11institutetext: Paderborn University, Warburger Str. 100, 33098 Germany
11email<EMAIL_ADDRESS>
https://www.uni-paderborn.de/en/person/85279

# Learning discrete Lagrangians for variational PDEs from data and detection
of travelling waves

Christian Offen ${\href https://orcid.org/0000-0002-5940-8057}$ Sina Ober-
Blöbaum

###### Abstract

The article shows how to learn models of dynamical systems from data which are
governed by an unknown variational PDE. Rather than employing reduction
techniques, we learn a discrete field theory governed by a discrete Lagrangian
density $L_{d}$ that is modelled as a neural network. Careful regularisation
of the loss function for training $L_{d}$ is necessary to obtain a field
theory that is suitable for numerical computations: we derive a regularisation
term which optimises the solvability of the discrete Euler–Lagrange equations.
Secondly, we develop a method to find solutions to machine learned discrete
field theories which constitute travelling waves of the underlying continuous
PDE.

###### Keywords:

System identification discrete Lagrangians travelling waves

#### Publication information

For the published version of the article refer to [15].

##  1 Introduction

In data-driven system identification, a model is fitted to observational data
of a dynamical system. The quality of the learned model can greatly be
improved when prior geometric knowledge about the dynamical system is taken
into account such as conservation laws [16, 14, 5, 9, 2], symmetries [10, 8,
7], equilibrium points [19], or asymptotic behaviour of its motions.

One of the most fundamental principles in physics is the variational
principle: it says that motions constitute stationary points of an action
functional. The presence of variational structure is related to many
qualitative features of the dynamics such as the validity of Noether’s
theorem: symmetries of the action functional are in correspondence with
conservation laws. To guarantee that these fundamental laws of physics hold
true for learned models, Greydanus et al propose to learn the action
functional from observational data [5] (Lagrangian neural network) and base
prediction on numerical integrations of Euler–Lagrange equations. Quin
proposes to learn a discrete action instead [18]. An ansatz of a discrete
model has the advantage that it can be trained with position data of motions
only. In contrast, learning a continuous theory typically requires information
about higher derivatives (corresponding to velocity, acceleration, momenta,
for instance) which are typically not observed but only approximated.
Moreover, the discrete action functional (once it is learned) can naturally be
used to compute motions numerically.

However, in [14] the authors demonstrate that care needs to be taken when
learned action functionals are used to compute motions: even if the data-
driven action functional is perfectly consistent with the training data (i.e.
the machine learning part of the job is successfully completed), minimal
errors in the initialisation process of numerical computations get amplified.
As a remedy, the authors develop Lagrangian Shadow Integrators which mitigate
these amplified numerical errors based on a technique called backward error
analysis. Moreover, using backward error analysis they relate the discrete
quantities to their continuous analogues and show how to analyse qualitative
aspects of the machine learned model. Action functionals are not uniquely
determined by the motions of a dynamical system. Therefore, regularisation is
needed to avoid learning degenerate theories. While in [14] the authors
develop a regularisation strategy when the action functional is modelled as a
Gaussian Process, Lishkova et al develop a corresponding regularisation
technique for artificial neural networks in [10] in the context of ordinary
differential equations (ODEs).

In this article we show how to learn a discrete action functional from
discrete data which governs solutions to partial differential equations (PDEs)
using artificial neural networks extending the regularisation strategy which
we have developed in [10]. Our technique to learn (discrete) densities of
action functionals can be contrasted to approaches where a spatial
discretisation of the problem is considered first, followed by structure-
preserving model reduction techniques (data-driven or analytical) [3, 4] and
then a model for the reduced system of ODEs is learned from data [20, 12, 1].

Travelling waves solutions of PDEs are of special interest due to their simple
structure. When a discrete field theory for a continuous process described by
a PDE is learned, they typically "get lost" because the mesh of the discrete
theory is incompatible with certain wave speeds. In this article, we introduce
a technique to find the solutions of data-driven discrete theories that
correspond to travelling waves in the underlying continuous dynamics (shadow
travelling waves). The article contains the following novelties:

  * •

We transfer our Lagrangian ODE regularising strategy [10] to data-driven
discrete field theories in a PDE setting and provide a justification using
numerical analysis.

  * •

The development of a technique to detect travelling waves in data-driven
discrete field theories.

The article proceeds with a review of variational principles (Section 2), an
introduction of our machine learning architecture and derivation of the
regularisation strategy (Section 3). In Section 4, we define the notion of
shadow travelling waves and show how to find them in data-driven models. The
article concludes with numerical examples relating to the wave equation
(Section 5).

##  2 Discrete and continuous variational principles

###  2.0.1 Continuous variational principles

Many differential equations describing physical phenomena such as waves, the
state of a quantum system, or the evolution of a relativistic fields are
derived from a variational principle: solutions are characterised as critical
points of a (non-linear) functional $S$ defined on a suitable space of
functions $u\colon X\to\mathbb{R}^{d}$ and has the form

$S(u)=\int_{X}L(\bm{x},u(\bm{x}),u_{x_{0}}(\bm{x}),u_{x_{1}}(\bm{x}),\ldots,u_{x_{n}}(\bm{x}))\mathrm{d}\bm{x},$
(1)

where $\bm{x}=(x_{0},x_{1},\ldots,x_{n})\in X$ and $u_{x_{j}}$ denote partial
derivatives of $u$. This variational principle can be referred to as a first-
order field theory, since only derivatives to the first order of $u$ appear.
In many applications the free variable $x_{0}$ corresponds to time and is
denoted by $x_{0}=t$. The functional $S$ is stationary at $u$ with respect to
all variations $\delta u\colon X\to\mathbb{R}$ vanishing at the boundary (or
with the correct asymptotic behaviour) if and only if the Euler–Lagrange
equations

$0=\mathrm{EL}(L)=\frac{\partial L}{\partial
u}-\sum\nolimits_{j=0}^{n}\frac{\mathrm{d}}{\mathrm{d}x_{j}}\frac{\partial
L}{\partial u_{x_{j}}}$ (2)

are fulfilled on $X$.

###### Example 1

The wave equation

$u_{tt}(t,x)-u_{xx}(t,x)+\nabla V(u(t,x))=0$ (3)

is the Euler–Lagrange equation $0=\mathrm{EL}(L)$ to the Lagrangian

$L(u,u_{t},u_{x})=\frac{1}{2}(u_{t}^{2}-u_{x}^{2})-V(u).$ (4)

Here $\nabla V$ denotes the gradient of a potential $V$.

###### Remark 1

Lagrangians are not uniquely determined by the motions of a dynamical system:
two first order Lagrangians $L$ and $\tilde{L}$ yield equivalent
Euler–Lagrange equations if $sL-\tilde{L}$ ($s\in\mathbb{R}\setminus\\{0\\}$)
is a total divergence $\nabla_{\bm{x}}\cdot
F(\bm{x},u(\bm{x}))=\frac{\partial}{\partial
x_{1}}(F^{1}(\bm{x},u(\bm{x})))+\ldots+\frac{\partial}{\partial
x_{n}}F^{n}(\bm{x},u(\bm{x}))$ for $F=(F^{1},\ldots,F^{n})\colon
X\times\mathbb{R}^{d}\to\mathbb{R}^{n}$.

###  2.0.2 Discrete variational principle

For simplicity, we consider the two dimensional compact case: let
$X=[0,T]\times[0,l]/\\{0,l\\}$ with $T,l>0$. Here $[0,l]/\\{0,l\\}$ is the
real interval $[0,l]$ with identified endpoints (periodic boundary
conditions). Consider a uniform, rectangular mesh $X_{\Delta}$ on $X$ with
mesh widths $\Delta t=\frac{T}{N}$ and $\Delta x=\frac{l}{M}$ for
$N,M\in\mathbb{N}$. A discrete version of the action functional (1) is

$S_{d}\colon(\mathbb{R}^{d})^{(N-1)\times M}\to\mathbb{R},\quad
S_{d}(U)=\Delta t\Delta
x\sum_{i=1}^{N-1}\sum_{j=0}^{M-1}L_{d}(u_{j}^{i},u_{j}^{i+1},u_{j+1}^{i})$

for a discrete Lagrangian density
$L_{d}\colon(\mathbb{R}^{d})^{3}\to\mathbb{R}$ together with temporal boundary
conditions for $u^{0}_{j}\in\mathbb{R}^{d}$ and $u^{N}_{j}\in\mathbb{R}^{d}$
for $j=0,\ldots,M-1$. Above $U$ denotes the values $(u^{i}_{j})_{j=0,\ldots
N-1}^{i=1,\ldots M-1}$ on inner mesh points. We have $u^{i}_{M}=u^{i}_{0}$ by
the periodicity in space. Solutions of the variational principle are
$U\in(\mathbb{R}^{d})^{(N-1)\times M}$ such that $U$ is a critical point of
$S_{d}$. This is equivalent to the condition that for all $i=1,\ldots,N-1$ and
$j=0,\ldots,M-1$ the discrete Euler–Lagrange equations

$\begin{split}\frac{\partial}{\partial
u^{i}_{j}}\big{(}L_{d}(u^{i}_{j},u^{i+1}_{j},u^{i}_{j+1})+L_{d}(u^{i-1}_{j},u^{i}_{j},u^{i-1}_{j+1})+L_{d}(u^{i}_{j-1},u^{i+1}_{j-1},u^{i}_{j})\big{)}=0\end{split}$
(5)

are fulfilled. The expression on the left of (5) is abbreviated as
$\mathrm{DEL}(L_{d})^{i}_{j}(U)$ in the following.

###### Remark 2

Instead of periodic boundary conditions in space, $S_{d}$ can be adapted to
other types of boundary conditions such as Dirichlet- or Neumann conditions.

###### Example 2

The discretised wave equation

$\frac{(u^{i-1}_{j}-2u^{i}_{j}+u^{i+1}_{j})}{\Delta
t^{2}}-\frac{(u^{i}_{j-1}-2u^{i}_{j}+u^{i}_{j+1})}{\Delta x^{2}}+\nabla
V(u^{i}_{j})=0$ (6)

is the discrete Euler–Lagrange equations to the discrete Lagrangian

$L_{d}(u^{i}_{j},u^{i+1}_{j},u^{i}_{j+1})=\frac{1}{2}\left(\frac{u^{i+1}_{j}-u^{i}_{j}}{\Delta
t^{2}}\right)^{2}-\frac{1}{2}\left(\frac{u^{i}_{j+1}-u^{i}_{j}}{\Delta
x^{2}}\right)^{2}-V(u^{i}_{j}).$

###### Remark 3

In analogy to Remark 1, notice that $L_{d}$ and $\tilde{L}_{d}$ yield the same
discrete Euler–Lagrange equations (5) if

$L_{d}(a,b,c)-s\tilde{L}_{d}(a,b,c)=\chi_{1}(a)-\chi_{1}(b)+\chi_{2}(a)-\chi_{2}(c)+\chi_{3}(b)-\chi_{3}(c)$
(7)

for differentiable functions $\chi_{1},\chi_{2},\chi_{3}\colon X\to\mathbb{R}$
and $s\in\mathbb{R}\setminus\\{0\\}$.

###### Remark 4

If $\mathrm{DEL}(L_{d})^{i}_{j}(U)=0$ (see (5)) and if
$\frac{\partial^{2}L_{d}}{\partial u^{i}_{j}\partial
u^{i+1}_{j}}(u^{i}_{j},u^{i+1}_{j},u^{i}_{j+1})$ is of full rank, then (5) is
solvable for $u^{i+1}_{j}$ as a function of $u^{i}_{j}$, $u^{i}_{j+1}$,
$u^{i-1}_{j}$, $u^{i-1}_{j+1}$, $u^{i}_{j-1}$, $u^{i+1}_{j-1}$ by the implicit
function theorem locally around a solution of (5). All of these points
correspond to mesh points that either lie to the left or below the point with
indices $(i,j)$. If $u^{1}_{j}$ is known for $0\leq j\leq M-1$, then utilising
the boundary conditions $u^{0}_{j}\in\mathbb{R}^{d}$ and $u^{i}_{0}=u^{i}_{M}$
we can compute $U$ by subsequently solving (5). This corresponds to the
computation of a time propagation.

The following Proposition analyses the convergence of Newton-Iterations when
solving (5) for $u^{i+1}_{j}$, as is required to compute time propagations. It
introduces a quantity $\rho^{\ast}$ that relates to how well the iterations
converge. We will make use of this quantity in the design of our machine
learning framework.

###### Proposition 1

Let $u^{i}_{j}$, $u^{i+1}_{j}$, $u^{i}_{j+1}$, $u^{i-1}_{j}$, $u^{i-1}_{j+1}$,
$u^{i}_{j-1}$, $u^{i+1}_{j-1}$ such that (5) holds. Let
$O\subset\mathbb{R}^{d}$ be a convex, neighbourhood of $u^{\ast}=u^{i+1}_{j}$,
$\|\cdot\|$ a norm of $\mathbb{R}^{d}$ inducing an operator norm on
$\mathbb{R}^{d\times d}$. Define $p(u):=\frac{\partial^{2}L_{d}}{\partial
u^{i}_{j}\partial u}(u^{i}_{j},u,u^{i}_{j+1})$ and let $\theta$ and
$\overline{\theta}$ be Lipschitz constants on $O$ for $p$ and for
$\mathrm{inv}\circ p$, respectively, where $\mathrm{inv}$ denotes matrix
inversion. Let

$\rho^{\ast}:=\left\|\mathrm{inv}(p(u^{\ast}))\right\|=\left\|\left(\frac{\partial^{2}L_{d}}{\partial
u^{i}_{j}\partial
u^{\ast}}(u^{i}_{j},u^{\ast},u^{i}_{j+1})\right)^{-1}\right\|$ (8)

and let $f(u^{(n)})$ denote the left hand side of (5) with $u_{j}^{i+1}$
replaced by $u^{(n)}$. If
$\|{u}^{(0)}-{u}^{\ast}\|\leq\min\left(\frac{\rho^{\ast}}{\overline{\theta}},\frac{1}{2\theta\rho^{\ast}}\right)$
for ${u}^{(0)}\in\mathcal{O}$, then the Newton Iterations
${u}^{(n+1)}:={u}^{(n)}-\mathrm{inv}(p(u^{(n)}))f(u^{(n)})$ converge
quadratically against ${u}^{\ast}$, i.e.

$\|{u}^{(n+1)}-{u}^{\ast}\|\leq\rho^{\ast}\theta\|{u}^{(n)}-{u}^{\ast}\|^{2}.$
(9)

###### Proof

The statement follows from an adaption of the standard estimates for Newton’s
method (see [6, §4], for instance) to the considered setting. A detailed proof
of the Proposition is contained in the Appendix (Preprint/ArXiv version only).

###### Remark 5

The assumptions formulated in Proposition 1 are sufficient but not sharp. The
main purpose of the proposition is to identify quantities that are related to
the efficiency of our numerical solvers and to use this knowledge in the
design of machine learning architectures.

##  3 Machine learning architecture for discrete field theories

We model a discrete Lagrangian $L_{d}$ as a neural network and fit its
parameters

  * •

such that the discrete Euler–Lagrange equations (5) for the learned $L_{d}$
are consistent with observed solutions $U=(u^{i}_{j})$ of (5)

  * •

and such that (5) is easily solvable for $u^{i+1}_{j}$ using iterative
numerical methods, so that we can use the discrete field theory to predict
solutions via forward propagation of initial conditions (see Remark 4).

For given observations $U^{(1)},\ldots,U^{(K)}$ with
$U^{(k)}=({u^{i}_{j}}^{(k)})$ on the interior mesh $X_{\Delta}$, we consider
the loss function $\ell=\ell_{\mathrm{DEL}}+\ell_{\mathrm{reg}}$ consisting of
a data consistency term $\ell_{\mathrm{DEL}}$ and a regularising term
$\ell_{\mathrm{reg}}$. We have

$\ell_{\mathrm{DEL}}=\sum_{k=1}^{K}\sum_{i=1}^{N-1}\sum_{j=0}^{M-1}\mathrm{DEL}(L_{d})^{i}_{j}(U^{(k)})^{2}$
(10)

with $\mathrm{DEL}(L_{d})^{i}_{j}$ from (5). $\ell_{\mathrm{DEL}}$ measures
how well $L_{d}$ fits to the training data.

Since a discrete Lagrangian $L_{d}$ is not uniquely determined by the system’s
motions by Remark 3 (indeed, $L_{d}\equiv\mathrm{const}$ is consistent with
any observed dynamics), careful regularisation is required. Indeed, in [14] we
demonstrate in an ode setting that if care is not taken, then machine learned
models for $L_{d}$ can be unsuitable for numerical purposes and amplify errors
of numerical integration schemes. In view of Proposition 1, we aim to minimize
$\rho^{\ast}$ (see (8)) and define the regularisation term

$\ell_{\mathrm{reg}}=\sum_{k=1}^{K}\sum_{i=1}^{N-1}\sum_{j=0}^{M-1}\left\|\left(\frac{\partial^{2}}{\partial
u^{i}_{j}\partial
u^{i+1}_{j}}L_{d}\left({u^{i}_{j}}^{(k)},{u^{i+1}_{j}}^{(k)},{u^{i}_{j+1}}^{(k)}\right)\right)^{-1}\right\|^{2}.$
(11)

In our experiments, we use the spectral norm in (11), which is the operator
norm induced by the standard Eucledian vector norm on $\mathbb{R}^{d}$. Let
$A_{\mathrm{reg}}^{i,j,k}:=\frac{\partial^{2}}{\partial u^{i}_{j}\partial
u^{i+1}_{j}}L_{d}\left({u^{i}_{j}}^{(k)},{u^{i+1}_{j}}^{(k)},{u^{i}_{j+1}}^{(k)}\right)$,
i.e. the summands of $\ell_{\mathrm{reg}}$ are
$\|(A_{\mathrm{reg}}^{i,j,k})^{-1}\|^{2}=\frac{1}{\lambda_{\min}^{2}}$, where
$\lambda_{\min}$ is the singular value $\lambda_{\min}$ of
$A_{\mathrm{reg}}^{i,j,k}$ with the smallest absolute norm. If
$u^{i}_{j}\in\mathbb{R}^{d}$ with $d=1$, then
$\|(A_{\mathrm{reg}}^{i,j,k})^{-1}\|$ can be evaluated without problems.
Otherwise, $\lambda_{\min}^{2}$ is computed as the smallest eigenvalue of the
symmetric matrix $(A_{\mathrm{reg}}^{i,j,k})^{\top}A_{\mathrm{reg}}^{i,j,k}$.
The eigenvalue can be approximated by inverse matrix vector iterations [6, §5]
or computed exactly if the dimension $d$ is small.

##  4 Periodic travelling waves

For simplicity, we continue within the two-dimensional space time domain
$X=[0,T]\times[0,l]/\\{0,l\\}$ with periodic boundary conditions in space
introduced in Section 2.0.2. A periodic travelling wave (TW) of a pde on $X$
is a solution of the form $u(t,x)=f(x-ct)$ for $c\in\mathbb{R}$ and with
$f\colon[0,l]/\\{0,l\\}\to\mathbb{R}^{d}$ defined on the periodic spatial
domain. Due to their simple structure, TWs are important solutions to pdes.
While the defining feature of a TW is its symmetry $u(t+s,x+sc)=u(t,x)$ for
$s\in\mathbb{R}$, evaluated on a mesh $X_{\Delta}$, no such structure is
evident unless the quotient $c\Delta t/\Delta x$ is rational and $T$
sufficiently large. However, after a discrete field theory is learned defined
by its discrete Lagrangian $L_{d}$, it is of interest, whether the underlying
continuous PDE has TWs. As in [13] we define shadow travelling waves (TWs) of
(5) as solutions to the functional equation

$\begin{split}0=&\partial_{1}L_{d}(f(\xi),f(\xi-c\Delta t),f(\xi+\Delta x))\\\
+&\partial_{2}L_{d}(f(\xi+c\Delta t),f(\xi),f(\xi+c\Delta t+\Delta x))\\\
+&\partial_{3}L_{d}(f(\xi-\Delta x),f(\xi-c\Delta t-\Delta
x),f(\xi))\end{split}$ (12)

where $\partial_{j}L_{d}$ denotes the partial derivative of $L_{d}$ with
respect to its $j$th slot.

###### Example 3

A Fourier series ansatz for $f$ reveals that the discrete wave equation (6)
with potential $V(u)=\frac{1}{2}u^{2}$ away from resonant cases TWs are
$u(t,x)=f(x-c_{n}t)$ with

$f(\xi)=\alpha\sin(\kappa_{n}\xi)+\beta\cos(\kappa_{n}\xi),\;\kappa_{n}=\frac{2\pi
n}{l},\;n\in\mathbb{Z},\alpha,\beta\in\mathbb{R}$ (13)

and with wave speed $c_{n}$ a real solution of

$\cos(\kappa_{n}c_{n}\Delta t)=1-\frac{\Delta t^{2}}{2}+\frac{\Delta
t^{2}}{\Delta x^{2}}(\cos(\kappa_{n}\Delta x))-1).$ (14)

A contour plot for $n=1$ is shown to the left of Fig. 2.

###### Remark 6

The TW equation (12) inherits variational structure from the underlying PDE:
an application of Palais’ principle of criticality [17] of the action of
$(\mathbb{R},+)$ on the Sobolev space $H^{1}(X_{c},\mathbb{R})$ with
$X_{c}=[0,l/c]\times[0,l]$ defined by $(s.u)(t,x):=u(t+s,x+cs)$ to the
functional $S(u)=\int_{X_{c}}L_{d}(u(t,x),u(t+\Delta t,x),u(t,x+\Delta
x))\mathrm{d}t$ reveals that (12) is governed by a formal 1st order
variational principle. This is investigated more closely in [13].

To identify TWs in a machine learned model of a discrete field theory, we make
an ansatz of a discrete Fourier series
$f(\xi)=\sum_{m=-\frac{M-1}{2}}^{-\frac{M}{2}}\hat{f}_{|m|}\exp(m\frac{2\pi
i}{l}\xi)$, where bounds of the sum are rounded such that we have $M$
summands. To locate a TW, the loss function
$\ell_{\mathrm{TW}}(c,\bm{\hat{f}})+\ell_{\mathrm{TW}}^{\mathrm{reg}}(c,\bm{\hat{f}})$
is minimised with

$\ell_{\mathrm{TW}}(c,\bm{\hat{f}})=\sum_{i=1}^{N-1}\sum_{j=0}^{M-1}\|\mathrm{DEL}^{i}_{j}(U)\|^{2},\qquad
U=\big{(}f(i\Delta t-cj\Delta x)\big{)}_{0\leq i\leq N}^{0\leq j\leq M-1}$
(15)

and regularisation
$\ell_{\mathrm{TW}}^{\mathrm{reg}}=\exp(-100\|U\|_{l^{2}}^{2})$ with discrete
$l^{2}$-norm $\|\cdot\|_{l^{2}}$ to avoid trivial solutions. Here
$\bm{\hat{f}}=(\hat{f}_{m})_{m}$.

##  5 Experiment

###  5.0.1 Creation of training data

We use the space-time domain $X$ (Section 2.0.2) with $T=0.5$, $l=1$, $\Delta
x=0.05$, $\Delta t=0.025$. To obtain training data that behaves like
discretised smooth functions, we compute $K=80$ solutions to the discrete wave
equation (Example 2) with potential $V(u)=\frac{1}{2}u^{2}$ on the mesh
$X_{\Delta}$ from initial data ${\bm{u}^{0}}=(u^{0}_{j})_{0\leq j\leq M-1}$
and ${\bm{u}^{1}}=(u^{1}_{j})_{0\leq j\leq M-1}$. To obtain ${\bm{u}^{0}}$ we
sample $r$ values from a standard normal distribution. Here $r$ is the
dimension of the output of a real discrete Fourier transformation of an
$M$-dimensional vector. These are weighted by the function $m\mapsto
M\exp(-2j^{4})$, where $m=0,\ldots,r-1$ is the frequency number. The vector
${\bm{u}^{0}}$ is then obtained as the inverse real discrete Fourier transform
of the weighted frequencies. To obtain ${\bm{u}^{1}}$ an initial velocity
field ${\bm{v}^{0}}=(v^{0}_{j})_{0\leq j\leq M-1}$ is sampled from a standard
normal distribution. Then we proceed as in a variational discretisation scheme
[11] applied to the Lagrangian density $L$ of the continuous wave equation
(Example 1): to compute conjugate momenta we set
$L_{\Sigma}({\bm{u}},{\bm{v}})=\sum_{j=0}^{M-1}\Delta xL(u_{j},v_{j})$ and
compute ${\bm{p}^{0}}=\frac{\partial
L_{\Sigma}}{\partial{\bm{v}^{0}}}({\bm{u}^{0}},{\bm{v}^{0}})$. Then
${\bm{p}^{0}}=-L_{\Sigma}({\bm{u}^{1}},({\bm{u}^{1}}-{\bm{u}^{0}})/\Delta t)$
is solved for ${\bm{u}^{1}}$. A plot of an element of the training data set is
displayed in Fig. 1. (TWs are not part of the training data.)

Figure 1: Left: Element of training data set. Centre: Predicted solution to
unseen initial values. Right: Continued solution from centre plot outside
training domain

###  5.0.2 Training and Evaluation

A discrete Lagrangian
$L_{d}\colon\mathbb{R}\times\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ is
modelled as a three layer feed-forward neural network, where the interior
layer has 10 nodes (160 parameters in total). It is trained on the
aforementioned training data set and loss function
$\ell=\ell_{\mathrm{DEL}}+\ell_{\mathrm{reg}}$ using the optimiser adam. We
perform 1320 epochs of batch training with batch size 10. For the trained
model we have $\ell_{\mathrm{DEL}}\approx 8.6\cdot 10^{-8}$ and
$\ell_{\mathrm{reg}}\approx 1.4\cdot 10^{-7}$. To evaluate the performance of
the trained model for $L_{d}$, we compute solutions to initial data by forward
propagation (Remark 4) and compare with solutions to the discrete wave
equation (Example 2). For initial data ${\bm{u}^{0}}$, ${\bm{u}^{1}}$ not seen
during training, the model recovers the exact solution up to an absolute error
$\|U-U_{\mathrm{ref}}\|_{\infty}<0.012$ on $X_{\Delta}$ and up to
$\|U-U_{\mathrm{ref}}\|_{\infty}<0.043$ on an extended grid with
$T_{\mathrm{ext}}=2.5$ (Fig. 1).

Figure 2: Left: Reference TW. Centre and Right: Identified TW in learned model

We have
$\max_{i,j}\|\mathrm{DEL}(L_{d})_{i,j}(U_{\mathrm{ref}}^{\mathrm{TW}})\|<0.004$,
where $U_{\mathrm{ref}}^{\mathrm{TW}}$ is the TW from Example 3 ($n=1$). This
shows that the exact TW is a solution of the learned discrete field theory.
This is remarkable since TWs are not part of the training data. However,
Remark 6 hints that the ansatz of an autonomous $L_{d}$ favours TWs as it
contains the right symmetries. Using the method of Section 4, a TW
$U^{\mathrm{TW}}$ and speed $c$ can be found numerically: with
$(c_{1},U_{\mathrm{ref}}^{\mathrm{TW}})$ as an initial guess with normally
distributed random noise ($\sigma=0.5$) added to the Fourier coefficients of
$U_{\mathrm{ref}}^{\mathrm{TW}}$ and to $c_{1}$ , we find $U^{\mathrm{TW}}$
and $c$ for the learned $L_{d}$ with errors
$\|U^{\mathrm{TW}}-U_{\mathrm{ref}}^{\mathrm{TW}}\|_{\infty}<0.12$ and
$|c-c_{1}|<0.001$ (using $10^{4}$ epochs of adam). (Fig. 2)

##  6 Conclusion and future work

We present an approach to learn models of dynamical systems that are governed
by an (a priori unknown) variational PDE from data. This is done by learning a
neural-network model of a discrete Lagrangian such that the discrete
Euler–Lagrange equations (DELs) are fulfilled on the training data. As DELs
are local, the model can be efficiently trained and used in simulations. Even
though the underlying system is infinite dimensional, model order reduction is
not required. It would be interesting to relate the implicit locality
assumption of our data-driven model to the more widely used approach to fit a
dynamical system on a low-dimensional latent space that is identified using
model order reduction techniques [1, 3, 4, 12, 20].

Our approach fits in the context of physics-informed machine learning because
the data-driven model has (discrete) variational structure by design. However,
our model is numerical analysis-informed as well: since our model is discrete
by design, it can be used in simulations without an additional discretisation
step. Based on an analysis of Newton’s method when used to solve DELs, we
develop a regulariser that rewards numerical regularity of the model. The
regulariser is employed during the training phase. It plays a crucial role to
obtain a non-degenerate discrete Lagrangian.

Our work provides a proof of concept illustrated on the wave equation. It is
partly tailored to the hyperbolic character of the underlying PDE. It is
future work to adapt this approach to dynamical systems of fundamentally
different character (such as parabolic or elliptic behaviour) by employing
discrete Lagrangians and regularisers that are adapted to the information flow
within such PDEs.

Finally, we clarify the notion of travelling waves (TWs) in discrete models
and show how to locate TWs in data-driven models numerically. Indeed, in our
numerical experiment the data-driven model contains the correct TWs even
though the training data does not contain any TWs. In future work it would be
interesting to develop techniques to identify more general highly symmetric
solutions in data-driven models and use them to evaluate qualitative aspects
of learned models of dynamical systems.

###  6.0.1 Source Code

https://github.com/Christian-Offen/LagrangianDensityML

###  6.0.2 Acknowledgements

C. Offen acknowledges the Ministerium für Kultur und Wissenschaft des Landes
Nordrhein-Westfalen and computing time provided by the Paderborn Center for
Parallel Computing (PC2).

##  Appendix 0.A Proofs

######  Proof (Proposition 1)

We adapt the standard estimates for Newton’s method (see [6, §4], for
instance) to the considered setting. Let $f\colon\mathcal{O}\to\mathbb{R}^{d}$
with

$f(u)=\frac{\partial}{\partial
u^{i}_{j}}\big{(}L_{d}(u^{i}_{j},u,u^{i}_{j+1})+L_{d}(u^{i-1}_{j},u^{i}_{j},u^{i-1}_{j+1})+L_{d}(u^{i}_{j-1},u^{i+1}_{j-1},u^{i}_{j})\big{)}.$

With this definition, $f(u^{\ast})=0$, $\theta$ is a Lipschitz constant for
$\mathrm{D}f\colon\mathcal{O}\to\mathbb{R}^{d\times d}$,
$u\mapsto\mathrm{D}f(u)$, $\overline{\theta}$ is a Lipschitz constant for
$\mathrm{inv}\circ\mathrm{D}f\colon\mathcal{O}\to\mathbb{R}^{d\times d}$,
$u\mapsto\mathrm{D}f(u)^{-1}$, and
$\rho^{\ast}=\|\mathrm{D}f(u^{\ast})^{-1}\|$. Here $\mathrm{D}f(u)$ denotes
the Jacobian matrix of $f$ at $u\in\mathcal{O}$.

Assume that for $n\in\mathbb{N}$ an iterate $u^{(n)}\in\mathcal{O}$ fulfils
$\|u^{(n)}-u^{\ast}\|\leq\min\left(\frac{\rho^{\ast}}{\overline{\theta}},\frac{1}{2\theta\rho^{\ast}}\right)$.
Then

$\begin{split}\|\mathrm{D}f(u^{(n)})^{-1}&\|=\|\mathrm{D}f(u^{(n)})^{-1}-\mathrm{D}f(u^{\ast})^{-1}+\mathrm{D}f(u^{\ast})^{-1}\|\\\
&\leq\|\mathrm{D}f(u^{(n)})^{-1}-\mathrm{D}f(u^{\ast})^{-1}\|+\rho^{\ast}\leq\underbrace{\overline{\theta}\|u^{(n)}-u^{\ast}\|}_{\leq\rho^{\ast}}+\rho^{\ast}\leq
2\rho^{\ast}.\end{split}$

The next iterate $u^{(n+1)}=u^{(n)}-\mathrm{D}f(u^{(n)})^{-1}f(u^{(n)})$ can
be bounded:

$\displaystyle\|$ $\displaystyle
u^{(n+1)}-u^{\ast}\|=\|u^{(n)}-u^{\ast}-\mathrm{D}f(u^{(n)})^{-1}(f(u^{(n)})-\underbrace{f(u^{\ast})}_{=0})\|$
$\displaystyle\leq\underbrace{\|\mathrm{D}f(u^{(n)})^{-1}\|}_{\leq
2\rho^{\ast}}\Big{\|}\underbrace{\mathrm{D}f(u^{(n)})(u^{(n)}-u^{\ast})}_{=\int_{0}^{1}\mathrm{D}f(u^{(n)})(u^{(n)}-u^{\ast})\mathrm{d}t}-\underbrace{(f(u^{(n)})-f(u^{\ast}))}_{\int_{0}^{1}\mathrm{D}f(u^{(n)}+t(u^{\ast}-u^{(n)}))(u^{(n)}-u^{\ast})\mathrm{d}t}\Big{\|}$
$\displaystyle\leq
2\rho^{\ast}\left\|\int_{0}^{1}\left(\mathrm{D}f(u^{(n)})-\mathrm{D}f(u^{(n)}+t(u^{\ast}-u^{(n)}))\right)(u^{(n)}-u^{\ast})\mathrm{d}t\right\|$
$\displaystyle\leq
2\rho^{\ast}\int_{0}^{1}\underbrace{\left\|\mathrm{D}f(u^{(n)})-\mathrm{D}f(u^{(n)}+t(u^{\ast}-u^{(n)}))\right\|}_{\leq\theta
t\|u^{(n)}-u^{\ast}\|}\left\|u^{(n)}-u^{\ast}\right\|\mathrm{d}t$
$\displaystyle\leq
2\rho^{\ast}\theta\left\|u^{(n)}-u^{\ast}\right\|^{2}\int_{0}^{1}t\mathrm{d}t$
$\displaystyle=\rho^{\ast}\theta\left\|u^{(n)}-u^{\ast}\right\|^{2}$

Moreover, since
$\left\|u^{(n)}-u^{\ast}\right\|\leq(2\theta\rho^{\ast})^{-1}$, we have
$\|u^{(n+1)}-u^{\ast}\|\leq\frac{1}{2}\|u^{(n)}-u^{\ast}\|$. By induction, the
Newton Iterations converge against $u^{\ast}$ and (9) holds true.

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Catalan Institute of Nanoscience and Nanotechnology - ICN2 (CSIC, BIST),
Campus UAB, 08193 Bellaterra, Spain Catalan Institute of Nanoscience and
Nanotechnology - ICN2 (CSIC, BIST), Campus UAB, 08193 Bellaterra, Spain

# Interference effects in one-dimensional moiré crystals.

Nils Wittemeier  Catalan Institute of Nanoscience and Nanotechnology - ICN2
(CSIC, BIST), Campus UAB, 08193 Bellaterra, Spain  Matthieu J. Verstraete
NanoMat/Q-Mat/CESAM and ETSF, Université de Liège (B5) and ETSF, B-4000 Liège,
Belgium  Pablo Ordejón  Catalan Institute of Nanoscience and Nanotechnology -
ICN2 (CSIC, BIST), Campus UAB, 08193 Bellaterra, Spain  Zeila Zanolli
Chemistry Department and, Debye Institute for Nanomaterials Science, Condensed
Matter and Interfaces, Utrecht University and ETSF, PO Box 80.000, 3508 TA
Utrecht, The Netherlands<EMAIL_ADDRESS>
###### Abstract

Interference effects in finite sections of one-dimensional moiré crystals are
investigated using a Landauer-Büttiker formalism within the tight-binding
approximation. We explain interlayer transport in double-wall carbon nanotubes
and design a predictive model. Wave function interference is visible at the
mesoscale: in the strong coupling regime, as a periodic modulation of quantum
conductance and emergent localized states; in the localized-insulating regime,
as a suppression of interlayer transport, and oscillations of the density of
states. These results could be exploited to design quantum electronic devices.

Quantum materials 1, 2 are a class of materials that exhibit quantum effects
at the macroscopic scale. They offer the opportunity to realize paradigm-
shifting quantum electronics. The controlled generation and manipulation of
quantum states by electrical, magnetic, or optical means is a key challenge in
bringing quantum materials to applications. Twisted bilayer graphene (tBLG) is
a prime example of a tunable quantum material. The twist angle, and the
resulting bi-dimensional moiré pattern, controls the emerging material
properties: for specific magic angles, strong interlayer coupling and flat
bands arise, resulting in superconductivity and strongly correlated phases 3,
4. One-dimensional moiré systems are realized in double-wall carbon nanotubes
(DWNT) and are determined by two parameters: the angle between the inner and
outer tubes’ chiral vectors (like tBLG) and the difference between their
radii. These degrees of freedom control the effective interlayer interaction
and the resulting electronic states. While the physics emerging in 2D moiré
crystals has been studied extensively, the understanding of its one-
dimensional counterpart is limited to the ideal infinite nanotube case 5, 6, 7
and the commensurate, telescopic nanotubes 8, 9, 10, 11, 12, 13. Koshino et
al. 5 devised a continuum model for the idealized infinite DWNT. They identify
three regimes with unique electronic properties (localized insulating, strong
and weak coupling) determined by the relative alignment of the chiral vectors
of the tubes. Dispersionless flat bands appear in the localized insulating
regime. The electronic structure of DWNT with weak coupling is given by the
superposition of the constituent nanotubes, whereas it is heavily perturbed in
the strong coupling regime: semiconducting tube combinations can produce a
finite density of states in their gap, and metallic tubes can become
semiconducting. Experimental evidence of DWNTs with strong interlayer coupling
has been recently reported 7, but the relation between the measurements and
the idealized infinite DWNT model by Koshino is not trivial. Experimental
conditions impose a finite tube length and electrical contacts on the outer
tube only. As the overlap region between inner and outer tubes is finite,
strong confinement effects can arise, as in SWNTs with finite length 14, 15.
It is not clear a priori whether the regimes will be visible, in what limits
of nanotube overlap they could be recovered, and whether the regime separation
is even valid for finite tubes. Answering these basic questions is essential
for the use of coaxial nanotubes in applications involving electronic
transport16.

In this work, we address these questions by studying interlayer transmission
between two concentric nanotubes that overlap in a finite region and extend
infinitely in opposite directions (telescopic double wall nanotubes, tDWNT,
Fig 1a). We predict the effects of the length of the overlap region in the
different regimes, going beyond previous works that only consider commensurate
tDWNT. The inter-layer transmission in tDWNTs composed of two armchair
nanotubes (armchair@armchair) oscillates as a function of the overlap length,
giving rise to regions with zero and maximal transmission. The overlap length
modulates the tDWNT electronic properties. We demonstrate that this tunability
extends to strongly coupled, chiral and incommensurate tDWNts. We explain that
dips in the transmission spectra of armchair@armchair tDWNTs emerge due to
back-scattering by localized states in the overlap region. These transmission
dips are absent in strongly coupled chiral tDWNT due the lack of rotational
symmetries. We devise a predictive model for the transmission, based on wave
interference in one dimension, that reproduces the tunability and position of
transmission dips. In the localized insulating regime we show that the
interlayer conductance is very small irrespective of the overlap length, in
agreement with Ref. 17. We recover the Koshino limit for sufficiently long
(but finite) overlap lengths, and provide benchmarks for experimental
realizations of 1D moirés and correlated states in DWNT. In the weak coupling
regime, we show that interlayer transmission for chiral or incommensurate
tDWNTs is suppressed. tDWNTs composed of metallic zigzag nanotubes
(zigzag@zigzag) also belong to the weak coupling regime, but are an exception
to this rule: the inter-layer transmission can be significant if states on the
inner and outer tube with different angular momentum couple. This coupling is
subject to selection rules based on the chiral indices of the tubes involved
8.

##  1 Results and Discussion

Figure 1: a) Transport set-up: The scattering region consists of overlapping
nanotubes, screening region, and part of the electrodes (light red/blue),
which extend in opposite semi-infinite directions, and function as leads
(red/blue). b) Projected TB band structure of an ideal (10,10)@(15,15) DWNT.
The bands with negative Fermi-velocity (gray) are hybridized between the
nanotubes, bands with positive Fermi-velocity (red/blue) are localized on one
tube each. c) TB electron transmission from a semi-infinite (10,10) into a
semi-infinite (15,15) SWNT through an overlap region with length
$34.08\text{\,}\mathrm{\SIUnitSymbolAngstrom}$. Horizontal lines visualize
energy levels for an isolated, finite section of the system with additional
$44\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ of leads on either side. The
yellow-to-blue scale indicates the weight of wave functions in the overlap
region. Each dip in conductance coincides (approximately) with one or several
localized eigenstates.

Conductance simulations are performed for two concentric nanotubes that
overlap over a finite length $L$, connected to semi-infinite SWNT leads (Fig.
1a). Electron transmission is only allowed at energies where conduction
channels are available in both electrodes (only metallic nanotube contacts
allow transmission close to the Fermi level). When electrodes consist of semi-
conducting tubes, transmission can be achieved for chemical potentials within
the valence or conduction bands of both nanotubes. In either case, the total
transmission $T$ through this asymmetrically contacted system can not exceed
the smallest of the two electrode transmissions. The magnitude of $T$ depends
on how electrode states couple through the overlap region, probing sensitively
the interlayer interaction.

###  1.1 Strong coupling regime

DWNTs are in the strong coupling regime when the constituting tubes’ chiral
vectors are nearly parallel and their difference points along the armchair
direction. DWNTs consisting of armchair tubes (n,n)@(m,m) fulfill both
conditions. Armchair single-wall nanotubes (SWNT) present two nearly linear
bands crossing at the Fermi energy, resulting in a metallic system with
$2\text{\,}\mathrm{\text{$G_{0}$}}$ conductance 18, 19. Without interlayer
interaction, the band structure and conductance of an ideal DWNT would simply
be the sum of the individual SWNT ones. Deviations from the ideal case can
only occur for perturbations of the sidewalls, for instance due to defects 20,
21 or functionalization 22, 23. The infinite (10,10)@(15,15) DWNT, instead,
features (Fig. 1b) a subtle band structure of states localized on either tube
(positive Fermi velocity, red/blue) and hybridized states (negative Fermi
velocity, gray). The conductance of the DWNT with finite overlap region shows
a maximum of $1\text{\,}\mathrm{\text{$G_{0}$}}$ computed in the set-up of
Fig. 1a. The transmission function presents a series of dips where conduction
is blocked (Fig. 1c). The number, position and width of the dips depend on $L$
(Fig. 2a). Each of these dips corresponds to a localized eigenstate in the
overlap region (Fig. 1c horizontal lines). At
$L\approx$50\text{\,}\mathrm{\SIUnitSymbolAngstrom}$$ and
$L\approx$150\text{\,}\mathrm{\SIUnitSymbolAngstrom}$$ the dips in
transmission are barely visible. At intermediate overlap lengths, the
interlayer conductance is reduced, the dips are broader, and transmission is
blocked completely for $L\approx$100\text{\,}\mathrm{\SIUnitSymbolAngstrom}$$
and $L\approx$200\text{\,}\mathrm{\SIUnitSymbolAngstrom}$$, following a
periodic behavior. This reduction of the maximum conduction as well as the
oscillation with overlap length was predicted in Ref. 10, and we can now show
that localized states cause the dips observed in the conductance at fixed
overlap lengths. We further show that the modulation of the transmission can
be observed in a wide range of energy and overlap lengths.

Figure 2: TB electron transmission from a (10,10) SWNT into a (15,15) SWNT
through a finite overlap region at different overlap lengths calculated with
a) LB+TB formalism, and b) our model (Eq.1). Overlaps are sampled with a high
density below $50\text{\,}\mathrm{\SIUnitSymbolAngstrom}$
($1/40\cdot$2.459\text{\,}\mathrm{\SIUnitSymbolAngstrom}$$) and lower density
above ($1/$2.459\text{\,}\mathrm{\SIUnitSymbolAngstrom}$$). The model
reproduces both trends in LB-transmission: the energy independent modulation
of the transmission with overlap length on long spatial period, and the
secondary modulation of the transmission dependent on energy and overlap.

The reduction of the maximum conductance near the Fermi energy with respect to
the pristine DWNT can be explained taking into account the nature of the
transport setup and the character of the bands in the DWNT. The asymmetric
setup limits the conductance from $4\text{\,}\mathrm{\text{$G_{0}$}}$ to the
minimum of the electrode conductances ($2\text{\,}\mathrm{\text{$G_{0}$}}$).
The reduction of the maximum conductance to
$1\text{\,}\mathrm{\text{$G_{0}$}}$ is traced back to the three-fold
rotational symmetry of the DWNT section, which results in the cancellation of
one of the two available transport channels in Ref. 10. We note here that this
reduction can already be inferred from the eigenstates of the pristine DWNT.
The electronic eigenstates with positive Fermi-velocity are unaffected by the
interlayer interaction, and have to scatter back into the electrode,
effectively limiting the possible available channels to the electrode bands
with negative Fermi velocity. These electrons couple to the two hybridized
DWNT states, and have the possibility to be transmitted through the device.
Depending on the overlap length this transmission occurs with almost no back-
scattering
($L\approx$50\text{\,}\mathrm{\SIUnitSymbolAngstrom}$,$150\text{\,}\mathrm{\SIUnitSymbolAngstrom}$$)
or is completely blocked
($L\approx$100\text{\,}\mathrm{\SIUnitSymbolAngstrom}$$). In order to
understand the origin of the transmission dips, we calculate the eigenstates
of finite segments of the open system including additional CNT-rings on each
side of the scattering region. In the limit of infinitely long electrode
regions, we would recover the states of the open system. We find that each dip
corresponds to a localized state (Fig. 1c), and that the agreement between
eigenenergies and dip energies varies slightly, depending on the exact number
of additional CNT-rings. The eigenstates localized in the overlap region
($L=$34.08\text{\,}\mathrm{\SIUnitSymbolAngstrom}$$) are all similar in shape:
the wave function weight on the atoms at the tube terminations are the
highest, and they decay towards the electrode regions (SI Fig. LABEL:SI-
fig:10-10_15-15-wfcs).

To explain the origin of the localized states and the modulation of the
transmission, we construct a simple model, assuming linear dispersion of the
SWNT and DWNT bands with one common Fermi wave vector $k_{F}$ and velocity
$v_{F}$. We further assume that the energy separation of the bands with
negative $v_{F}$ in the DWNT is symmetric, giving two new vectors at each
energy $k^{\pm}(E)=k(E)\pm\frac{1}{2}\delta k_{F}$. An incoming electron with
energy $E=-v_{F}(k-k_{F})$ needs to couple to the hybridized states with the
same energy to pass from one layer into the other. The resulting superposition
of overlap states propagates with two wave vector components: the average wave
vector $k$, and the wave vector difference $\delta k_{F}/2$ modulating the
incoming wave. As this superposition propagates through the scattering device
its weight oscillates between the two tubes. If the wavelength of the
modulation is commensurate with the overlap region ($\delta k_{F}L=2n\pi$) the
incoming electron will be reflected at the termination and scatters back into
the electrode. However, if the overlap length fits $(2n+1)/4$ wavelengths of
the modulation ($L=(2n+1)\pi/2\delta k_{F}$) the electron passes through the
overlap region without back-scattering (Fig. 3). At intermediate lengths, the
incoming wave is partially transmitted reducing the conductance without fully
blocking it. Similarly, at certain energies, which depend on the overlap
length, the primary component $k$ becomes commensurate with the overlap. This
allows a standing wave to form in the quantum box of the overlap region, which
blocks the transmission and explains the emergence of dips at specific
energies. We combine the two trends to model the transmission using sine
functions:

$\displaystyle T(E,L)=\sin(L\frac{\delta
k_{F}}{2})^{2}\cdot\sin(L(k_{F}-\frac{E}{v_{F}}))^{2}$ (1) $\displaystyle
T(E,\frac{2\pi n}{\delta k_{F}})=0$ $\displaystyle T(-v_{F}(\frac{\pi
n}{L}+k_{F}),L)=0$

Fig. 2 shows the impressive agreement between the transmission calculated with
this simple model and the tight-binding results. The model reproduces the
energy-independent global oscillation of the transmission with the overlap
length, and allows us to predict the periodicity of the transmission
oscillation from the shift of the Fermi vector ($\delta
k_{F}\approx$0.071\text{\,}{\mathrm{\SIUnitSymbolAngstrom}}^{-1}$$) for any
nanotube combination. For this tube shift ($\delta
k_{F}\approx$0.065\text{\,}{\mathrm{\SIUnitSymbolAngstrom}}^{-1}$$) the
corresponding periodicity is $89\text{\,}\mathrm{\SIUnitSymbolAngstrom}$. The
periodicity observed in our simulations is only slightly larger
($100\text{\,}\mathrm{\SIUnitSymbolAngstrom}$). The curvature of regions
without transmission (dark blue) can be recovered when the non-linearity of
the bands is included in the model (Supplementary information). The model also
reproduces the $E$ and $L$ dependent position of the smaller dips observed in
the transmission. Furthermore, the shape of localized eigenstates matches the
expected character (Fig. 3). The sine functions make the dips smoother
compared to the LB-calculation, but the general trends are accurately
reproduced. In fact this simple expression is an approximations of the more
accurate one derived in Ref. 8 when the linear-band approximation is applied.
Numerical interpolation of the band structure allows for very accurate
reproduction of the LB+TB transmission (SI Fig. LABEL:SI-
fig:10-10_15-15-trans+accurate). However, the simple approach is qualitatively
correct and can intuitively be connected to the image of 1D-waves.

Figure 3: Schematic of standing waves in overlap region between two semi-
infinite nanotubes.

Figure 4: Strong coupling (M@M): TB electron transmission from (18,15) into
(23,20) with different overlap lengths $L$. The transmission oscillates as
function of $L$. At $0\text{\,}\mathrm{eV}$ and $0.31\text{\,}\mathrm{eV}$
sharp features occur irrespective of $L$.

We now consider an example of an incommensurate and chiral, strongly coupled,
metallic DWNT: (18,15)@(23,20). The band structure can be computed by
artificially imposing periodicity using a commensurate supercell consisting of
4 and 3 repetitions of (18,15) and (23,20), respectively, and straining both
tubes by $\pm 1\%$ (SI Fig LABEL:SI-fig:18-15_23-20-bands). For the telescopic
setup, the maximum conductance ($2\text{\,}\mathrm{\text{$G_{0}$}}$) can be
attained due to the strong hybridization of both linear bands between the
tubes, resulting in two simultaneously available conduction channels. The
maximum of the conductance near $E_{F}$ (Fig 4a) oscillates with the overlap
length between $2\text{\,}\mathrm{\text{$G_{0}$}}$ and $0$, with a period of
$90\pm 1.\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ (Fig 4b). Our simple model
of wave function interference in a quantum box predicts a spatial period of
$96\text{\,}\mathrm{\SIUnitSymbolAngstrom}$, using a shift of the Fermi wave
vectors $\delta
k_{F}\approx$0.065\text{\,}{\mathrm{\SIUnitSymbolAngstrom}}^{-1}$$ extracted
from the band structure in 5. Despite its simplicity, our model is semi-
quantitative and catches the main physical mechanisms at play: the
oscillations of the transmission are caused by the difference in the Fermi
wave vectors of the two tubes, and will appear in all combinations of strong
coupling, metallic@metallic nanotubes. The suppression of transmission at
discrete energies observed in (10,10)@(15,15) results from achirality, and
does not occur in chiral nanotubes. The wave functions of chiral tubes possess
no rotational symmetry around the tube axis. Therefore they can only have
nodes at few, specific points along the tube circumference, and cannot form
localized states in the overlap region (SI Fig LABEL:SI-fig:18-15-wfc-2181).
Avoided crossings in the band structure 5 result in sharp dips at
$0\text{\,}\mathrm{eV}$ and $0.31\text{\,}\mathrm{eV}$ when the conductance is
substantial ($L\approx$44\text{\,}\mathrm{\SIUnitSymbolAngstrom}$$,
$L\approx$20\text{\,}\mathrm{\SIUnitSymbolAngstrom}$$), and peaks at $\pm
0.31\text{\,}\mathrm{eV}$ when conductance is suppressed
($L\approx$88\text{\,}\mathrm{\SIUnitSymbolAngstrom}$$) (Fig. 4).

Figure 5:  Localized insulating DWNT (M@SC): density of states in the
scattering region (a) and electron transmission (b) from (27,3) into (35,3)
for different overlap lengths. Interference between the tubes results in sharp
spikes and flat bands (SI Fig. LABEL:SI-fig:27-03_35-03-bands), which are more
pronounced with increasing overlap length. The transport gap of the outer tube
is highlighted in striped grey.  Figure 6: Localized quasi-insulating (M@M):
density of states in the scattering region (a) and electron transmission (b)
from (27,3) into (36,3) for different overlap lengths. Interference between
the tubes result in sharp spikes and flat bands at $E_{F}$ and at the edge of
the quasi-insulating gap, which are more pronounced with increasing overlap
length.

###  1.2 Localized insulating regime

The localized insulating regime is achieved when the chiral vectors of the
inner and outer tube are nearly parallel and their difference points along the
zig-zag direction. We consider two representatives of this family,
(27,3)@(36,3) and (27,3)@(35,3), which are metallic-metallic (M@M) and
metallic-semiconducting (M@SC), respectively. Interference effects are
observable as peaks in the (27,3)@(35,3) DOS in Fig. 5. The small
($3.12\text{\,}\mathrm{\SIUnitSymbolAngstrom}$) interlayer spacing enhances
the coupling and suppresses the increase in DOS near the onset of the outer
tube parabolic bands (gray area in Fig. 5 and SI Fig. LABEL:SI-
fig:27-03_35-03-bands_zoom). This effect can be observed already in short DWNT
segments ($L<$$81\text{\,}\mathrm{\SIUnitSymbolAngstrom}$) and is not
exclusive to the infinite tubes of Ref. 5. In (27,3)@(36,3), instead, the
interference is weaker due to the larger interlayer spacing
($3.51\text{\,}\mathrm{\SIUnitSymbolAngstrom}$) and the onset of the parabolic
bands is clearly visible at $\pm 0.35\text{\,}\mathrm{eV}$ (Fig. 6). A series
of peaks are observed in the DOS of both cases (Fig. 5 and 6). This is
consistent with the emergence of flat bands (SI Fig. LABEL:SI-
fig:27-03_35-03-bands) predicted by Koshino et al., who attribute them to the
localization of electrons in an effective potential with long spatial period
($\approx$1200\text{\,}\mathrm{\SIUnitSymbolAngstrom}$$ for these two cases).
However, our simulations indicate that localized states already emerge in much
shorter tube segments
($L\approx$244\text{\,}\mathrm{\SIUnitSymbolAngstrom}$$), and could be
experimentally observable even without requiring long, pristine DWNT samples.
While some of the localized states emerge at these short overlap lengths, the
peak density in the DOS increases with overlap length and is significantly
lower than predicted by Koshino. This shows that the minima of effective
potential that cause localization, emerge successively with increasing
overlap. The full set localized states will be observable only if the overlap
length is larger than the period of the effective potential.

We observe additional oscillations of lower magnitude in the DOS of
(27,3)@(36,3) DWNT, which can not be attributed to flat bands. These
oscillations occur in the energy ranges between
$-0.35\text{\,}\mathrm{eV}-0.05\text{\,}\mathrm{eV}$ and
$0.1\text{\,}\mathrm{eV}0.35\text{\,}\mathrm{eV}$ and are a result of the
finite overlap length which causes a quantization of the wave vectors of
states in the finite overlap region. Analogous to the discussion in the strong
coupling regime, we assume linear dispersion of the low energy bands and
determine the energy spacing of states commensurate with the overlap length:

$\delta
E=v_{F}\frac{\pi}{L}\approx$5.6\text{\,}\mathrm{eV}\text{\,}\mathrm{\SIUnitSymbolAngstrom}$\frac{\pi}{L},$
(2)

which matches the observed periods in the DOS ($\delta E_{DOS}$) quite well
(Table 1).

L [$\mathrm{\SIUnitSymbolAngstrom}$] | $\delta E_{DOS}[$\mathrm{eV}$]$ | $\delta E[$\mathrm{eV}$]$
---|---|---
1016 | 0.016 | 0.017
528 | 0.031 | 0.033
284 | 0.058 | 0.062
Table 1: Oscillation periods in DOS of a (27,3)@(36,3) tDWNT in the energy
ranges $-0.35\text{\,}\mathrm{eV}-0.05\text{\,}\mathrm{eV}$ and
$0.1\text{\,}\mathrm{eV}0.35\text{\,}\mathrm{eV}$ for three different overlap
length extracted from TB+LB calculations ($\delta E_{DOS}$) and calculated
using Eq. 2 ($\delta E$).

Given the small size of these oscillations, they will likely be challenging to
observe experimentally. These oscillations are not predicted by Koshino, since
they are caused by the finite length of the overlap. This quantization effect
is absent in the (27,3)@(35,3) DWNT, because its outer tube is semiconducting
and, therefore, only one set of linear bands is present. In both localized
insulating DWNTs considered here, the interlayer conductance is very small
($<$0.01\text{\,}\mathrm{\text{$G_{0}$}}$$) around the Fermi energy
($-0.35\text{\,}\mathrm{eV}$ to $0.35\text{\,}\mathrm{eV}$) and only becomes
relevant in those regions where non-linear bands are present in the inner and
outer nanotubes. This demonstrates that not only the metallic/semi-conducting
nature of the individual tubes is important for the conductance between
layers, but that the coupling regime also plays a significant role.
Consequently, metallic shells in multi-wall carbon nanotubes are likely to
contribute less to the overall conductance than previously expected due to the
low interlayer conductance, unless the strong coupling condition is fulfilled.
This explains the low interlayer conductivity reported in Ref. 11. Different
examples of localized-insulating tDWNT (metallic@metallic) with shorter
interlayer spacing ($\differential{R}$) can be found in SI Fig. LABEL:SI-
fig:45-18_54-18-trans
($\differential{R}=$3.4\text{\,}\mathrm{\SIUnitSymbolAngstrom}$$) and
LABEL:SI-fig:399-336_408-336-trans
($\differential{R}=$3.1\text{\,}\mathrm{\SIUnitSymbolAngstrom}$$) .

###  1.3 Weak coupling regime

In the weak coupling regime there is a distinction between DWNTs composed of
two zig-zag nanotubes (zig-zag@zig-zag) and all others. In zig-zag@zig-zag
tDWNTs the rotational symmetry plays an important role, specifically, the
three-fold rotational symmetry of (9,0)@(18,0) leads to significant interlayer
transmission near the Fermi-level 8. Similar to (10,10)@(15,15), the
commensurability allows the formation of localized states in the overlap
region and causes the transmission to be blocked at the corresponding
energies. For other tDWNTs in this regime, composed of either incommensurable
or chiral nanotubes, the interlayer transmission is negligible as a result of
weak coupling (SI Fig. LABEL:SI-fig:09-00_18-00-trans-LABEL:SI-
fig:39-00_36-18-trans).

###  1.4 Conclusion

In conclusion, we show that the two main coupling regimes predicted for double
wall carbon nanotubes (strong coupling and localized insulating) are present
already in finite DWNT segments. These regimes should be identifiable in
experimental set-ups even when DWNT segments are short. For strongly coupled
armchair@armchair metallic nanotubes, the interlayer transmission is
significant and modulated by two phenomena. It is blocked by localized states
at discrete energies for any given overlap region length, and it oscillates
with the overlap length due to wave interference in the overlap region. The
rotational symmetry of the tDWNT can reduce the maximal transmission, blocking
transmission through one of the electrode channels completely, as seen for
(10,10)@(15,15), or partially 10. In other strongly coupled tDWNTs, where at
least one nanotube is chiral, wave interference also lead to an oscillation of
the maximal transmission. However, the absence of rotational symmetry prevents
the formation of localized states and the corresponding back-scattering. Based
on the picture of 1D waves in a quantum box, we derive an expression for the
interlayer transmission of strongly coupled nanotubes. The simplified model
describes the modulation of transmission with excellent accuracy. In weakly
coupled zigzag@zigzag metallic tDWNTs, the interlayer transmission is strongly
affected by the rotational symmetry. This leads to significant transmission
between the layers of (9,0)@(18,0), while the interlayer-transmission in
(18,0)@(27,0) is negligible. In other weak coupling tDWNTs, where at least one
tube is chiral or the two tubes are incommensurate, the interlayer
transmission is negligible. Lastly, in the localized insulating regime the
interlayer transmission is also heavily suppressed near the Fermi-level. In
this regime the emergence of flat bands causes strong oscillations in the DOS.
Notably, the flat bands already appear in segments significantly shorter than
the predicted spatial periodicity of the underlying effective potential
($\approx$1200\text{\,}\mathrm{\SIUnitSymbolAngstrom}$$ for tubes discussed
above). The number of localized states is directly linked to the overlap
length. Our results highlight that applications like nano-eletronic switches
based on tDWNTs are very sensitive to their structure. Chiral tDWNTs are the
most promising candidates for such applications. These tubes preserve the
oscillating behavior found in all strongly coupled tDWNTs, while the absence
of rotational symmetries prevents back-scattering at localized states.

##  2 Methods

###  2.1 Tight-binding model

We model inter- and intra-layer interactions using a non-orthogonal tight-
binding (TB) model with one orbital per carbon site, based on Reich et al. 24,
Laissardiere et al. 25, and Bonnet et al. 6. We extend the model to include
curvature effects, using pp$\sigma$ interactions in addition to the pp$\pi$
interactions, and interlayer hopping to third nearest neighbors. The
interlayer interaction ranges up to a cut-off distance of
$5\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ and is described by exponential
decays:

$\displaystyle H_{pp\sigma/\pi}(d)$
$\displaystyle=H^{inter}_{pp\sigma/\pi}e^{\frac{(d-a_{0})}{g_{pp\sigma/\pi}}}$
(3) $\displaystyle S_{pp\sigma/\pi}(d)$
$\displaystyle=S^{inter}_{pp\sigma/\pi}e^{\frac{(d-a_{0})}{h_{pp\sigma/\pi}}},$
(4)

where $a_{0}=$3.35\text{\,}\mathrm{\SIUnitSymbolAngstrom}$$ is the interlayer
spacing of bulk graphite. The TB parameters were fit to electronic band
structures of single and double-wall carbon nanotubes obtained from first-
principles density functional theory. We first fix the pp$\pi$ parameters of
the intra-layer interaction by fitting to the band structure of graphene, to
reproduce correct behavior in the limit of very large nanotubes. Next, we fix
the pp$\sigma$ parameters of the intra-layer interaction to reproduce the ab
initio bands of nanotubes with radii between
$5.2\text{\,}\mathrm{\SIUnitSymbolAngstrom}6.2\text{\,}\mathrm{\SIUnitSymbolAngstrom}$:
(9,9), (16,0) and (9,6) (SI-Fig. LABEL:SI-fig:fit). Lastly, we optimize the
the interlayer parameters to reproduce the ab initio bands of two DWNTs:
(16,0)@(24,0) and (9,6)@(15,10). The optimized parameters are summarized in
Table 2. The transmission and density of states (DOS) are extracted using the
Landauer-Büttiker formalism (LB) formalism 26 as implemented in TBTrans 27.
All calculations are performed at zero voltage between the two electrodes.

Table 2: Optimized tight-binding parameters. onsite | intra-layer | inter-layer
---|---|---
|  |  | $pp\sigma$ | $pp\pi$ |  | $pp\sigma$ | $pp\pi$
$E_{p}$[$\mathrm{eV}$] | $-2.04$ |  $H^{1nn}$[$\mathrm{eV}$] | $3.93$ | $-2.81$ |  $H^{inter}$[$\mathrm{eV}$] | $0.505$ | $0.709$
|  | $S^{1nn}$ | $0.573$ | $0.301$ | $S^{inter}$ | $0.003$ | $0.062$
|  |  $H^{2nn}$[$\mathrm{eV}$] | $1.17$ | $-0.679$ |  $g$[$\mathrm{\SIUnitSymbolAngstrom}$] | $0.408$ | $0.387$
|  | $S^{2nn}$ | $0.018$ | $0.047$ |  $h$[$\mathrm{\SIUnitSymbolAngstrom}$] | $1.12$ | $0.620$
|  |  $H^{3nn}$[$\mathrm{eV}$] | $1.11$ | $-0.298$ |  |  |
|  | $S^{3nn}$ | $0.074$ | $0.040$ |  |  |

###  2.2 First principles calculations

First principles calculations were done with the siesta 28, 29 implementation
of the density functional theory (DFT) method. We employed the local density
approximation (LDA) exchange-correlation functional as parametrized J. P.
Perdew and Y. Wang 30 in conjunction with optimized norm-conserving Vanderbilt
(ONCV) scalar-relativistic pseudopotentials 31, 32 with $2s$ and $2p$ valence
electrons. The Brillouin zone was sampled using a $\Gamma$-centered, one-
dimensional grid with 78 k-points for the pristine armchair nanotubes.

##  3 Acknowledgments

The authors acknowledge enriching discussions on the tight-binding
parametrization with L. Henrard and P. Lambin. This work was supported by
Spanish MINECO (the Severo Ochoa Centers of Excellence Program under Grant No.
SEV- 2017-0706), Spanish MICIU, AEI and EU FEDER (Grants No.
PGC2018-096955-B-C4), Generalitat de Catalunya (Grant No. 2017SGR1506 and the
CERCA Programme), and the European Union MaX Center of Excellence (EU-H2020
Grant No. 824143). ZZ acknowledges financial support by the Ramon y Cajal
program RYC-2016-19344 (MINECO/AEI/FSE, UE) and the Netherlands Sector Plan
program 2019-2023. MJV acknowledges funding by the Belgian FNRS (PDR G.A.
T.1077.15-1/7 and a sabbatical “OUT” grant at ICN2), and computational
resources from the Consortium des Equipements de Calcul Intensif (CECI, FRS-
FNRS G.A. 2.5020.11) and Zenobe/CENAERO funded by the Walloon Region under
G.A. 1117545. NW has received funding from the European Union’s Horizon 2020
research and innovation programme under the Marie Skłodowska-Curie grant
agreement No. 754558. We acknowledge computer resources at MareNostrum4 at
Barcelona Supercomputing Center (BSC), provided through the PRACE Project
Access (OptoSpin project 2020225411) and RES (activity FI-2020-1-0014), and
technical support provided by the Barcelona Supercomputing Center. The authors
acknowledge use of the open-source project sisl33 used to generate atomic
structures and as basis of implementation for our tight-binding model.

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#  An Ultimate Approach of Mitigating Attacks in RPL Based Low Power Lossy
Networks

Jaspreet Kaur  PhD Scholar at CSE Department
Indian Institute of Technology Jodhpur
Jodhpur, India
<EMAIL_ADDRESS>
###### Abstract

The Routing Protocol for Low-Power and Lossy Networks (RPL) is the existing
routing protocol for Internet of Things (IoT). RPL is a proactive,lightweight,
Distance Vector protocol which offers security against various forms of
routing attacks. Still, there are various attacks(as rank, version attacks and
many more ) which is possible in this network due to problem of
unauthenticated or unencrypted control frames, centralized root controller,
compromised or unauthenticated devices and many more ways. There are various
solutions present in the literature but every solution has its pros and cons.
There is no appropriate system framework till now which completely solves
these all issues. So, we present an ultimate approach to mitigate these RPL
attacks more efficiently and effectively. We use IDS based system for internal
attacks and a mini-firewall for removing the external attacks. In IDS based
approach, we use intrusion detection system at multiple locations for
analyzing the behaviour of nodes. The final decision whether the node is
attacker or not depends on mainly three things as: trust between the
neighbouring nodes, local decision by multiple sink nodes and global decision
by root node. We also use some blockchain features in this framework for
better internal security. We use some threshold values and rules in mini-
firewall for removing external attacks. In this paper, we provide the proposed
approach and theoretical analysis of this approach which provide better
protection from these attacks than any other method.

###### Index Terms:

Internet of Things(IoT), Routing Protocol, RPL Attacks, Low Power Lossy
Networks, Multiple Sink Nodes, Trust, Local Decision, Global Decision,
Blockchain, Intrusion Detection System(IDS) , Mini-Firewall.

##  I Introduction

In today’s world, IoT is technical revolutionary area in mobile and wireless
communication field which deploy Low power and Lossy Networks (LLN). These
networks are typically composed of many heterogeneous embedded devices with
limited power, memory, and processing resources. Now, IoT is applicable in
many areas such as industrial monitoring, smart home, health care,
environmental monitoring, smart city, smart grid and many more. Due to the
huge number of applications of these networks, security become a critical part
for privacy of personal data. RPL is default routing protocol in these network
which is susceptible from various attacks. Now, We briefly describe the
related technologies use in this paper.

###  I-A Routing Protocol as RPL and it’s Attacks

RPL is distance based proactive protocol used for routing in IoT network. At
beginning, a RPL protocol creates a graph-like structure called the
Destination Oriented Directed Acyclic Graph (DODAG). The DODAG consists of
paths from the sender nodes(IoT devices) to the sink node(6LBR). During
routing, every node maintains its rank relative to its position in the DODAG
tree, and every DODAG is maintained by control and route information . The
control frames are used by DODAG are DODAG Information Object
(DIO),Destination Advertisement Object (DAO) and DODAG Information
Solicitation (DIS) for transmitting the DODAG information. Route path
selection is a key factor for RPL, RPL use the Various routing metrics, route
constraints and objective functions (OF) such as hop count, energy
minimization and latency to compute the best route path. The basic framework
of RPL network is shown below.

Figure 1: RPL Based Low Power Lossy Network

There are various attacks possible in RPL network which significantly impact
the network resources and its performance. These attacks are possible due to
the problem of unauthenticated or unencrypted control frames, centralized root
controller, compromised and unauthenticated devices and many more ways. Some
of these attacks are shown in figure 2 and briefly describe below.

Figure 2: RPL Network Attacks

####  I-A1 Sinkhole Attack

In this internal attack, attacker or compromised node advertises beneficial
path to attract many nearby nodes to route traffic through it. This attack
disrupt the network topology and can become very stronger when combined with
another attacks[2].

####  I-A2 Version Number Modification Attack

This internal attack is occurred by changing version number(lower to higher)
of a DODAG tree. When nodes receive the new higher version number DIO message
they start the formation of new DODAG tree. This results unoptimized or
inconsistency of network topology, increases control overhead and higher
packet loss[2].

####  I-A3 Denial of Service Attack

Denial of service or Distributed denial of service attack is attempt to make
resources unavailable to its valuable user [12-13]. In RPL this attack can be
done by the IPv6 UDP packet flooding[2].

####  I-A4 Neighbor Attack

In this attack, intruder broadcast DIO messages that it received without
adding information of himself.The node who receives this messages may conclude
that new neighbor node send this DIO message. The victim nodes select the out
range neighbors node for routing purpose which affects network quality
parameters[2].

####  I-A5 Wormhole Attack

This attack can occurred by creating tunnel between the two attackers and
transmitting the selective traffic through it which Disrupt the network
topology and traffic flow[2].

####  I-A6 Decreased Rank Attacks

In a DODAG, rank is used for positions of nodes relative to the sink node. It
means lower the rank is close to the root and vice verse. When a malicious
node advertises a lower rank value, it means, it wants majority of traffic
pass through it. As a result, many legitimate nodes connect to the attacker.
An attacker can change its rank value through the falsification of DIO
messages. finally, an attacker either drop this messages or selectively
forward these messages. It becomes more powerful when combined with other
attacks[2].

####  I-A7 Identity Attacks

Identity attacks contain clone ID and sybil attacks. In a clone ID attack, an
attacker copies the identities of a valid node onto another physical node. In
a sybil attack, which is similar to a clone ID attack, an attacker uses
several logical entities on the same physical node. These attacks can be used
to take control over large parts of a network without deploying physical
nodes[2,12-13].

####  I-A8 Sniffing Attacks

A sniffing attack perform by the listening of the packets transmitted over the
network. This attack compromises the confidentiality of communications. It
also check the pattern of traffic for major attacks[12-13].

###  I-B Blockchain

Blockchain is fundamentally a decentralized, distributed, shared, and
immutable database ledger that stores data across a peer-to-peer (P2P)
network. It has chained blocks of data that have been timestamped and
validated by miners. Fundamentally, the block data contains a list of all
transactions and a hash to the previous block. The blockchain has a full
history of all transactions and provides a global distributed trust[1].

###  I-C Intrusion Detection System(IDS)

An intrusion detection system (IDS) is a system that monitors network traffic
for suspicious activity and issues alerts when such activity is occurred.
These can be signature based or anomaly based and network or host based
depending upon the need of application. These mainly used for the detection of
internal attacks.

###  I-D Firewall

Firewall is used for protection from outsider attacks. This use some threshold
values or specific rules to filter unwanted traffic such as filtering is based
on port numbers, ip addresses and many more parameters.

###  I-E Trust

Trust means integrity, strength, ability, confidence of one person on the
other person or thing. A trust is a relationship or agreement which one party,
known as a trustor, gives another party, the trustee.

The rest of the paper is organized as follows: section 2 describes the Related
Work and motivation for our work. Section 3 mentions our proposed framework
system followed by the theoretical analysis of work in section 4. In Section
5, a conclusion of our findings is presented with future extensions to our
work.

##  II Related Work & Motivation

For detection and mitigation of RPL attacks, various mechanisms are presented
in the literature. But they have their own pros and cons. They are useful for
one attack but not for others. There are no standardized framework system for
security of RPL network. Specific literature survey for some of the attacks
are present below.

In paper[1], They describe various IoT security issues, open challenges and
provide blockchain as a solution for these attacks. They also review the RPL
attacks and their solutions present in the literature. In paper[2], the
detailed survey of various RPL attacks and their solutions are presented. They
describe taxonomy of RPL attacks based on resources, topology and traffic.
They also give risk management process for these attacks.

In paper[3], They discuss and implement various RPL attacks in cooja
simulator. They also present various solution present in the literature and
give a new solution as Lightweight heartbeat protocol for RPL network. This
new protocol is based on the successful transmission of ICMPv6 messages from
the root node to nodes and vice versa. This new protocol gives less overhead
but will work with its full potential if IPsec with ESP is used.

In paper[4], They describe the blackhole attack and present a solution for
this in a 2-step process. In the first step, local decision is made by a node
by observing the behaviour of neighbouring nodes. If any node is found to be
suspicious, then the final decision whether this node is blackhole node or not
taken by the root node, this process is called as global verification process.
This solution is very effective but in this, every node observes the behaviour
of its neighbours which increases the memory overload in these constrained
devices and final decision is made by the root node which is a single point of
failure(if compromised).

In paper[5], They describe the wormhole attack and present a solution for this
as markle tree authentication. This solution is effective but increases the
network complexity and control overhead due to the hashing and encryption
techniques. In paper[6], They describe sinkhole attacks and give a solution
which is the combination of two techniques as: Rank authentication
technique(one way hash chaining) and parent fail-over technique(end to end
acknowledgement). This solution is very promising but sybil attack is not
managed by the parent fail-over technique and rank authentication is secure
untill a random number given to the root node is not hacked by the attacker.

In paper[7], Authors presented SVELTE which is Real-time intrusion detection
system for the Internet of Things. This IDS is extensible and also uses the
feature of firewall. But at a time implemented only few number of attacks
detection, rest are still waiting to implement. But what if attackers hacked
the security features of SVELTE? In paper[8], A hybrid routing protocol which
is the combination of proactive and reactive approach is presented for
wireless sensor networks with mobile sinks. This approach is very useful for
network life time and maintaining cost of DAGs.

In paper[9], authors present a solution for rank and version number attacks as
VeRA-version number and rank authentication in rpl. This solution is
cryptographically secure But their is some faulty results due to the missing
correlation between rank hash chain and version number hash chain. In
paper[10], authors present solutions for mitigating topological attacks in RPL
such as Vera++ and TRAIL. But both technology uses cryptography which makes
these solutions highly complex and Distributed attackers communicated out-of-
band channel cannot be detected in this model.

In paper[11], authors present a solution as SecTrust-RPL for mitigating the
rank and sybil attacks. They use trust(direct or recommended) as a defence
mechanism. This solution is very powerful but they assume the nodes are
stationary and use location as a defence metric. They only simulate the direct
trust in this model not the indirect one. They also assume only 10% attacking
nodes from the total nodes and what if trust maintenance database is hacked by
the attacker?

As shown from the above discussion, we conclude that every solution has some
limitation. This is the base of my motivation for doing this research because
there is no standard framework or method to providing security in RPL network.
Various security modes such as unsecured, preinstalled, authenticated are
theoretically presented in standard RPL protocol. But, the real time IoT
products(produced by cisco and many more) and simulation tools (cooja, RIOT
and many more) are still support only unsecured mode. Security modes are still
remains for practical implementation due to the constrained nature of Iot
devices.

##  III Proposed Framework System

We propose an ultimate approach (structure of proposed framework is shown in
figure 3) for removing all internal and external RPL attacks. For removing
internal attacks, we use a IDS or trust based approach and for mitigating
external attacks, we have use some threshold values or rules in mini-firewall.
We use intrusion detection system(combination of signature and anomaly based)
at multiple locations (global and local) for analyzing the behaviour of nodes.
Global IDS must be placed at 6LBR (sink node of rpl network) node and we take
multiple fixed local sink node (approximately 10-15 % of total nodes) along
with local IDS, finally local sink nodes attach to the global sink node.

Figure 3: Proposed Framework

Iot devices or nodes are either mobile or stationary attach to the closest
(closeness depends on the objective functions used as hop count, latency and
energy minimization) local sink node. We also include the trust
parameter(successful messages exchange ratio) between local nodes for better
security. The final decision whether the node is attacker or not depends on
mainly three things as: trust between the neighbouring nodes, local decision
by multiple sink nodes and global decision by root node. Mini-firewall is
placed on every node as well as on every sink node and contains the list of
authenticated nodes along with appropriate parameters such as ip address and
rank for mitigating the external attacks.

In figure 3, every node including sink node maintain a database. For normal
nodes, database contains the trust values and list of authenticated nodes
(firewall rules) of each of their neighbour nodes(database is write protected
or secure by the private key). Trust value is defined by successful packets
exchange ratio such as more than 70 % ratio is treated as best path, less than
40% is treated as worst and in between them we check exactly three
times(threshold value) for improving result or path. In each of local sink
node, their are various tables or list with different functionalists. First
table contain all of signature IDS rule, firewall rules and threshold values.
Second thing we maintains a singly link list of blocks in which each block
contains the transactions or packets of their local network according to
particular time interval and block size which is useful for tracking of real
time data(anomaly IDS).

For maintain immutability, each block contains the hash of the previous block
same as blockchain. These block hashes and first table calculated hash are
saved in second table which is write protected or secure by the private key.
Both of the table update periodically whenever any new hash or rules are
occurred correspondingly. For backup of data, we save the each of local sink
link list to the immediate neighbour(unidirectional) local sink along with
their hashes(update instantly). For global sink node, First table is same as
the local sink node. Secondly, we maintain the combination of all link
lists(update instantly) of each of local sink node for global view of the RPL
network. Second table contains hash of first table along with hashes of all
link lists for each of local network and rest is same as local sink node. Due
to the resource constrained nature of devices, after some particular time
interval we remove particular number of blocks from starting of link list
along with their hashes in both local as well as global sink nodes while
maintaining the summary of that records in the first table as adding some
rules.

From the above we also say that, our proposed approach also follows the
blockchain features such as immutability, decentralization, distributed and
shared ledger. The more explanatory database structure is shown in figure 4:

Figure 4: Database Structure

##  IV Observations & Theoretical Results

The above proposed approach is theoretically analyzed in the smart environment
for mitigating the RPL attacks. Some of the important analysis results &
observations are as follows:

  * •

It Removes the single point of failure(root node). As in the proposed
decentralized approach, The final decision whether the node is attacker or not
depends on mainly three things as: trust between the neighbouring nodes, local
decision by multiple sink nodes and global decision by root node.

  * •

Due to the multiple local sink, number of transmission is reduced from entire
network to only closest sink and every local sink node maintains DAG only upto
limited nodes in the main RPL network. So, it reduces network load in terms of
control message overhead and increased throughput.

  * •

Sink discovery by sensor nodes almost eliminate use of old routes which
results in re-transmission of packets. Thus, reduces network overhead.

  * •

For removing unauthenticated devices and unauthenticated control
packets(external attacks), we use mini-firewall at every node(including sink
node) which maintains a database of authenticated devices using some
parameters as ip address, rank, version number and sequence number.

  * •

For mitigating the attacks occur from unencrypted control frames or
compromised nodes (internal attacks), we use IDS which check the attacker at
three phases. At first phase, we take trust parameter between nodes as mention
above for selecting the best path (best parent node) towards sink. In the
second phase, each local sink node maintain the link list of blocks for
tracking the history of their network. Finally the global sink node maintain
the history of DODAG (all local sink node) for mitigating any attack. The sink
nodes also uses signature rules(as rank of parent node should be less than
child node and many more rules) for easily detecting of known attacks.

  * •

The link list of blocks is used for unknown attacks. For maintaining
immutability of data, hash of previous block is contained in the current block
same as blockchain by which we can detect any changes in that blocks at very
less complexity. If we detect any changes, then from that block to the current
block in that link list is replaced by the backup data(stored in the immediate
neighbour sink node).

  * •

Signature based rules, threshold values and firewall rules are periodically
updated by which we take less time for detecting already happened attacks
previously.

  * •

Due to the resource limitations of sink nodes, after a particular time
interval we remove certain number of blocks along with their hashes(in second
table) from starting point of list while maintaining or adding that blocks
important features as rules in the first table.

  * •

All of the attacks are mitigated by this proposed approach by seeing behavior
of a node up to joining of nodes into this network. After recognizing of
malicious node, it can be easily removed from the network. Thus reduces DAG
inconsistency and improve network performance.

  * •

We assume all sink nodes including 6LBR are stationary. So, that we can check
their geographical location for more better protection.

  * •

Overall, we use semi supervised IDS, Blockchain features, Trust parameters and
mini-Firewall for protecting RPL network from various attacks.

##  V Conclusion & Future work

Finally, we say that RPL network(one of the most used routing protocol in
IoT)is prone to various kind of attacks as rank, version modification attack
and sybil attack etc. These attacks are very dangerous for the network
resources and performance. These attacks are occur due to problem of
unauthenticated or unencrypted control frames, centralized root controller,
compromised or unauthenticated devices and many more ways. There is no
standard protection framework developed yet in the survey. So, we develop an
ultimate framework for mitigating RPL attacks in smart environment. In this
approach we use multiple sinks, Trust, semi supervised IDS and firewall
mechanisms from protection to these attacks. We also maintain blockchain
features for more better security. Then, we theoretically analyzed our
approach which is very effective and reliable for mitigating these attacks. It
also reduces network overhead along with increases the performance and
throughput of the network due to the multiple sinks. These node’s information
are never compromised by the attacker due to the IDS, Firewall techniques and
distributed blockchain features.

This is our ongoing research work. In the future work, Firstly we will do
simulation or real time implementation of this approach using network
simulator (NS3) for RPL Secure network along with Ethereum blockchain. After
that we will focus on more attacks such as Zero day attacks in RPL protocol
and other layers attacks in IoT stack. We also want to reduce the complexity
of our approach as low as possible through new ideas such as how to remove
blocks of data at more effective way so that it can not loss any important
information. Lastly we must ensure, our approach is backward compatible to
original protocol.

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