abstract

Graphene quantum dots provide a platform for manipulating electron behaviors
in two-dimensional (2D) Dirac materials. Most previous works were of the
“forward” type in that the objective was to solve various confinement,
transport and scattering problems with given structures that can be generated
by, e.g., applying an external electrical field. There are applications such
as cloaking or superscattering where the challenging problem of inverse design
needs to be solved: finding a quantum-dot structure according to certain
desired functional characteristics. A brute-force search of the system
configuration based directly on the solutions of the Dirac equation is
computational infeasible. We articulate a machine-learning approach to
addressing the inverse-design problem where artificial neural networks subject
to physical constraints are exploited to replace the rigorous Dirac equation
solver. In particular, we focus on the problem of designing a quantum dot
structure to generate both cloaking and superscattering in terms of the
scattering efficiency as a function of the energy. We construct a physical
loss function that enables accurate prediction of the scattering
characteristics. We demonstrate that, in the regime of Klein tunneling, the
scattering efficiency can be designed to vary over two orders of magnitudes,
allowing any scattering curve to be generated from a proper combination of the
gate potentials. Our physics-based machine-learning approach can be a powerful
design tool for 2D Dirac material-based electronics.

##  I Introduction

Two-dimensional (2D) Dirac material systems such as graphene, topological
insulators, molybdenum disulfied, and topological Dirac semimetals [1, 2, 3,
4, 5] constitute a research frontier in condensed matter physics and materials
science. A common feature of these materials is that their energy bands
contain a Dirac cone structure that gives rise to a linear energy-momentum
relation (dispersion relation) for low-energy excitations, which is
characteristic of relativistic quantum motions governed by the Dirac equation.
The Dirac cone structure can be exploited for storing and transferring
information with applications in Dirac-material based electronics and
spintronics [6, 7, 8, 9].

Given a 2D Dirac material, the system structure and the applied gate
potentials, a combination of these can generate rich quantum behaviors. For
example, in the field of relativistic quantum chaos [10, 11, 12, 13],
electrons can be confined in a graphene sheet of certain geometric shape,
generating billiard systems that provide a platform to study the relativistic
quantum manifestations of different kinds of classical dynamics. In
scattering, a combination of gate potentials can be applied to a graphene
sheet to create a quantum-dot structure, which can be experimentally realized
through STM tips or doping [14, 15, 16, 17, 18, 19]. Graphene quantum dots are
also a paradigm to study various transport phenomena [20, 21, 22]. Given a
scattering structure, the tight-binding Hamiltonian or the Dirac equation can
be solved to yield the scattering functions so that the performance of the
device can be assessed. A common feature among all these previous works is
their “forward” nature: studying various quantum behaviors or performance of
the underlying system with a given structure.

Inverse design addresses the opposite problem: how to design a Dirac material
system to generate certain desired functional characteristics. For example,
suppose we wish to design a graphene device to generate the desired scattering
functions by using a 2D multilayer structure of concentric circular graphene
ribbons, where the layers can have different widths and each layer is made
physically distinct from the others through a vertical electric field (gate
potential). A brute-force approach to searching the optimal multilayer
structure to generate the desired scattering functions would be to test a
large number of combinations of the geometric parameters of the various layers
as well as the values of the gate potentials. With luck, it may be possible to
find a specific structure and a set of gate potentials such that the resulting
scattering curves approximately match the desired functions. However, such a
brute-force approach is generally practically infeasible because of the
extremely large parameter space that needs to be searched. There is in fact no
guarantee that this approach would be successful, as the existence of a device
with the desired response is unknown a priori. Besides quantum scattering
[23], inverse-design problems of this kind occur in other fields such as
quantum information [24], biology [25], molecular design [26], and photonics
[27].

Recently, machine learning has been introduced to inverse optical design where
the goal is to find the best structure of a multilayer dielectric sphere to
generate the desired electromagnetic response by approximating the Maxwell
equations with a trained artificial neural network [28]. The approximation can
simplify the original optimization problem without loss of details.
Subsequently, the approach has been extended to designing optical metasurfaces
[29], metagrating [30], and multifunctional devices [31, 32]. Machine learning
has thus offered a general solution to the inverse-design problem.

This work concerns designing systems that can generate extreme scattering
characteristics. In particular, the performance of any scattering system,
optical or electronic, can be conveniently characterized by the scattering
cross section as a function of a basic quantity such as the wavelength (in
optics) or the Fermi energy (in electronic systems). The two opposite extremes
correspond to a small and a large scattering cross section, respectively, with
the former representing cloaking and the latter signifying superscattering. In
optics, one approach to cloaking is through scattering cancellation [33] based
on the the idea that, in a multilayer structure, polarization from different
layers can cancel each other to generate an exceedingly small scattering cross
section. Multilayer structures can also be exploited to produce
superscattering [34]. There were experimental results on cloaking or
superscattering in optics [35, 36, 34]. For those problems, a basic physical
constraint is that it is generally not possible to generate cloaking or
superscattering for all kinds of incident waves [37, 38], so these exotic
phenomena can occur only for a certain type of incident waves. In optics, both
cloaking and superscattering can occur in a multilayer structure of dielectric
materials, so it is possible to generate cloaking and superscattering in the
same device. For example, it has been recently demonstrated that a single
device based on Ag-Semiconductor multilayer sphere can produce large and small
scattering cross sections [28, 39]. Without changing the dimension of the
device, a superscattering-cloaking transition was reported in a two-layer
system where a significant change in the scattering cross section can occur
with respect to the incident angle [40]. One optical material capable of such
a transition is GeTe [41, 42], where, under different environmental conditions
such as the temperature [43], the material exhibits different physical
properties. Related optical scattering problems were investigated with
graphene coated nanosphere [44, 45] and its design [46]. The analogy between
matter and electromagnetic waves stimulated the idea of generating electron
cloaking through a multilayer structure [47, 48, 49], which subsequently was
extended to graphene for cloaking relativistic electrons [50, 51, 52].

Our study focuses on the following question: is it possible to generate the
two extreme scattering behaviors, cloaking and superscattering, in the same 2D
Dirac material system? Because of the relative easiness to experimentally tune
the gate potential through varying an applied electrical field [53], graphene
stands out as a viable candidate for generate superscattering and cloaking as
well as a transition between them in the same fixed structure. To be concrete,
we consider a multilayer scattering structure that consists of $N$ concentric
circles, where a different gate potential is applied to each distinct circular
layer. To reduce the dimension of the parameter space so as to make the
inverse-design problem feasible, we fix the geometric structure of the
scattering system and allow the set of gate potentials to be tunable. We
develop a class of neural networks subject to physical constraints with the
goal to find a set of gate potentials to generate electronic cloaking and
another set to generate superscattering in the same device. A key component of
the inverse design is our articulation of a physical loss function to
significantly reduce the training and testing errors and to eliminate
nonphysical solutions. We demonstrate that, even in the regime of Klein
tunneling, the scattering efficiency (to be defined in Sec. II.1) can vary
over two orders of magnitude. Compared with the corresponding optical system
where the material for each layer is fixed and the width for each shell can be
changed, in our graphene scattering system only the gate potential is changed
with the device structure intact, rendering it experimentally feasible. Our
physics-based machine-learning approach can be a powerful design tool for
graphene-based electronic devices and is generalizable to solving inverse
problems in other areas of science.

##  II Methods: multilayer graphene quantum dot, machine learning and inverse
design

We first clarify the physical meanings of the terms “cloaking” and
“superscattering” adopted in this work. Our goal of inverse design is to
realize a structure that can generate these two opposite extremes of quantum
scattering. Ideal cloaking is characterized by zero scattering cross section
or efficiency. Since near-zero scattering efficiency can trivially occur for a
scatterer of arbitrarily weak strength: $\bar{V}\bar{R}\rightarrow 0$, where
$\bar{V}$ and $\bar{R}$ are the average gate potential and the mean size of
the quantum dot, respectively, cloaking is meaningful only for scatterers with
a reasonably large scattering strength, e.g., the quantum dot of size $R_{B}$
in Fig. 1(b). For this case to be qualified as cloaking, the resulting
scattering cross section should be as small as that from a scatterer of much
smaller size such as the dot of radius $R_{C}$ in Fig. 1(b). In the opposite
limit, strong scattering can naturally occur if the scattering strength is
high. By superscattering we mean that, even when the scatterer is relatively
weak, the resulting scattering cross section can be as large as that from a
scatterer of much larger size, such as the quantum dot of size $R_{A}$ in Fig.
1(b).

Figure 1: Illustration of a multilayer graphene quantum dot, scattering
efficiency, and the physics-constrained machine learning scheme. (a) A
multilayer circular graphene quantum dot generated by the gate potential
profile $V(r)$ in Eq. (2). (b) For the special case of a single-layer quantum
dot, scattering efficiency versus the Fermi energy $E$ for three different
sizes of the quantum dot: $R_{A}=63$ nm, $R_{B}=30$ nm, and $R_{c}=3$ nm.
Because of the simple structure, the overall scattering efficiency increases
as the dot becomes larger, as expected, indicating naturally that a single-
layer structure is not capable of generating extreme scattering behaviors. (c)
The neural-network architecture underlying our proposed scheme of physics-
constrained machine learning. The basic component of the neural network is a
multilayer perceptron, where the input is the set of constant gate potential
values $(V_{1},V_{2},\ldots,V_{N})$ and the output is the properly discretized
scattering efficiency function as exemplified in (b). The function is defined
in a fixed energy range and is uniformly sampled with $M$ discrete points,
leading to an $M$-dimensional output vector:
$\bm{Q}\equiv(q_{1},q_{2},\ldots,q_{M})^{T}$. The rightmost layer with the
output vector
$\bm{\hat{Q}}\equiv(\hat{q}_{1},\hat{q}_{2},\ldots,\hat{q}_{M})^{T}$ is
introduced to ensure that the whole neural-network architecture “respects” the
basic physics of quantum scattering (see text for details).

###  II.1 Graphene quantum dot

We consider a 2D circular graphene quantum dot of radius $R_{N}$ subject to a
gate potential profile $V(\mathbf{r})$. The Hamiltonian of the scattering
system is

$\displaystyle H=v_{g}\bm{\sigma}\cdot\mathbf{p}+V(\mathbf{r}),$ (1)

where $v_{g}$ is the Fermi velocity and
$\bm{\sigma}\equiv[\sigma_{x},\sigma_{y}]$ is the vector of Pauli matrices.
For simplicity, we assume that the gate potential has no angular dependence
and write $V(\mathbf{r})=V(r)$, for $0<r\leq R_{N}$. We further assume that
$V(r)$ is a piecewise constant function with $N$ distinct values, which
experimentally can be realized by STM tips or doping in the substrate [14, 15,
16, 17, 18, 19]. The potential profile is

$\displaystyle V(r)=\begin{dcases}V_{1},&r<R_{1}\\\ V_{2},&R_{1}<r<R_{2}\\\
\cdots&\cdots\\\ V_{N},&R_{N-1}<r<R_{N}\\\ 0,&r>R_{N}\end{dcases},$ (2)

leading to a multilayer scattering structure with $N$ distinct concentric
circular layers, where the gate potential in each layer is a constant, as
schematically illustrated in Fig. 1(a). An advantage of the multilayer
structure is that the structural parameters
$(\mathbf{R},\mathbf{V})\equiv([R_{1},R_{2},\cdots,R_{N}]^{T},[V_{1},V_{2},\cdots,V_{N}]^{T})$
can be relatively readily adjusted in experiments. This should be contrasted
to a multilayer structure in optics [28, 29, 30, 31, 32], where only the set
of radii can be changed once the dielectric materials are fixed.

Our goal is to design the potential profile $V(r)$ to generate extreme
scattering behaviors for a fixed set of radii $(R_{1},R_{2},\ldots,R_{N})$. To
gain insights, we first consider the relatively simple case of a single-layer
scatterer: $N=1$. Matching the boundary condition at $r=R_{1}$ and setting the
Fermi velocity to be $v_{g}=10^{6}$ m/s, we obtain the solutions of the 2D
Dirac equation [54] in terms of the scattering spinor wavefunction (Appendix
A). The scattering behavior can be conveniently characterized by the
scattering efficiency defined as

$\displaystyle Q=\frac{\text{Scattering Cross Section}}{\text{Geometric
Size}}=\frac{2}{kR_{1}}\sum_{l=-\infty}^{\infty}|A_{l}|^{2},$ (3)

where $A_{l}$ is the coefficient for scattering wave in the polar coordinates
associated with angular momentum $l$. Intuitively, a large scattering
efficiency indicates that the object is more “visible.” Three examples of the
scattering efficiency are shown in Fig. 1(b) for $V_{1}=87.5$ meV and for dot
size $R_{A}=63$ nm, $R_{B}=30$ nm, and $R_{C}=3$ nm, corresponding to strong,
intermediate, and weak (all in the relative sense) scattering, respectively.
Note that the $Q$ value for $R_{B}$ is approximately two orders of magnitude
larger than that for $R_{C}$. Note also that the incident energy is
approximately one-quarter to three-quarter of the potential height, so the
particle is in the Klein tunneling regime [55]. A recent work [56] indicated
that, for a graphene quantum dot, in the limit $(kR_{1})\rightarrow 0$, for a
fixed Fermi energy the scattering efficiency scales with the size of the
scatterer as $Q\propto(VR_{1})^{3}$. However, the scaling breaks down for
$R_{1}\approx 10$ nm at which the scattering efficiency tends to saturate and
increases only slowly with further increase in $R$. Overall, Fig. 1(b)
indicates that, as the size of the scatterer increases, scattering is
enhanced, as can be intuitively anticipated. The simple single-layer structure
is thus not capable of generating extreme scattering behaviors.

For a graphene multilayer circular quantum dot with given parameters
$(\mathbf{R},\mathbf{V})$, the scattering problem can be solved by the method
of transfer matrix [57] through matching the spinor wavefunctions at all the
layer boundaries (Appendix A). After the spinor wavefunctions in all layers
have been obtained, we can calculate the scattering efficiency according to

$Q=\frac{2}{k_{N+1}R_{N}}\sum_{l=-\infty}^{\infty}\left|A_{l}^{N+1}\right|^{2},$
(4)

where $k_{N}\equiv|E|/v_{g}$. Given a scattering structure, it is thus
straightforward to calculate the scattering efficiency (the solutions of the
forward problem).

To generate extreme scattering behaviors, i.e., cloaking or superscattering,
it is necessary to define the input properly by specifying a range for the
gate potentials because, if the potentials are too weak, the scattering
efficiency will be trivially near zero and this does not lead to cloaking.
Likewise, unreasonably large potential values can lead to strong scattering,
but this is not superscattering. Making use of the scaling relation for the
scattering efficiency for weak scatters [56]:

$\displaystyle Q\propto\left[\sum{|V_{i}|(R_{i+1}-R_{i})}\right]^{3},$

we have $Q\propto(\langle|\mathbf{V}|\rangle R_{B})^{3}$ because
$R_{i}=iR_{N}/N$. It is thus reasonable to choose the gate potentials such
that $\langle|\mathbf{V}|\rangle\approx V_{0}$, where
$\langle|\mathbf{V}|\rangle$ is the average gate potential applied to the
multilayer structure and $V_{0}$ is the potential applied to a single-layer
structure of the same size.

For a single-layer scatterer, the problem of finding a scattering
configuration to generate the desired scattering efficiency
$Q_{\text{desired}}$ can be formulated as the following optimization problem

$\underset{\mathbf{V}}{\min}\text{
}\left\|\frac{Q(\mathbf{V})-Q_{\text{desired}}}{Q_{\text{desired}}}\right\|^{2},$
(5)

where $\|\cdot\|^{2}$ denotes the mean-square error (MSE). Because of the
simple geometry, it is only necessary to optimize a small number of
parameters, making it relatively straightforward to obtain the solutions of
the optimization problem. In particular, it is only necessary to discretize
each parameter dimension and perform a grid search. Difficulties arise when
the scatterer has many layers, making the function $Q(\mathbf{V})$ highly
nonlinear with many parameters that need to be optimized. For example, for a
multilayer sphere of $10$ layers, even if only four parameters need to be
determined for each layer, the number of parameter combinations will be larger
than $4^{10}\sim 10^{6}$. This issue of computational complexity motivated us
to exploit machine learning to solve the optimization problem.

###  II.2 Physics-constrained machine learning

Machine learning provides an effective platform to approximate mathematical
functions arising from physics [58]. For example, in photonic design, neural
networks can be used to approximately represent the scattering cross section
from a variety of devices [28, 29]. To exploit machine learning to generate
extreme scattering behaviors in a graphene quantum dot, an essential step is
to train a multilayer perceptron such that it mimics the scattering process.
To achieve this, we introduce a physics-constrained neural network
architecture, as illustrated in Fig. 1(c). The concrete design of the
architecture is as follows. The input vector has the dimension $N$ \- a set of
constant gate potential values $\bm{V}\equiv(V_{1},V_{2},\ldots,V_{N})^{T}$.
The output is the scattering efficiency function as exemplified in Fig. 1(b),
which is defined in a given energy range and is uniformly sampled with $M$
discrete points, leading to an $M$-dimensional output vector:
$\bm{Q}\equiv(q_{1},q_{2},\ldots,q_{M})^{T}$. In this work, we use $M=200$ (a
rather arbitrary choice). Besides the input and output layers, the neural
network has four hidden layers, each containing $200$ neurons. There are thus
five layer-to-layer transforms from the input to the output. The activation
for the first four transforms is set to be the standard Rectified Linear Unit
(ReLU) and there is no activation from the fourth hidden layer to the output
layer (a simple linear transform). Note that, before any activation, the
neural network uses a linear transformation between two adjacent layers, which
has approximately $4\times 10^{4}$ parameters. Thus the number of training
parameters in the architecture from the input vector $\bm{V}$ to the output
vector $\bm{Q}$ is approximately $10^{5}$.

To train the neural network, a certain amount of ground truth data from the
graphene scattering system is required. To generate the data, it is necessary
to determine the range of the input - which in our case is the range of the
externally applied gate potential. To be concrete, we assume that the mean
value of the gate potential in the whole quantum-dot scattering region is
$\langle V(r)\rangle\approx V_{0}$, where $V_{0}=87.5$ meV, so we choose
$|V_{i}|\in(75\text{ meV},100\text{ meV})$. The training data are generated by
randomly select $V_{i}$ in this range, where $V_{i}$ can be either positive or
negative, i.e., $-100\ \text{meV}<V_{i}<-75\ \text{meV}$ or $75\
\text{meV}<V_{i}<100\ \text{meV}$. For a larger number $N$ of layers, the
multilayer quantum-dot structure is geometrically more sophisticated, thereby
requiring more training data. Empirically, we choose the number of training
data points to be $10^{3}N$. Say the scattering structure has 10 circular
layers: $N=10$. Compared with a brute-force grid search in 10 dimensions, our
choice of $10^{4}$ data points corresponds to search in a parameter space of a
significantly reduced dimension: about three. Our machine-learning approach
thus requires far less data amount than a brute-force search would.

The training process is rather standard and is briefly described, as follows.
The training datasets are generated by directly simulating quantum scattering
based on the solutions of the Dirac equation in the setting of a multilayer
quantum dot. Training is conducted by employing the standard stochastic
gradient descent algorithm with batch size $128$. The method of adaptive
momentum (Adam) is used to minimize the loss function to find the parameters.
The whole process from network construction and training to predicting the
scattering-efficiency function is accomplished by using the open source
package Tensorflow and Keras [59, 60].

A key to the success of a machine-learning architecture is the training loss.
A commonly used loss function is the mean square error (MSE):

$\mathcal{L}_{\text{MSE}}=\|Q_{\text{pred}}-Q_{\text{true}}\|^{2},$ (6)

which typically works well when the data values are within the same order of
magnitude. For the multilayer perceptron in Fig. 1(c), the weights and biases
can be solved by the optimization algorithm based on the loss function. The
structure from the input vector $\bm{V}$ to the output vector $\bm{Q}$ is
mathematically designed without taking into account physical constraints of
the underlying quantum scattering system. As a result, even with extensive
training, nonphysical results can arise, which are in fact not uncommon. For
example, to realize cloaking requires that the scattering efficiency have near
zero values, but some components of the output vector $\bm{Q}$ can be
negative, which is not physical, as illustrated in Fig. 2(a) and the inset for
a graphene quantum dot that has $N=9$ layers. This problem cannot be fixed
through training. It is thus necessary to take physical constraints into
account in the design of the neural-network architecture.

Figure 2: The advantages of invoking a physics-constrained loss function. The
performance of a given loss function can be conveniently visualized by the
plot of the neural-network predicted values of the scattering efficiency
function versus the true value. (a) The plot based on the mathematical loss
function Eq. (6) (without physical considerations). There is a relatively wide
spread of the predicted values about the true values and some predicted values
of the scattering efficiency are negative (non-physical). (b) With the
physics-constrained loss function Eq. (7), the predicted values are all
positive and are closer to the true values. (c) Relative prediction error
versus the number of layers in the scattering structure. Compared with the
case of loss function Eq. (6), the relative error associated with the physics-
constrained loss function Eq. (7) is markedly reduced (by about 50$\%$).

Incorporating physical principles and constraints into designing neural
networks has become a recent subarea of research in machine learning. For
example, the Hamiltonian structure has been built into the neural-network
architectures to predict the dynamical behaviors of classical mechanical
systems [61, 62, 63, 64, 65, 66, 67, 68, 69]. Physics-based neural networks
have also been extended to other fields such as fluid dynamics [70, 71] and
meta surface design in optical [72] or quantum systems [73, 74]. For our
quantum scattering system, a basic requirement is that the scattering cross
sections or efficiency not be negative. We address the issue of physical
constraints by incorporating an additional output layer - the rightmost layer
in Fig. 1(c) with the output vector
$\bm{\hat{Q}}\equiv(\hat{q}_{1},\hat{q}_{2},\ldots,\hat{q}_{M})^{T}$. More
specifically, the additional output layer takes the vector $\bm{Q}$ as input
and performs an exponential operation on each component of $\bm{Q}$ to
generate the vector $\bm{\hat{Q}}$. Mathematically, the operation can be
represented as $\mathcal{R}^{-1}(\cdot)\equiv\exp(\cdot)$, which maps $Q$ to a
strictly positive function $\hat{Q}$. The consideration has led us to
introduce the following two-component loss function:

$\mathcal{L}_{\text{Physical
Loss}}=\|Q_{\text{pred}}-\mathcal{R}(Q_{\text{true}})\|^{2}+\|\hat{Q}_{\text{pred}}-Q_{\text{true}}\|^{2},$
(7)

where $Q_{\text{pred}}$ is the scattering efficiency calculated from the
vector $\bm{Q}$ and $\hat{Q}_{\text{pred}}=\mathcal{R}^{-1}(Q_{\text{pred}})$.
Note that the squared difference between the true $Q$ function and the
transformed function $\hat{Q}$ is the conventional loss function. The idea to
enforce the physical rule of the nonnegativeness of the scattering cross
section is to supply an additional term: the squared difference between the
transformed true output function $\mathcal{R}(Q_{\rm true})$ and the original
output function $Q$. From the point of view of optimization, this loss
function works as follows. If $Q_{\text{true}}$ is close to zero, the
operation $\mathcal{R}$ will return a relatively large value. When
$Q_{\text{true}}$ is large, the second term in the loss function will
dominate. For small and large values of $Q_{\text{true}}$, the predicted
values of the scattering efficiency will be positive.

The performance of the physics-constrained loss function is illustrated in
Fig. 2(b), where various values of the predicted scattering efficiency
function is plotted against the corresponding true values. Compared with Fig.
2(a) based on the mathematical loss function in Eq. (6), we see that the
predicted values are all positive (as they should be) and are closer to the
true values. The prediction performance can be characterized by the following
relative error:

$\text{Relative
Error}=\left|\frac{Q_{\text{true}}-Q_{\text{pred}}}{Q_{\text{true}}}\right|.$
(8)

Figure 2(c) shows the error versus the number of layers in the graphene
quantum-dot structure, where each data point is generated from an ensemble
average of $100$ independent neural-network realizations and, for each
realization, the testing dataset contains 1000 points. Two sets of results are
shown: one according to the physics-constrained loss function Eq. (7) (blue
points) and another based on the loss function Eq. (6) (red). We see that, as
the number of layers in the quantum dot increases, the architecture becomes
more complex, leading to some increase in the relative error. However, the
relative error can be markedly reduced by employing the physics based loss
function Eq. (7).

###  II.3 Principle of inverse design

Figure 3: Flow chart of machine-learning based solution to the inverse-design
problem of quantum scattering. A multilayer neural network is first trained
using a number of functions $Q(E)$ of the scattering efficiency versus the
electron energy for scattering from a multilayer graphene quantum dot subject
to externally applied gate potentials, one to each graphene layer. For any
given set of gate potentials, the corresponding function of the scattering
efficiency is obtained from the exact solutions of the Dirac equation. A well
trained neural network will generate solutions of the Dirac equation. Solving
the inverse-design problem requires the following steps: (1) choosing a random
initial set of gate potentials as the input to the neural network to obtain
the $Q$ function, (2) calculating the difference between this and the desired
$Q$ function, (3) if the difference is sufficiently small (e.g., less than a
pre-defined threshold), then a solution is deemed to be found, (4) otherwise
calculating a change in each gate potential through the standard gradient
approach applied to the neural network to yield a new set of input variables,
and (5) repeat steps (3) and (4) until a solution is found.

The physics-constrained neural network as shown in Fig. 1(c), once trained,
takes an input vector $\bm{V}$ of the gate potentials to generate an output
vector $\hat{\bm{Q}}$ representing the scattering efficiency that is the
solution of the 2D Dirac equation. A well trained neural network thus
effectively functions as a Dirac equation solver. The main advantage of this
substitution lies in the computational efficiency, as what it takes for the
well trained neural network to generate the solutions of the Dirac equation is
simply matrix multiplications through a gradient descent. More specifically,
let $\bm{Q}_{\text{desired}}$ be the desired function of the scattering
efficiency in the same energy range as the training datasets, represented as a
vector. The objective is to solve the following optimization problem:

$\underset{\mathbf{V}}{\min}\text{
}\left\|\frac{\hat{\bm{Q}}(\mathbf{V})-{\bm{Q}}_{\text{desired}}}{{\bm{Q}}_{\text{desired}}}\right\|^{2},$
(9)

where $\hat{\bm{Q}}$ is the output vector from the neural network and
$\bm{Q}_{\text{desired}}$ is normalized to accommodate the behavior of the
scattering efficiency on multiple energy scales. The error in (9) to be
minimized measures the functional distance between neural-network generated
scattering efficiency and the desired one. In general, there is no guarantee
that a particular scattering configuration can be found to yield the desired
function $\bm{Q}_{\text{desired}}$, so the error can be large. The goal is to
find the optimal input vector $\bm{V}$ to minimize the error in (9). As we
will describe below, this can be done in a computationally efficient manner.
Note that the error in (9) is different from the training error in Eq. (8),
which measures the difference between the neural network output to the ground
truth, i.e., the scattering efficiency function from a given set of input gate
potentials.

To better describe the solutions of the optimization problem, we employ a
concrete parameter setting. To generate either cloaking or superscattering, we
use a small or a large quantum dot, respectively, as a reference to obtain the
desired function ${\bm{Q}}_{\text{desired}}$, subject to the average gate
potential $87.5$ meV. Initially, the various gate potentials are chosen
randomly from the two symmetric intervals defined by $|V_{i}|\in[75,100]$ meV.
Let the desired energy region be from $25$ meV to $70$ meV, which is
discretized into $200$ points, on which both the $\bm{Q}_{\text{desired}}$ and
$\hat{\bm{Q}}(\mathbf{V})$ functions are evaluated. The number $N$ of layers
in the graphene quantum dot depends on the design imperative, i.e., whether
cloaking or superscattering is to be realized. Given a task, once $N$ has been
determined, it is fixed so that training and inverse design are carried out on
the same neural-network architecture. After training, the error function in
(9) is a deterministic function, whose global minimum can be found by using,
e.g., the standard interior point method [75, 76]. We incorporate a gradient
descent procedure into the interior point method as it was established
previously that this can significantly improve the computation speed [28].

A detailed description of the steps involved in finding an optimal solution to
the inverse-design problem is presented in Fig. 3.

Four remarks are in order. First, using the trained neural network in Fig. 3
as a Dirac-equation solver has the advantage of being extremely computational
efficient. If the original Dirac equation were used for calculating the
scattering-efficiency $Q$ function and for finding the optimal set of gate
potentials, a vast number of combinations of the potentials would typically be
needed and, for each combination, the calculation of the $Q$ involves
calculating a large number of sophisticated mathematical functions. With the
neural network, the standard gradient-descent method can be invoked to find
the changes in the gate potentials, quickly yielding the optimal solution. For
each given set of input variables, all needed for the neural network is matrix
multiplications, which can be done extremely efficiently using well-developed
packages.

Second, compared with designing a multilayer spherical scattering structure in
optics where the input vector is chosen from a single interval of the physical
parameter (e.g., dielectric constant), for graphene the input gate potentials
are chosen from two symmetric intervals: one negative and another positive.
There is then a gap in the input parameter space. To our knowledge, this type
of inverse design has not been studied. Our solution is to test all possible
combinations for $V_{i}\in[-100,-75]\cup[75,100]$ meV. For each $V_{i}$, there
are two possibilities: positive or negative. For an $N$-layer circular quantum
dot, altogether we have $2^{N}$ possible combinations. Take $N=2$ as an
example. If we wish to find the best $V_{(1,2)}$ such that
$|V_{(1,2)}|\in[75,100]$ meV, there are four combinations: $V_{1}\in[75,100]$
meV $\cap$ $V_{2}\in[75,100]$ meV, $V_{1}\in[75,100]$ meV $\cap$
$V_{2}\in[-100,-75]$ meV, $V_{1}\in[-100,-75]$ meV $\cap$ $V_{2}\in[75,100]$
meV, and $V_{1}\in[-100,-75]$ meV $\cap$ $V_{2}\in[-100,-75]$ meV. For each
combination, an optimal solution can be found using one or two optimization
steps. We find that, for a larger value of $N$, the best scattering structure
can be found with one or two optimization steps for each combination of the
gate potentials. As a result, for any reasonable value of $N$ (e.g., $N=10$),
the required computational complexity is well manageable.

Third, to find the global minimum in (9), including derivatives can increase
the computational efficiency [28]. In particular, to optimize $\mathbf{V}$
from (9), we calculate the derivative
$\partial\text{Loss}/\partial\mathbf{V}$. Since $Q_{\text{desired}}$ is fixed,
it is only necessary to evaluate the derivative of $\hat{Q}(\mathbf{V})$ with
respect to $\mathbf{V}$. Once the neural network has been trained, such
derivative can be evaluated through back propagation.

Fourth, there is no guarantee that the “best” scattering configuration can be
found to make the scattering efficiency function match the desired function.
Empirically, we find that increasing the number $N$ of graphene layers
typically results in a smaller optimization loss. This issue will be addressed
in detail in Sec. IV.

##  III Results

###  III.1 Cloaking

The goal of designing a graphene-based cloaking system is to generate near-
zero scattering cross section or efficiency from a relatively large quantum
dot structure. Since a small structure tends to generate minuscule scattering
efficiency, the desired vector ${\bm{Q}}_{\text{desired}}$ can be found by
solving the relativistic quantum scattering problem over a small single-layer
structure, e.g., a circular cavity of radius $R_{C}=3$ nm subject to a gate
potential of $V=87.5$ meV, which produces near-zero scattering efficiency, as
shown in Fig. 1(b). Now consider a nine-layer ($N=9$) structure of size ten
times larger: $R_{N}=30$ nm. The question can be stated as: is it possible to
find a set of suitable gate potentials represented by input vector ${\bm{V}}$
such that the resulting scattering efficiency function is arbitrarily close to
${\bm{Q}}_{\text{desired}}$, subject to the constraint $\sum|V_{i}|/N\approx
87.5$ meV?

Figure 4(a) presents three curves of the resulting scattering efficiency
function: the desired function (dashed blue), the function from the optimal
multilayer graphene structure found by our physics-constrained neural network
(solid blue curve), and a training curve that is the nearest to the target
curve from the training dataset (dotted orange curve). The optimal physical
structure of the multilayer graphene quantum dot found by our physics-
constrained neural network is shown in the lower right inset, where the gate
potential values applied to different layers are specified by the color bar
with $\sum|V_{i}|/N=85.7$ meV. We see that the optimal structure produces a
scattering efficiency function that well approximates the desired function.
The purpose of the dotted orange curve is to demonstrate that the optimal
structure is not a trivial interpolation of some of structures in the training
dataset. Apparently, this “best” training function deviates from the desired
function as the energy increases. The structure found by the neural network is
thus one that produces real cloaking in that the scattering efficiency is as
small as that produced by a structure of ten times smaller in the entire
energy range of interest.

Figure 4: Optimal multilayer graphene structure exhibiting cloaking found by
machine learning. The parameter requirements of inverse design are $R_{N}=30$
nm and $\sum|V_{i}|/N\approx 87.5$ meV. (a) Comparison of three curves of
scattering efficiency: the desired function from a single-layer scatterer of
ten times smaller (dashed blue curve), the function underlying cloaking from
the neural network (solid blue curve), and a curve closest to the target curve
from the training dataset (dotted orange curve). The inset illustrates the
optimal quantum-dot structure predicted by machine learning. That the two blue
curves stay close to each other in the entire energy range is indicative of
cloaking, due to the much larger size of the neural-network produced
multilayer structure. (b) Spatial patterns of the spinor wavefunction and the
local currents associated with cloaking, with the quantum-dot structure at the
center and an incident plane wave from the left. The wavefunction intensity is
nearly constant (unity) in the relevant spatial domain and the current vectors
hardly change their directions, signifying cloaking. (c) Conventional
scattering patterns from a single-layer quantum dot of the same size. There is
no cloaking due to the significant changes in both the wavefunction intensity
and the current direction across the domain.

Intuitively, the emergence of cloaking from a large scattering structure
requires destructive quantum interference. If the gate potentials applied to
nearby layers have the same sign, it will be difficult to produce such
interference. However, alternating signs of the gate potentials between
neighboring layers can facilitate the occurrence of destructive interference.
The pattern of gate potentials of the optimal multilayer structure predicted
by the neural network indeed has this feature, as demonstrated in Fig. 4(a).

To visualize the cloaking phenomenon, we show the spatial distribution of the
scattering wavefunction. In particular, we compare two cases: the optimal
nine-layer structure found by the neural network and a single-layer quantum
dot of the same size ($R=30$ nm), in terms of the wavefunction intensity
defined as $|\psi_{1}|^{2}+|\psi_{2}|^{2}$, where $\psi_{1}$ and $\psi_{2}$
are the two spinor components and the directions of the local current vectors
are given by $[\langle\sigma_{x}\rangle,\langle\sigma_{y}\rangle]$. For a
planar incident wave of unit intensity from the left, if cloaking has indeed
occurred, the wavefunction intensity should be nearly constant in the spatial
domain of interest and the current vectors should not change their directions,
i.e., to maintain their horizontal directions. Such behaviors characteristic
of cloaking are exemplified in Fig. 4(b) for the optimal multilayer structure.
In contrast, for the single-layer structure where no cloaking occurs, the
wavefunction intensity varies across the domain and there are significant
changes in the directions of the local current vectors, as shown in Fig. 4(c).

For a graphene quantum-dot scatter with a weak scattering strength as measured
by $\bar{V}R$, where $\bar{V}$ is the average potential and $R$ is the dot
size, the scattering efficiency with the strength exhibits the following
scaling relation [56]: $Q\sim(\bar{V}R)^{3}$. The target quantum dot used for
generating the cloaking behavior has the size of 3 nm. According to this
scaling law, the scattering efficiency should be quite small. For a relatively
large circular structure of size, e.g., $10$ nm, the scattering efficiency
saturates and the scaling law no longer holds, but it can still be used to
obtain an order-of-magnitude estimate of the scattering efficiency. We find
that for a conventional structure of size $30$ nm without cloaking, the
scattering efficiency is typically two orders of magnitude larger than that
from a structure of size ten times smaller. It is remarkable that our physics-
constrained machine learning approach is able to predict a large quantum-dot
structure but with scattering efficiency two orders of magnitude smaller than
it “should be” in the conventional sense, effectively realizing cloaking.

###  III.2 Superscattering

Figure 5: Optimal multilayer graphene structure exhibiting superscattering
produced by machine learning. As for the case of cloaking in Fig. 4, the
structure has the size $R_{N}=30$ nm and the gate potentials satisfy the
constraint $\sum|V_{i}|/N\approx 87.5$ meV. (a) Comparison of three curves of
scattering efficiency: the desired function from a single-layer scatterer of
twice larger size (dashed blue curve), the function associated with
superscattering from machine learning (solid blue curve), and a training curve
closest to the target curve from the training dataset (dotted orange curve).
The upper right inset illustrates the optimal quantum-dot structure predicted
by the neural network. (b) Spatial patterns of the spinor wavefunction and the
local currents associated with superscattering at the energy $E=30$ meV, with
the quantum-dot structure at the center and an incident plane wave from the
left. There are significant variations in the wavefunction intensity and in
the directions of the local current vectors across the spatial domain of
interest. (c) Conventional scattering patterns from a single-layer quantum dot
of the same size.

We now address the opposite (to cloaking) problem: is it possible to design a
multilayer scattering structure of certain size to generate scattering
efficiency that conventionally would be achieved only with a much larger
scatterer? To be concrete, we use a nine-layer structure of size $R_{N}=30$ nm
with gate potentials subject to $\sum|V_{i}|/N\approx 87.5$ meV and generate
the target scattering efficiency function ${\bm{Q}}_{\text{desired}}$ using a
single-layer scatterer of size $R_{A}=63$ nm, as illustrated in Fig. 1(b). The
actual curve of ${\bm{Q}}_{\text{desired}}$ is shown by the dashed blue curve
in Fig. 5(a), where a scattering resonance arises at energy about $E=30$ meV.
Applying our machine-learning based inverse design algorithm, we obtain the
quantum-dot structure as illustrated in the upper right corner of Fig. 5(a)
with $\sum|V_{i}|/N=89.8$ meV, where the resulting scattering efficiency
function is represented by the solid blue curve that stays near the target
curve in the entire energy range. This means that the optimal structure of
size $30$ nm is able to produce scattering efficiency as large as that from a
structure of much larger size. For reference, the “best” scattering efficiency
function from the training dataset is shown (the dotted orange curve). The
optimal quantum-dot structure discovered by the neural network for
superscattering requires a negative gate potential for majority of the layers.
The spatial distribution of the wavefunction intensity and the local current
vectors associated with superscattering are shown in Fig. 5(b), and the same
quantities from a single-layer scatterer of the same size is shown in Fig.
5(c). It can be seen that scattering from the optimal multilayer structure is
markedly enhanced. An advantage of our machine-learning design is that no
additional training of the neural network is required here, insofar as it has
been trained for cloaking with a predefined set of scattering configurations,
each with a distinct scattering efficiency curve. That is, the same neural
network used to realize cloaking can be used to find a system that exhibits
superscattering.

Comparing Fig. 5 with Fig. 4, we see that the effect of scattering enhancement
is not as pronounced as that associated with scattering suppression in
cloaking. The physical reason is Klein tunneling. In particular, the median
Fermi energy is about half of the maximum gate potential: $E\approx|V|/2$, for
which Klein tunneling is significant in graphene [55]. As a result, a spinor
wave tends to change little in its wavevector after entering the scattering
region. To generate strong cloaking in graphene is thus much more likely than
to produce superscattering. We note that Klein tunneling is a unique feature
of Dirac waves.

##  IV Issues pertinent to inverse design

###  IV.1 Effect of the number of quantum-dot layers

Physically, the objective of machine-learning based inverse design is to find
a multilayer quantum dot structure to generate cloaking or superscattering by
tuning a set of experimentally feasible, external parameters. From a
mathematical point of view, the task is to find a complicated multilayer
scattering structure to achieve a desired function. While increasing the
number of layers in general allows more complicated scattering functions, the
computational complexity increases as well. From the point of view of machine
learning, a more sophisticated scattering structure requires more training
data.

Figure 6(a) shows, when designing a cloaking device, the relative training and
inverse-design errors versus $N$, where the shaded area denotes all possible
error values from training, with the solid orange curve denoting the minimally
possible training error. Physically, a circular quantum dot of too few layers
is unable to lead to cloaking, so the errors are large for small $N$ values.
As $N$ increases, it is more likely to generate destructive interference to
realize cloaking, leading to a decrease in the errors. The solid blue curve in
Fig. 6(a) represents the error in the scattering function from the optimal dot
structure found by inverse design, which is below the minimum training-error
curve. That is, the neural network is able to find a quantum-dot structure
whose scattering function is closer to the desired one than any function in
the training dataset. A similar behavior arises for the task of generating
superscattering, as shown in Fig. 6(b).

The results in Fig. 6 indicate that there is no guarantee that an increase in
the number of layers can lead to better inverse design. In our work, the set
of radii of the graphene quantum dot is fixed to be $R_{i}=iR_{N}/N$. Suppose
$N$ is increased by some integer factor. The resulting quantum dot would lead
to a better chance of realizing scattering possibilities but the structure
becomes more sophisticated. In general, the performance of the new structure
should be improved, but empirically this is true only when the numbers of
layers of the two structures are commensurate. From another angle, a
perturbation on the gate potential of a graphene layer will affect the
resulting error in the scattering efficiency but, as will be shown below in
Fig. 7, and modifying the gate potential of an outer layer has a stronger
effect on augmenting the error than disturbing the potential of an inner
layer. This suggests that the non-monotonous behavior in the error be an
intrinsic physical property of the scattering system that has little
dependence on the machine learning procedure.

Due to the physical nature of the error behavior, choosing the “best” number
of graphene layers is difficult. In fact, the same problem arises in optical
inverse-design problems. For example, in Refs. [28, 39], it was necessary to
pre-assign the number of neural-network layers. A mathematical difficulty is
that this number is not a continuous variable. In Refs. [77, 78],
reinforcement learning was used to find the best device whose dimension is
only allowed to take on some discrete value. Another difficulty is that the
multilayer perceptron employed in our work is suitable for inputs with a fixed
dimension. In the field of computer vision, a method called Spatial Pyramid
Pooling [79] was proposed to address this difficulty. The idea was extended to
adaptive pooling layers in PyTorch [80]. While for our quantum-dot inverse-
design problem, increasing the number of layers can lead to better performance
in most cases, to the best of our knowledge to predetermine the optimal number
of layers before training remains to be an open problem.

Figure 6: Effect of the number of quantum-dot layers on training and inverse-
design errors. (a) For designing a cloaking device, the errors versus $N$,
where the shaded region indicates the normalized mean square error from the
training dataset and the solid blue curve represents the error between the
scattering function from the optimal structure and the desired function. As
the number of quantum-dot layers increases, the errors decrease. (b) Similar
behaviors arise when designing a superscattering device.

###  IV.2 Robustness of inverse design

The main idea underlying machine-learning based inverse design is that a
properly trained neural network is able to generate an approximation of a
desired mathematical function. In general, the optimal physical structure
found by machine learning is close to but not necessarily the global minimum
of the loss function Eq. (9). It is useful to examine how “far away” the loss
associated with the optimal structure is from the global minimum. Consider the
“best” quantum-dot structures for cloaking and superscattering design, as
shown in Figs. 4 and 5, respectively, where the structure contains nine layers
so the loss function depends on nine gate-potential variables and is thus
nine-dimensional. Suppose the optimal structure is described by the following
set of $N$ potential values: $[V_{1},V_{2},\cdots,V_{N}]$. We apply a small
perturbation e.g., $V_{1}$, which leads to a new set of potentials:
$[V_{1}\delta,V_{2},\cdots,V_{N}]$, and examine the dependence of the loss
function on $\delta$. If the original solution $[V_{1},V_{2},\cdots,V_{N}]$
corresponds to the global minimum, the loss function versus $\delta$ should
exhibit a minimum at $\delta=1$. Figure 7(a) shows the loss
$\mathcal{L}^{(i)}$ versus $\delta$ for different layer index $i$. The
functions are approximately parabolic with its minimum close to $\delta=1$ but
not exactly at $\delta=1$, because both the neural network and the
optimization algorithm are approximate solvers and the resulting scattering
structure is not the absolute best. When the potential change occurs on one of
the innermost layers (corresponding to small layer indices), the parabolic
functions are rather shallow, but they are steep for large layer indices,
indicating that modifying the gate potential of an outer layer has a stronger
effect on augmenting the error. A plausible reason is that an outer layer has
a larger area than that of an inner layer for a fixed layer width.

Figure 7: Accuracy and robustness of inverse design. (a) For a graphene
quantum dot of $N=9$ layers to realize cloaking, normalized mean square error
in the machine generated scattering efficiency function versus the
perturbation parameter $\delta$ in the gate potential as applied to different
layers of the graphene quantum dot, where $\delta=1$ corresponds to zero
perturbation. The error is calculated from the exact solution of the Dirac
equation. The ideal global minimum of the loss function should occur at
$\delta=1$. The actual minima are quite close to $\delta=1$. (b) Histogram of
the relative error in the machine-generated scattering efficiency function
from an ensemble of $10^{4}$ combinations of the input parameters. More than
$80\%$ of times, the global minima are within $3\%$ of $\delta=1$, indicating
robustness of the inverse-design scheme. The inset shows two examples of the
scattering efficiency function from the machine-predicted optimal dot
structure, with the relative error of $3\%$ and $5\%$, respectively, where the
desired function is indicated by the dashed curve.

As discussed in Sec. II.3, in comparison with the inverse problem of
electromagnetic wave scattering, a difficulty in solving the inverse problem
of quantum scattering in graphene is that the input parameters can be selected
from a positive and a negative interval, leading to a gap in the between the
two allowed parameter intervals. Implementing an optimization algorithm, e.g.,
the interior point method, requires examining a large number of parameter
combinations from the two intervals. For an $N$-layer graphene quantum dot,
the optimization needs to be done $2^{N}$ times, requiring that the algorithm
be computationally efficient. In general, even based on a fixed nonlinear
function, the optimization result would depend on the initial condition.
Figure 7(b) shows, for $N=9$ and cloaking design, the statistical distribution
of the relative error of the optimization algorithm from $10^{4}$ different
initial conditions. It can be seen that for more than $80\%$ of the initial
conditions, the optimization errors are about $3\%$, indicating the robustness
of our machine-learning based inverse-design scheme.

###  IV.3 Computational time required for training and convergence

Figure 8: Computational time required for training and convergence. The data
points associated with the left and right $y$-axis represent, respectively,
the average time (with variance) of the training time and the number of
iterations required for the neural network to converge as the number of layers
constituting the quantum-dot structure increases from four to ten. The
training time increases linearly with $N$ but the number of iterations changes
little due to the fact that the number of parameters in the neural network is
approximately invariant as $N$ increases in this range.

Two main factors determine the time required for training: the amount of
training data and the number of training parameters. Empirically, the amount
of training data is proportional to the number $N$ of the layers constituting
the quantum-dot structure. With regard to the training parameters, our neural
network contains four hidden layers. For different values of $N$, the number
of neural-network parameters remains approximately constant. Figure 8
exemplifies how the average training time and variance depend on $N$. With CPU
i7-6850k without GPU acceleration, the average training time ranges from $100$
seconds to $400$ seconds as the number of layers of the scattering structure
increases from four to ten.

About the convergence of the training process, we observe that the training
error depends on the initial condition for the neural network and data
sampling, as demonstrated in Fig. 2(c). However, the variance in the error is
small, indicating that convergence is always granted for training.

The computational speed for the inverse design depends on two factors: the
number of iterations required to achieve convergence and the time cost of each
iteration. Because the number of parameters involved in the neural network
does not change significantly with $N$, for each iteration approximately a
similar amount of time is required for calculating the gradient for different
values of $N$. As $N$ increases, while more iterations are needed for
optimizing the neural network, we find that the required increase in the
number of iterations is insignificant. For example, as $N$ increases from four
to ten, the average number of iterations required increases only from about 21
to about 23, as shown in Fig. 8 as well as in Fig. 7(b) where more than $80\%$
of the initial conditions lead to satisfactory convergence.

##  V Discussion

To develop quantum systems to realize drastically distinct functions under a
different set of externally imposed parameter values is a challenging problem.
For example, for a multilayer graphene quantum-dot system in which each layer
is subject to a gate potential, one may wish to make the system “invisible” by
generating cloaking with weak scattering under one set of gate potentials, but
as the environment changes the opposite extreme may be desired: strong
scattering (or superscattering defined in a broad sense). Would it be possible
to simply change the set of external gate potentials, while keeping the
physical structure of the system intact, so as to induce a metamorphic
transition in the quantum scattering dynamics from one extreme to the opposite
extreme? In optics, if the goal is to realize one of the two extremes, then
the method of scattering cancellation can be effective [33, 36, 46, 34]. The
challenge is whether it is possible to create a system that is capable of the
two extremes of scattering - superscattering and cloaking, in certain energy
range by tuning a set of experimentally adjustable parameters. In principle,
by exploiting a multilayer scattering structure to induce unusually strong
destructive or constructive quantum interference, it should be possible to
meet the challenge. However, the computational complexity associated with a
brute-force search of the parameter space, aided by the exact solution of the
Dirac equation, tends to grow exponentially with the number of layers
constituting the scatterer, making the problem NP hard.

The main idea underlying this work is that machine learning provides an
effective and computationally efficient approach to solving the challenging
inverse-design problem, i.e., to generate the two opposite extremes of quantum
scattering by tuning a set of experimentally accessible parameters. For the
concrete setting of a multiple-ring graphene quantum dot structure, the basic
component of the machine learning architecture is a conventional multilayer,
feed forward neural network that takes a set of gate potentials (one applied
to each ring layer of the quantum dot) as input and generate the scattering
cross section or efficiency versus the Fermi energy (scattering function) as
the output. Straightforwardly training this baseline neural-network
architecture using an adequate number of pre-determined scattering functions
obtained from directly solving the Dirac equation reveals that a non-
negligible fraction of the output functions can be nonphysical in that they
contain negative values of the scattering efficiency. Imposing the physical
constraint that the scattering efficiency must be positive through an
additional output layer leads to a class of physics-constrained neural
networks. The main accomplishment of this paper is a successful demonstration
that physics-constrained machine learning can effectively solve the inverse
design problem of generating quantum scattering dynamics at the opposite
extremes in the same system.

The key advantage and appealing feature of our machine-learning approach to
the extreme inverse-design problem can be appreciated using the paradigmatic
multilayer graphene quantum dot structure studied in this paper. Given a
desired scattering function, the starting point is a random set of gate
potentials. It is practically impossible for this initial set of input
parameter values to produce a scattering function closed to the desired one,
so adjustments to the inputs are necessary. Solving the Dirac equation for
random parameter perturbations is computationally infeasible not only because
the solutions of the Dirac equation require evaluating a large number of
sophisticated mathematical functions but, more importantly, because of the NP-
hard nature of the search problem. These fundamental difficulties can be
overcome by exploiting machine learning for the following reasons: (1) all
mathematical operations needed is matrix multiplication that can be done
extremely efficiently and (2) the standard gradient descent can guild the
solution in the input-parameter space to approach the optimal one in only a
few steps. We have demonstrated that, typically, the optimal solution is close
to the global minimum of the physics-constrained loss function.

In our present work, we have studied quantum-dot systems in the weak
scattering regime, where the wavevector inside the cavity has the same order
of magnitude as the wavevector outside the cavity. In this case, the
scattering efficiency does not exhibit sharp resonant peaks, in contrast to
the case of strong resonances where the wavevector inside the cavity is much
larger than the wavevector outside [81]. Empirically, a change in the
potential profile requires that the neural network be retrained.

In general, inverse problems in physical sciences are challenging, but to
design a system that can generate physical behaviors of two opposite extremes
is even harder. We emphasize that no feasible conventional methods exist which
can be used to solve this type of extreme inverse problems due to the
following difficulties: the need to search through a high-dimensional
parameter space, the complex relation between the system structure and
functions, and the uncertainty about the existence of a configuration that can
deliver the desired system functions. Our machine-learning based solution
represents a step forward in this area.

## Acknowledgment

We thank Dr. H.-Y. Xu for providing great inputs during the initial stage of
this project. This work was supported by AFOSR under Grant No.
FA9550-21-1-0438.

##  Appendix A Scattering cross section for circular graphene quantum dot

###  A.1 Single-layer circular graphene quantum dot

Consider a single-layer graphene quantum dot of radius $R_{1}$, subject to
gate potential $V_{1}$, where the circular boundary separates the whole space
into two regions: inside ($r<R_{1}$) and outside ($r>R_{1}$) the dot, denoted
as regions 1 and 2, respectively. The energies inside and outside the dot are
$E-V$ and $E$, respectively, giving the corresponding wavevectors as
$k_{1}=|E-V|/v_{g}$ and $k_{2}=|E|/v_{g}$. For convenience, we define two sign
quantities: $\tau_{1}\equiv\text{Sign}(E-V)$ and
$\tau_{2}\equiv\text{Sign}(E)$.

The incident wave is

$\psi_{\text{in}}(\mathbf{r})=\frac{1}{\sqrt{2}}\binom{1}{\tau_{2}}e^{ik_{2}x}.$
(10)

Using the Jacobi-Anger identity

$e^{iz\cos\theta}\equiv\sum_{l=-\infty}^{\infty}i^{l}J_{l}(z)e^{il\theta},$
(11)

we expand the incident wave in the polar coordinates as

$\psi_{\text{in}}(\mathbf{r})=\sum_{l=-\infty}^{\infty}\frac{i^{l}}{\sqrt{2}}\binom{J_{l}(k_{2}r)}{i\tau_{2}J_{l+1}(k_{2}r)}e^{il\theta}.$
(12)

Inside the quantum dot, we have

$\psi_{1}(\mathbf{r})=\sum_{l=-\infty}^{\infty}A^{(1)}_{l}\binom{J_{l}(k_{1}r)}{i\tau_{1}J_{l+1}(k_{1}r)}e^{il\theta},$
(13)

and the wavefunction outside the dot is given by

$\psi_{2}(\mathbf{r})=\psi_{\text{in}}+\sum_{l=-\infty}^{\infty}A^{(2)}_{l}\binom{H^{(1)}_{l}(k_{2}r)}{i\tau_{2}H^{(1)}_{l+1}(k_{2}r)}e^{il\theta}.$
(14)

Matching the boundary conditions for each angular momentum $l$, we get

$\begin{split}\begin{bmatrix}J_{l}(k_{1}R_{1})&-H_{l}^{(1)}(k_{2}R_{1})\\\
\tau_{1}J_{l+1}(k_{1}R_{1})&-\tau_{2}H_{l+1}^{(1)}(k_{2}R_{1})\end{bmatrix}\begin{pmatrix}A^{(1)}_{l}\\\
A^{(2)}_{l}\end{pmatrix}=\frac{1}{\sqrt{2}}\begin{pmatrix}i^{l}J_{l}(k_{2}R_{2})\\\
\tau_{2}i^{l}J_{l+1}(k_{2}R_{2})\end{pmatrix}.\end{split}$ (15)

###  A.2 Multilayer circular graphene quantum dot

Consider an $N$-layer circular graphene quantum dot, where all the layers are
concentric and have radii $R_{1}$, $R_{2}$, $\cdots$, and $R_{N}$ with the
respective gate potentials $V_{1}$, $V_{2}$, $\cdots$, and $V_{N}$. Let
$k_{i}=|E-V_{i}|/v_{g}$ and $\tau_{i}\equiv\text{Sign }(E-V_{i})$. In region
2, we have $k_{N+1}=|E|/v_{g}$ and $\tau_{N+1}=\text{Sign}(E)$. To represent
the wavefunction in each region, we use $A^{(1)}$ for $r<R_{1}$ and

$\psi_{1}(\mathbf{r})=\sum_{l=-\infty}^{\infty}A^{(1)}_{l}\binom{J_{l}(k_{1}r)}{i\tau_{1}J_{l+1}(k_{1}r)}e^{il\theta},$
(16)

where $r<R_{1}$. For $R_{i}<r<R_{i+1}$ we have

$\begin{split}\psi_{i+1}(\mathbf{r})=\sum_{l=-\infty}^{\infty}A^{(i+1)}_{l}\binom{H_{l}^{(1)}(k_{i+1}r)}{i\tau_{i+1}H_{l+1}^{(1)}(k_{i+1}r)}e^{il\theta}+\sum_{l=-\infty}^{\infty}B^{(i+1)}_{l}\binom{H_{l}^{(2)}(k_{i+1}r)}{i\tau_{i+1}H_{l+1}^{(2)}(k_{i+1}r)}e^{il\theta}.\end{split}$
(17)

For $r>R_{N}$ we have

$\begin{split}\psi_{N+1}(\mathbf{r})=\sum_{l=-\infty}^{\infty}\frac{i^{l}}{\sqrt{2}}\binom{J_{l}(k_{N+1}r)}{i\tau_{N+1}J_{l+1}(k_{N+1}r)}e^{il\theta}+\sum_{l=-\infty}^{\infty}A^{(N+1)}_{l}\binom{H_{l}^{(1)}(k_{N+1}r)}{i\tau_{N+1}H_{l+1}^{(1)}(k_{N+1}r)}e^{il\theta}.\end{split}$
(18)

The unknown coefficients are $A_{l}^{(1)}$, $A_{l}^{(2)}$, $\cdots$,
$A_{l}^{N}$ and $A_{l}^{(N+1)}$ as well as $B_{l}^{(2)}$, $B_{l}^{(3)}$,
$\cdots$, and $B_{l}^{(N)}$. Altogether, there are $2N$ unknown parameters to
be determined. We use the $N$ boundary conditions to generate $2N$ equations,
as follows. For the boundary at $r=R_{1}$, we have

$\begin{split}A^{(1)}_{l}J_{l}(k_{1}R_{1})-A^{(2)}_{l}H_{l}^{(1)}(k_{2}R_{1})-B^{(2)}_{l}H_{l}^{(2)}(k_{2}R_{1})&=0,\\\
A^{(1)}_{l}\tau_{1}J_{l+1}(k_{1}R_{1})-A^{(2)}_{l}\tau_{2}H_{l+1}^{(1)}(k_{2}R_{1})-B^{(2)}_{l}\tau_{2}H_{l+1}^{(2)}(k_{2}R_{1})&=0.\\\
\end{split}$ (19)

For the boundary at $r=R_{i}$, where $1<i<N$, we have

$\begin{split}A^{(i)}_{l}H_{l}^{(1)}(k_{i}R_{i})+B^{(i)}_{l}H_{l}^{(2)}(k_{i}R_{i})-A^{(i+1)}_{l}H_{l}^{(1)}(k_{i+1}R_{i})-B^{(i+1)}_{l}H_{l}^{(2)}(k_{i+1}R_{i})&=0,\\\
A^{(i)}_{l}\tau_{i}H_{l+1}^{(1)}(k_{i}R_{i})+B^{(i)}_{l}\tau_{i}H_{l+1}^{(2)}(k_{i}R_{i})-A^{(i+1)}_{l}\tau_{i+1}H_{l+1}^{(1)}(k_{i+1}R_{i})-B^{(i+1)}_{l}\tau_{i+1}H_{l+1}^{(2)}(k_{i+1}R_{i})&=0.\end{split}$
(20)

For the boundary at $r=R_{N}$, we have

$\begin{split}A^{(N)}_{l}H_{l}^{(1)}(k_{N}R_{N})+B^{(N)}_{l}H_{l}^{(2)}(k_{N}R_{N})-A^{(N+1)}_{l}H_{l}^{(1)}(k_{N+1}R_{N})&=\frac{i^{l}}{\sqrt{2}}J_{l}(k_{N+1}R_{N}),\\\
A^{(N)}_{l}\tau_{N}H_{l+1}^{(1)}(k_{N}R_{N})+B^{(N)}_{l}\tau_{N}H_{l+1}^{(2)}(k_{N}R_{N})-A^{(N+1)}_{l}\tau_{N+1}H_{l+1}^{(1)}(k_{N+1}R_{N})&=\frac{i^{l}}{\sqrt{2}}\tau_{N+1}J_{l+1}(k_{N+1}R_{N}).\end{split}$
(21)

Solving this set of linear equations, we get the coefficients $A^{(N+1)}$. The
scattering efficiency is given by

$Q=\frac{2}{k_{N+1}R_{N}}\sum_{l=-\infty}^{\infty}\left(A^{(N+1)}_{l}\right)^{2}.$
(22)

In numerical computations, we truncate the summation at $\max|l|=7$ to achieve
the desired accuracy.

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