††footnotetext: † These authors contributed equally to this work

# Phonon drag-electric current generation at the liquid-graphite interface

Alice Marcotte† Laboratoire de Physique de l’École Normale Supérieure, ENS,
Université PSL, CNRS, 3 Sorbonne Université, Université Paris Cité, 75005
Paris, France  Mathieu Lizee† Laboratoire de Physique de l’École Normale
Supérieure, ENS, Université PSL, CNRS, 3 Sorbonne Université, Université Paris
Cité, 75005 Paris, France  Baptiste Coquinot  Laboratoire de Physique de
l’École Normale Supérieure, ENS, Université PSL, CNRS, 3 Sorbonne Université,
Université Paris Cité, 75005 Paris, France  Center for Computational Quantum
Physics, Flatiron Institute, New York, USA  Nikita Kavokine  Center for
Computational Quantum Physics, Flatiron Institute, New York, USA  Karen
Sobnath  Université Paris Cité, Laboratoire Matériaux et Phénomènes
Quantiques, CNRS, F-75013 Paris, France  Clément Barraud  Université Paris
Cité, Laboratoire Matériaux et Phénomènes Quantiques, CNRS, F-75013 Paris,
France  Ankit Bhardwaj  National Graphene Institute, The University of
Manchester, UK  Department of Physics and Astronomy, The University of
Manchester, UK  Boya Radha  National Graphene Institute, The University of
Manchester, UK  Department of Physics and Astronomy, The University of
Manchester, UK  Antoine Niguès  Laboratoire de Physique de l’École Normale
Supérieure, ENS, Université PSL, CNRS, 3 Sorbonne Université, Université Paris
Cité, 75005 Paris, France  Lydéric Bocquet  Laboratoire de Physique de l’École
Normale Supérieure, ENS, Université PSL, CNRS, 3 Sorbonne Université,
Université Paris Cité, 75005 Paris, France  Alessandro Siria  Laboratoire de
Physique de l’École Normale Supérieure, ENS, Université PSL, CNRS, 3 Sorbonne
Université, Université Paris Cité, 75005 Paris, France

# Rising electronic winds under liquid flows on carbon surfaces

Alice Marcotte† Laboratoire de Physique de l’École Normale Supérieure, ENS,
Université PSL, CNRS, 3 Sorbonne Université, Université Paris Cité, 75005
Paris, France  Mathieu Lizee† Laboratoire de Physique de l’École Normale
Supérieure, ENS, Université PSL, CNRS, 3 Sorbonne Université, Université Paris
Cité, 75005 Paris, France  Baptiste Coquinot  Laboratoire de Physique de
l’École Normale Supérieure, ENS, Université PSL, CNRS, 3 Sorbonne Université,
Université Paris Cité, 75005 Paris, France  Center for Computational Quantum
Physics, Flatiron Institute, New York, USA  Nikita Kavokine  Center for
Computational Quantum Physics, Flatiron Institute, New York, USA  Karen
Sobnath  Université Paris Cité, Laboratoire Matériaux et Phénomènes
Quantiques, CNRS, F-75013 Paris, France  Clément Barraud  Université Paris
Cité, Laboratoire Matériaux et Phénomènes Quantiques, CNRS, F-75013 Paris,
France  Ankit Bhardwaj  National Graphene Institute, The University of
Manchester, UK  Department of Physics and Astronomy, The University of
Manchester, UK  Boya Radha  National Graphene Institute, The University of
Manchester, UK  Department of Physics and Astronomy, The University of
Manchester, UK  Antoine Niguès  Laboratoire de Physique de l’École Normale
Supérieure, ENS, Université PSL, CNRS, 3 Sorbonne Université, Université Paris
Cité, 75005 Paris, France  Lydéric Bocquet  Laboratoire de Physique de l’École
Normale Supérieure, ENS, Université PSL, CNRS, 3 Sorbonne Université,
Université Paris Cité, 75005 Paris, France  Alessandro Siria  Laboratoire de
Physique de l’École Normale Supérieure, ENS, Université PSL, CNRS, 3 Sorbonne
Université, Université Paris Cité, 75005 Paris, France

# Electronic winds risen under liquid flows on carbon surfaces

Alice Marcotte† Laboratoire de Physique de l’École Normale Supérieure, ENS,
Université PSL, CNRS, 3 Sorbonne Université, Université Paris Cité, 75005
Paris, France  Mathieu Lizee† Laboratoire de Physique de l’École Normale
Supérieure, ENS, Université PSL, CNRS, 3 Sorbonne Université, Université Paris
Cité, 75005 Paris, France  Baptiste Coquinot  Laboratoire de Physique de
l’École Normale Supérieure, ENS, Université PSL, CNRS, 3 Sorbonne Université,
Université Paris Cité, 75005 Paris, France  Center for Computational Quantum
Physics, Flatiron Institute, New York, USA  Nikita Kavokine  Center for
Computational Quantum Physics, Flatiron Institute, New York, USA  Karen
Sobnath  Université Paris Cité, Laboratoire Matériaux et Phénomènes
Quantiques, CNRS, F-75013 Paris, France  Clément Barraud  Université Paris
Cité, Laboratoire Matériaux et Phénomènes Quantiques, CNRS, F-75013 Paris,
France  Ankit Bhardwaj  National Graphene Institute, The University of
Manchester, UK  Department of Physics and Astronomy, The University of
Manchester, UK  Boya Radha  National Graphene Institute, The University of
Manchester, UK  Department of Physics and Astronomy, The University of
Manchester, UK  Antoine Niguès  Laboratoire de Physique de l’École Normale
Supérieure, ENS, Université PSL, CNRS, 3 Sorbonne Université, Université Paris
Cité, 75005 Paris, France  Lydéric Bocquet  Laboratoire de Physique de l’École
Normale Supérieure, ENS, Université PSL, CNRS, 3 Sorbonne Université,
Université Paris Cité, 75005 Paris, France  Alessandro Siria  Laboratoire de
Physique de l’École Normale Supérieure, ENS, Université PSL, CNRS, 3 Sorbonne
Université, Université Paris Cité, 75005 Paris, France

# Strong electronic winds blowing under liquid flows on carbon surfaces

Alice Marcotte† Laboratoire de Physique de l’École Normale Supérieure, ENS,
Université PSL, CNRS, 3 Sorbonne Université, Université Paris Cité, 75005
Paris, France  Mathieu Lizee† Laboratoire de Physique de l’École Normale
Supérieure, ENS, Université PSL, CNRS, 3 Sorbonne Université, Université Paris
Cité, 75005 Paris, France  Baptiste Coquinot  Laboratoire de Physique de
l’École Normale Supérieure, ENS, Université PSL, CNRS, 3 Sorbonne Université,
Université Paris Cité, 75005 Paris, France  Center for Computational Quantum
Physics, Flatiron Institute, New York, USA  Nikita Kavokine  Center for
Computational Quantum Physics, Flatiron Institute, New York, USA  Karen
Sobnath  Université Paris Cité, Laboratoire Matériaux et Phénomènes
Quantiques, CNRS, F-75013 Paris, France  Clément Barraud  Université Paris
Cité, Laboratoire Matériaux et Phénomènes Quantiques, CNRS, F-75013 Paris,
France  Ankit Bhardwaj  National Graphene Institute, The University of
Manchester, UK  Department of Physics and Astronomy, The University of
Manchester, UK  Boya Radha  National Graphene Institute, The University of
Manchester, UK  Department of Physics and Astronomy, The University of
Manchester, UK  Antoine Niguès  Laboratoire de Physique de l’École Normale
Supérieure, ENS, Université PSL, CNRS, 3 Sorbonne Université, Université Paris
Cité, 75005 Paris, France  Lydéric Bocquet  Laboratoire de Physique de l’École
Normale Supérieure, ENS, Université PSL, CNRS, 3 Sorbonne Université,
Université Paris Cité, 75005 Paris, France  Alessandro Siria  Laboratoire de
Physique de l’École Normale Supérieure, ENS, Université PSL, CNRS, 3 Sorbonne
Université, Université Paris Cité, 75005 Paris, France

###### Abstract

The interface between a liquid and a solid is the location of plethora of
intrincate mechanisms at the nanoscale, at the root of their specific emerging
properties in natural processes or technological applications Gonella2021 ;
Kavokine2021 . However, while the structural properties and chemistry of
interfaces have been intensively explored, the effect of the solid-state
electronic transport at the fluid interface has been broadly overlooked up to
now. It has been reported that water flowing against carbon-based
nanomaterials, such as carbon nanotubes Kral2001 ; Gosh2003 ; Cohen2003 ;
Persson2004 ; Zhao2008 ; Rabinowitz2020  or graphene sheets Dhiman2011 ;
Yin2012 ; Lee2013 ; Kuriya2020 ; Yin2014 ; Park2017 , does induce electronic
currents, but the mechanism at stake remains controversial review2019 ;
Dhiman2011 ; Yin2012 . Here, we unveil the molecular mechanisms underlying the
hydro-electronic couplings by investigating the electronic conversion under
flow at the nanoscale. We use a tuning fork-Atomic Force Microscope (AFM) to
deposit and displace a micrometric droplet of both ionic and non-ionic liquids
on a multilayer graphene sample, while recording the electrical current across
the carbon flake. We report measurements of an oscillation-induced current
which is several orders of magnitude larger than previously reported for water
on carbon Yin2014 , and further boosted by the presence of surface wrinkles on
the carbon layer. Our results point to a peculiar momentum transfer mechanism
between fluid molecules and charge carriers in the carbon walls mediated by
phonon excitations in the solid Kral2001 ; Lee2013 . Our findings pave the way
for active control of fluid transfer at the nanoscale by harnessing the
complex interplay between collective excitations in the solid and the
molecules in the fluid.

Transport of fluid across or at the interface of carbon unveiled unexpected
properties in term of very fast permeation, non-linear ionic transport and
sieving and even energy storage and harnessing Kavokine2021 . These results
highlight the subtle interactions of water and charges with the carbon
surface. Classically, the interaction of a fluid with a solid wall is based on
picturing the solid as a static periodic potential that acts on the fluid
molecules: interactions result from collisions on the surface roughness, as
flows induced on the roughness scale dissipate mechanical energy. However,
recent results for fluid and ion transport in carbon nanotubes and
nanochannels made of graphitic materials have pointed out the limits of such
standard description for fluid-solid interactions and existing couplings of
liquid transport with electronic degrees of freedom inside the confining
walls. This takes the form of electronic currents induced in the solid surface
under flowing liquid Gosh2003 ; Cohen2003 ; Zhao2008 ; Rabinowitz2020 ;
Dhiman2011 ; Yin2012 ; Lee2013 ; Kuriya2020 ; Yin2014 ; Park2017 , the
modification of wetting by electronic screening Comtet2017 ; Schlaich2022 , or
anomalies in the water-carbon friction Kavokine2022 . The interaction
mechanisms at stake remain largely unexplained and to be properly understood.

Figure 1:  Experimental set-up. a, Schematic of the experiment. A quartz
capillary of tip diameter of 500 nm is filled with liquid and glued on the
lower prong of a millimetric quartz tuning fork. Once the tip is in contact
with the graphite substrate, a liquid droplet is formed at its extremity. The
scanner allows for droplet relative motion on the graphite flake in the XY-
plane, while current is measured between both gold electrodes. An additional
electric potential drop $\Delta V$ can be imposed between the electrodes.
Inset: The resonance curve of the tuning fork whose frequency shift is used to
deposit the droplet and maintain a gentle mechanical contact. b, Frequency
shift response of the AFM during a typical experiment, indicating the
stiffness of the pipette-sample contact. First, the pipette is brought into
contact with the sample (1). After the contact, a slow relaxation (2)
indicates the formation of the droplet. Finally, the sample is put into
horizontal motion with respect to the droplet while a PID controller keeps a
constant frequency shift (3). c, AFM image of a multilayer graphene sample
(3.6 nm thick) exhibiting a high density of wrinkles. d, AFM image of a flat
multilayer graphene sample (1.5 nm-thick). e, Schematic of droplet motion. f,
Current (nA) as a function of time (s) resulting from droplet oscillations of
2 $\mu$m amplitude and 0.5 Hz frequency, on the wrinkled sample displayed on c
(black dots) and on the flat graphite sample displayed on d (red dots). g, The
corresponding current spectra as a function of frequency. For the wrinkled
sample (black curve), the spectrum displays a peak at the droplet oscillation
frequency (0.5 Hz).

In this article, we report the generation of electric current in a few-layer
graphene sample by the displacement of a sub-micrometer scale liquid droplet
along its surface. We deposited a thin graphite flake (1 to 70 nm in
thickness) onto a Si/SiO2 substrate; two gold electrodes, separated by a
distance of $\sim 10\mu$m and connected to a low noise current amplifier,
allowed us to measure the electric current passing through the flake (see
Figure S1). The sample was placed in a tuning fork atomic force microscope
(AFM), that was specifically designed for controlled liquid deposition, see
Figure 1a. The liquid was introduced into a quartz capillary, which was then
glued to the quartz tuning fork (see Figure S2). A piezo-dither induced a
nanometric oscillation of the tuning fork at its resonant frequency ($f_{0}$
$\approx$ 32 kHz), a typical resonance curve is shown in the inset of Figure
1a. By monitoring precisely the tuning fork oscillation’s phase and amplitude
as the pipette tip approach the sample, we achieved a fully controlled contact
between the carbon surface and the pipette’s aperture, leading to the
formation of a liquid capillary bridge. The capillary bridge then evolved into
a droplet, of radius fixed by the outer diameter of the tip (500 nm) Fang2006
; Fabie2012 . This novel experimental set-up allows to deposit very viscous
liquid at nano and micro scale over a broad range of substrates hummink . We
used a room temperature ionic liquid (BMIM-PF6) and a neutral and apolar
silicon oil (Polyphenyl-methylsiloxane), both very viscous with a viscosity of
0.3 Pa$\cdot$s and 0.1 Pa$\cdot$s, respectively, and with very small vapor
pressures, allowing us to safely neglect evaporation. The continuous
monitoring of the contact with both the in-situ optical microscope and the AFM
frequency shift signal allows us to ensure that the droplet remained hooked to
the tip, sliding over the carbon surface. Moreover, motion does preserve the
roughly spherical shape of the droplet (see Figure S2d).

In a typical experiment, the droplet is put into oscillatory motion along the
carbon surface by a piezo-scanner with an amplitude of a few microns at a
frequency of the order of 1 Hz. An electric current through the carbon surface
is then measured at the droplet oscillation frequency (Figure 1e and Figure
S3). As shown on Figure 1a, our set-up enables to further impose a constant
potential drop $\Delta V$ (typically between -50 mV and 50 mV) between the
electrodes. We report electric currents in the tens of nano-Ampere range
induced by a droplet moving at a few micrometers per second. To allow for a
comparison with other current generation experiments, we define the current
density as the AC-current amplitude divided by the diameter of the droplet (of
order $1\rm\mu m$) : $j=I/(2R_{\rm drop})$ typically in the tens of
$\rm{nA/\mu m}$ range.

Firstly, several benchmark measurements allowed us to ensure that the current
generation is due to the specific interaction of the liquid droplet with the
few-layer graphene. We checked that no alternating current is recorded when
the deposited droplet oscillates over the SiO2 substrate between the
electrodes but without the multilayer graphene substrate. We further
controlled that no current is generated when the pipette is not in contact
with the graphite but slightly above or even when a tungsten tip is used
instead of the liquid filled pipette, see Figure S4. Altogether, we can safely
conclude that the measured effect is due to the displacement of the liquid
droplet on the carbon surface.

Figure 2:  Generation of a droplet motion-induced electronic current. a,
Current density (nA/$\mu m$) as a function of fluid velocity $v=2\pi fa$ for 4
different samples. Each curve is measured at a bias $\Delta V$ indicated in
the corresponding plot. The multi-layer graphene subtrates have thickness 3.3,
1.7, 3.6 and 8.2nm, respectively and their corresponding surface roughness
measured using AFM are [55,50,40,20] nm respectively. b, Time-dependent
current under droplet oscillation for various voltage bias $\Delta V$. c,
Effect of the voltage bias $\Delta V$ on the electro-fluidic conductivity
$\sigma_{\rm ef}=j/v$, showing a linear dependency. The dotted lines are
guides to the eye. d, Electro-fluidic conductivity $\sigma_{\rm ef}$ (at
$\Delta V=20$ mV) versus the substrate corrugation, measured using AFM images.
The black diamonds correspond to current measurements below the detection
level.

On Figure 2a, we report results of several samples (with thickness varying
between 1.7 and 8.2 nm), where an electro-fluidic current is measured. We
observe that for a fixed oscillation frequency $f$ and bias voltage $\Delta
V$, the amplitude of the AC generated current increases linearly with the
droplet oscillation amplitude $a$, or equivalently, with the droplet peak
velocity ($v=2\pi fa$). Taking advantage of the demonstrated linear dependency
of the current density with respect to the droplet peak velocity, we define
the electro-fluidic conductivity of our samples as

$\sigma_{\rm ef}={j\over v},$

quantifying the cross-coupling between electronic current $j$ and liquid flow
$v$. Among the five samples presented here, we show an electro-fluidic
conductivity ranging from 2.8 to 21.6 $\rm{nA.s/\mu m^{2}}$ for a droplet size
in the micrometer range.

To understand the mechanism driving this electro-fluidic current and having in
mind the strongly energy-dependent electronic density of graphene close to the
Fermi level (Neto2009, ), we tune the bias voltage $\Delta V$ between the
electrodes and measure its influence on the electro-fluidic current. We show
that the alternating current induced by the droplet oscillation strongly
depends on $\Delta V$, with a linear dependency. This is highlighted in
Figures 2b-c, where the electro-fluidic conductivity is shown to scale roughly
linearly with the potential drop $\Delta V$, with a small or negligeable shift
at zero potential drop and a sign reversal of the current for negative versus
positive $\Delta V$. Finally, the slope of this linear dependency $\sigma_{\rm
ef}/\Delta V$ is in the range of 0.05 to 0.2 $\rm{nA.s/\mu m^{2}/mV}$, see
Figure 2c, and slightly varies across samples.

Now, an unexpected feature of the experimental results is the sharp
distinction among carbon samples regarding the amplitude of the the electro-
fluidic conductivity. Over all few-layer-graphene measured, with thicknesses
ranging from 1 nm up to 70 nm, we observe that the thinner ones tends to
exhibit the largest electro-fluidic conductivity, see Figure S7a. However,
investigating the surface topography of the carbon samples using AFM imaging,
such as on Figure 1c-d, we rather reveal a correlation between the amplitude
of the electro-fluidic conductivity and the existence of surface corrugation
in the forms of folds and wrinkles. Notably, thinner samples exhibit a large
number of folds and wrinkles whereas thicker samples present a flat surface
without folds or defects (see Figure S1). This difference is due to the
fabrication process involving the transfer of the exfoliated graphene flake
onto the oxidized silicon wafer (See Sec. S-I) : the thinner flakes are softer
and more flexible than the thicker ones, and more likely to exhibit wrinkles
Liu2011 ; Deng2016 , see Figure 1a and Figure S7. In order to disentangle the
role of wrinkles from that of sample thickness, we implemented an alternative
fabrication method in which all transfer steps are avoided and few-layer
graphene flakes were directly exfoliated on the Si/SiO2 wafer. This allowed us
to obtain sub-10 nm, un-transferred and smooth graphene samples with a uniform
wrinkle-free surface (see Figure 1b and Figure S1) to compare with the
transferred and crumpled flakes of similar thickness showing a much larger
surface corrugation (see Figure S9 for a Raman analysis of wrinkles). As shown
on Figure 2d, droplet motion on smooth samples results in no detectable
current. A contrario, crumpled flakes of the same thickness have a high
wrinkle density and show the strongest electro-fluidic conductivity, in the
$\rm{nA.s/\mu m^{2}}$ range. Our results reveal that wrinkles are instrumental
to the current generation mechanism, while there is no evidence of an effect
of graphene thickness. Further evidence of the crucial role of wrinkles is
provided by the dependence of the generated current on the droplet oscillation
direction, see Figure S8. As expected, the current amplitude is 180$\degree$
periodic with respect to the angle between the droplet trajectory and the
electrodes. However, the maximal amplitude is not necessarily reached when the
droplet oscillation is perpendicular to the electrodes, and the direction
yielding the maximum current strongly depends on the position on the sample’s
surface. We argue that this spatial and directional variability is related to
the random distribution in height, position and direction of the wrinkles on
the carbon surface. This further highlights the role of wrinkles as current
amplifiers. Let us note for completeness that the data displayed on Figure 2
are obtained along the direction with maximal amplitude.

Summarizing our experimental results, we have shown that the motion of a
micrometric droplet on a crumpled, few-layer graphene surface generates a
strong electro-fluidic current in the flake, which is proportional to the
velocity of the droplet, as well as to the bias voltage applied between the
electrodes. A remarkable feature is the strong dependence of the magnitude of
the generated current on the surface corrugation of the flake.

While our results echo previous reports of electric potential differences
induced by water flow in carbon-based nanomaterials Kral2001 ; Gosh2003 ;
Cohen2003 ; Persson2004 ; Zhao2008 ; Rabinowitz2020 ; Dhiman2011 ; Yin2012 ;
Lee2013 ; Kuriya2020 ; Yin2014 ; Park2017 , our nanoscale investigation puts
us in position to disentangle various effects at the root of electronic
current generation under fluid flow, and get much insights into the molecular
mechanism underlying the phenomenon. Previously, several mechanisms have been
invoked to rationalize such conversion, including streaming potential effects
Cohen2003 , adsorbed/desorbed ion hopping Persson2004 ; Dhiman2011 , Coulomb
drag Gosh2003 ; Rabinowitz2020  or charging/discharging of a pseudo-
capacitance Yin2014 ; Park2017 . These mechanisms involve Coulomb interactions
between the polar liquid – exhibiting molecular partial charges – and the
carbon surface as the main pathway for momentum transfer from the liquid to
the electrons. As we now show, these previous explanations fail to account for
the phenomenon at stake. Indeed, the strong effect of surface corrugation
reported here suggests that current generation proceeds through a ’mechanical’
transfer of momentum of the fluid towards the electronic degrees of freedom.
In order to confirm this hypothesis and rule out any effect caused by the
charges in the ionic liquid, we first repeated the experiment, now with an
apolar silicone oil replacing the ionic liquid (and a wrinkled few-layer
graphene as a substrate). We do observe qualitatively and quantitatively
similar current generation for the silicone oil and the ionic liquid, see
Figure S5. This discards coulombic couplings between the liquid molecules and
the electrons and all corresponding mechanisms based on charge-charge
interactions across the interface. These considerations naturally lead us to
reconsider phonon-drag mechanisms, as first proposed theoretically in 2001
Kral2001  as the driving mechanism for the measured electro-fluidic current.
The phonon drag relies on hydrodynamic (viscous) friction at the solid-liquid
interface, which transfers momentum to the solid’s phonons, thereby exciting a
’phonon wind’. Through the electron-phonon interaction, the phonon wind
’blows’ on the Fermi sea, so that electrons near the Fermi surface are dragged
along the direction of the liquid flow, hence producing an electric current
(see Figure 3a). A complete theory of phonon drag at the quantum level, within
the non-equilibrium Keldysh framework is developed in Coquinot2022 . Here, we
give only a semi quantitative description that allows us to capture the main
physical ingredients.

First, assuming a Couette flow within the liquid droplet, the hydrodynamic
friction force is $\textbf{F}_{0}\sim\eta\mathcal{A}_{\rm drop}\textbf{v}_{\rm
drop}/h_{\rm drop}$, $v_{\rm drop}$ is the drop velocity, $\eta$ its
viscosity, $\mathcal{A}_{\rm drop}$ its surface and $h_{\rm drop}$ its height
(here typically, $\eta=0.3\,\textnormal{Pa}\cdot\textnormal{s}$ and $h_{\rm
drop}\approx 1\,\mu\textnormal{m}$). In line with observations, wrinkles are
expected to enhance this hydrodynamics friction force. In a simple picture,
wrinkles are modelled as protusions on which the flowing liquid exerts a
Stokes force scaling (per wrinkle) as $\textbf{F}_{\rm W}\propto\eta h_{\rm
W}\textbf{v}_{\rm drop}$, where $h_{\rm W}$ is the typical height of the
wrinkles ($h_{\rm W}\sim 10-30\leavevmode\nobreak\ \rm nm$). One then obtains
a total friction force $\textbf{F}_{\rm W}\sim\textbf{F}_{0}(1+W)$, where we
introduced the dimensionless wrinkling number, $W=3\pi n_{\rm W}h_{\rm
W}h_{\rm drop}$, with $n_{\rm W}$ the wrinkle density to account for the
increase in viscous friction due to surface corrugation.

Figure 3:  Phonon drag mechanism. The flowing liquid a exerts a friction force
$F_{\rm W}$ on the graphite sample. This force is related to a momentum
transfer to the phonons, that generates an asymmetry in the phonon
distribution b which is biased in the direction of the liquid flow. Finally
this asymmetry propagates to the electronic distribution c by electron-phonon
interaction. This asymmetry in the electronic distribution translates into the
generation of a net electric current, whose sign depends on the nature of
charge carriers at the Fermi level (electrons or holes). We stress that this
is an out of equilibrium phenomenon in which most of the momentum is relaxed
to the crystal d by the Umklapp process with a rate $1/\tau_{\rm um}$.
Finally, on e, we compare the theoretical derivation of the electro-fluidic
conductivity $\sigma_{\rm ef}$ versus the bias voltage $\Delta V$. We stress
that the experimental data is the same as in Figure 2c and that the solid
lines are plotted using Eq. (4) assuming the bulk graphite effective mass
$m=0.1m_{\rm e}$ and $W$ the wrinkling number as the only fitting parameter
chosen between 5 and 20. This wrinkling number characterizes the enhancement
of Stokes friction due to wrinkles on the graphene surface.

Our model is sketched on Figure 3a-d : the hydrodynamic friction force
transfers momentum from the liquid (Figure 3a) to the solid exciting acoustic
phonons with dispersion relation $\epsilon_{\mathbf{q}}=\hbar qc$ where $c$ is
the phonon (sound) velocity ($c\approx 2\times 10^{4}\leavevmode\nobreak\ \rm
m\cdot s^{-1}$ Nika2009 ; Ochoa2011 ; Cong2019 ). The phonons are mainly
relaxed by transferring their momentum to the crystal (Figure 3d) through
Umklapp processes Nika2009 , on a typical timescale $\tau_{\rm um}\sim$ 10 ps
Klemens1994 . As sketched on Figure 3b, the constant influx of momentum
modifies the phononic distribution by shifting the energetic cost of a phonon
of wavevector $\mathbf{q}$ according to
$\epsilon_{\mathbf{q}}\mapsto\epsilon_{\mathbf{q}}-\hbar\mathbf{q}\cdot\mathbf{v}_{\rm
eff}$, which is lower along the direction of the liquid flow. Here
$\mathbf{v}_{\rm eff}$ is the effective velocity of the phonon wind: it is
determined by balancing the momentum fluxes in and out of the phonon system,
according to $\tau_{\rm um}\textbf{F}_{\rm W}/\mathcal{A}_{\rm
drop}=\Delta\textbf{P}_{\rm ph}/\mathcal{A}_{\rm
drop}=\int\frac{\mathrm{d}\mathbf{q}}{2\pi}\,\hbar\mathbf{q}\,n_{\rm B}(\hbar
qc-\hbar\mathbf{q}\cdot\textbf{v}_{\textnormal{eff}})$, with $n_{\rm B}$ the
Bose-Einstein distribution and $\mathbf{q}$ is the phonon wavevector, that we
assume two-dimensional. This yields

$\textbf{v}_{\textnormal{eff}}\simeq\frac{2\pi}{3\zeta(3)}\frac{\hbar^{2}c^{4}}{(k_{B}T)^{3}}\times(1+W)\eta\frac{\textbf{v}_{\rm
drop}}{h_{\rm drop}}\times\tau_{\rm um}$ (1)

Then, $v_{\textnormal{eff}}\approx 9\cdot 10^{4}\times(1+W)\times v_{\rm
drop}$. As the electrons scatter on the phonons, they acquire an average
velocity equal to the phonon wind velocity; our complete theory shows that
this is true whatever the form of the electron-phonon interaction Coquinot2022
. As sketched on Figure 3c, in a 2d model of the solid, this amounts to
shifting the band energies according to
$\epsilon_{\mathbf{q}}\mapsto\epsilon_{\mathbf{q}}-\hbar\mathbf{q}\cdot\mathbf{v}_{\rm
eff}$. Assuming for concreteness that the liquid flows to the right, this
roughly means that in an energy window of width $\hbar k_{\rm F}v_{\rm eff}$
around the Fermi level ($k_{\rm F}$ is the Fermi wavevector), the electrons
move to the right at the Fermi velocity $v_{\rm F}$. The corresponding current
density reads

$\textbf{j}=ev_{\rm F}\times N(\epsilon_{\rm F})\times[\hbar k_{\rm
F}\textbf{v}_{\rm eff}],$ (2)

where $N(\epsilon_{\rm F})$ is the density of states at the Fermi level
$\epsilon_{\rm F}$. We obtain a current density proportional to the droplet
velocity, to the liquid viscosity and to the wrinkling number W in excellent
agreement with the experiment. Moreover, the current is independent of the
chemical nature of the liquid as observed in experiment (ionic liquid and
silicone oil generate similar current). According to the complete derivation
presented in Coquinot2022  (see Sec. S-III for the main ingredients), we add
in the following a correction factor of order $1/2$ to the current density.

Electronic bands in graphene have a linear dispersion,
$\epsilon_{\mathbf{q}}=\hbar v_{\rm F}q$ ; however, in our multilayer samples
of few-nanometer thickness (see Figure S7a) one expects the electronic
properties to be better described by a parabolic band with effective mass $m$
Gruneis2008 ; Guinea2008 ; Charlier1992 :
$\epsilon_{\mathbf{q}}=\hbar^{2}q^{2}/2m$. The density of states is then
independent of energy: $N(\epsilon)=N(\epsilon_{\rm F})=m/\hbar^{2}$.
Altogether, we obtain the electro-fluidic cross conductivity

$\sigma_{\rm ef}=e\frac{m}{\hbar^{2}}\epsilon_{\rm
F}\times\frac{v_{\textnormal{eff}}}{v_{\rm drop}}.$ (3)

The prediction in Eq. (3) can now be compared with experimental results.
Experimentally, the applied bias voltage, sets the Fermi level $\epsilon_{\rm
F}\approx e\Delta V/2$, so that

$\sigma_{\rm ef}\approx 0.1\times m/m_{\rm e}\times(1+W)\times\Delta
V\,[\textnormal{nA.s/}\mu\textnormal{m}^{2}]$ (4)

with $\Delta V$ in mV. This estimate quantitatively reproduces the
conductivity measured experimentally of a few nA.s/$\mu$m2 with the correct
linear scalings of the electro-fluidic conductivity with the droplet velocity
and the bias voltage. In addition, this current is proportional to the
wrinkling number, in line with the observed impact of the presence of surface
wrinkles as current amplifiers. On Figure 3e, we compare this theoretical
prediction with the experimental measurement of $\sigma_{\rm ef}$ as a
function of the bias voltage. For the electronic effective mass, we consider
graphite in a parabolic approximation, $m\sim 0.1m_{\rm e}$, where $m_{\rm e}$
is the electron mass Gruneis2008 . The theoretical estimate involves no
fitting parameter other than the wrinkling number W chosen between 5 and 20 to
reproduce the experimental data.)
Note that although the above discussion focussed on electrons as charge
carriers, one expects that the current will change sign if the charge carriers
are holes. This explains why the generated current flips sign along with
$\Delta V$: for positive bias, the charge carriers are electrons in the
conduction band, whereas for negative bias, the charge carriers are holes in
the valence band (a seamless transition is ensured by the fact that the bands
are touching). Altogether, the proposed modeling strongly supports the
hypothesis that the experimentally observed electric current is due to charge
carriers being dragged by a phonon wind. Our model describes the momentum
transfer from the liquid to the phonon distribution where an imbalance in the
distribution develops. Scattering on the electron bath then results in an
electronic current.

In conclusion we report the generation of a very large electronic current
induced by the controlled motion of a liquid droplet at the surface of carbon
nanomaterials. Our results suggest that the current generation is mediated by
hydrodynamic momentum transfer from the liquid molecules to the phonons in the
solid. Our specifically tailored system is particularly well suited to
evidence the phonon-drag induced electronic current. Indeed, the combination
of a local micrometric flow on strongly wrinkled graphene samples along with
the choice of a very viscous and neutral liquid allows the phonon-drag as a
direct momentum transfer to dominate any other current generation mechanism.
The versatility of our setup allows to tune the fluid velocity as well as the
solid’s electronic properties using the bias between the electrodes to fully
characterize the phenomenon. Finally, all experimental results were
rationalized with our new dedicated model of the phonon-drag. Our findings
pave the way for the active control of fluid transport at the nanoscale by
harnessing the interplay between fluid molecules and the collective excitation
in the solid suggesting that novel nanofluidic functionalities can be proposed
where a flow of fluid at the interface with tunable solid can be generated by
externally excitable phonons or structural deformation waves. Further, the
coupling of electronic and micro-fluidic currents bears the promise of
ubiquitous and environmentally friendly energy production as demonstrated
recently in titanium carbide MXene based transpiration-driven electrokinetic
power generators (Bae2022, ).

## Acknowledgements

L.B. acknowledges funding from the EU H2020 Framework Programme/ERC Advanced
Grant agreement number 785911-Shadoks. A.S. acknowledges funding from FET-open
ProID. The Flatiron Institute is a division of the Simons Foundation. B.C.
acknowledges funding from a J.-P. Aguilar grant of the CFM Foundation, and
thanks Olivier Parcollet for hosting him at the Flatiron Institute. B.R.
acknowledges the Royal Society University Research Fellowship (URF/R1/180127)
and European Research Council Starting Grant (agreement number 852674 -
AngstroCAP). A.B. thanks Dr Ashok Keerthi for help in fabrication.

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H I     8-20     6.9468     0     6.89e+02     107965     109404     2.18     1.57     1.38     1.89     0.31     0.76     0.28     0.58     0.57     0.64

Ar II     $^2P^o\! -\! ^2P^o$ 3/2 - 1/2     6.9853     4     4.23e-02     0     1432     1105.30     10.23     1423.34     8.85     174.53     1.25     167.85     0.99     147.86     0.94

H I     8-19     7.0927     3     9.07e+02     107965     109375     2.41     0.22     –     –     –     –     –     –     –     –

H I     8-18     7.2717     4     1.22e+03     107965     109340     2.62     0.13     1.09     0.53     0.09     0.22     0.18     0.29     0.20     0.20

H$_2$     1-1 $S(5)$     7.2801     4     5.44e-08     5814     7187     0.16     0.19     0.12     0.79     0.40     0.31     0.91     0.43     0.66     0.28

Ni III     $^3F\! -\! ^3F$ 4-3     7.3492     35     6.50e-02     0     1361     12.66     0.19     7.54     0.52     3.53     0.38     2.78     0.21     2.59     0.21

He I     $^3D\! -\! ^3F^o$ multiplet     7.4334     3     –     193917     195262     0.40     0.02     –     –     –     –     –     –     –     –

H I     5-6     7.4599     2     1.02e+06     105292     106632     237.55     0.71     185.67     1.39     69.24     0.63     61.44     0.23     50.09     0.13

H I     8-17     7.4951     4     1.66e+03     107965     109299     1.48     0.52     –     –     –     –     –     –     –     –

H I     6-8     7.5025     4     1.56e+05     106632     107965     64.75     0.49     48.65     1.36     18.14     0.74     16.22     0.42     13.29     0.22

H I     7-11     7.5081     3     2.12e+04     107440     108772     15.37     0.81     11.08     3.11     3.95     2.03     3.73     1.47     3.09     0.60

H I     8-16     7.7804     2     2.33e+03     107965     109250     4.21     0.28     5.03     1.20     –     –     –     –     –     –

UID     –     7.8780     0     –     –     –     2.54     0.41     –     –     –     –     –     –     –     –

H$_2$     0-0 $S(4)$     8.0250     5     2.64e-08     1169     2415     16.95     0.16     16.26     0.64     57.39     0.47     96.98     0.68     85.43     0.34

H I     8-15     8.1549     4     3.36e+03     107965     109191     4.85     0.13     3.45     0.61     1.36     0.36     1.31     0.37     0.88     0.18

H I     9-27     8.3084     0     1.30e+02     108325     109528     0.92     0.15     –     –     –     –     –     –     –     –

H I     9-26     8.3907     4     1.59e+02     108325     109516     0.54     0.11     –     –     –     –     –     –     –     –

UID     –     8.4095     0     –     –     –     0.76     0.31     –     –     –     –     –     –     –     –

H I     9-25     8.4849     3     1.95e+02     108325     109503     0.57     0.40     –     –     –     –     –     –     –     –

H I     9-24     8.5938     -4     2.42e+02     108325     109488     1.04     0.48     –     –     –     –     –     –     –     –

H I     8-14     8.6645     3     5.01e+03     107965     109119     4.38     0.55     2.75     0.95     –     0.77     0.19     0.53     0.26     0.44

H I     9-23     8.7206     3     3.04e+02     108325     109471     0.74     0.33     –     –     –     –     –     –     –     –

H I     7-10     8.7601     3     3.90e+04     107440     108582     24.65     0.53     18.23     0.54     8.08     0.66     6.76     0.65     5.25     0.35

H I     9-22     8.8697     5     3.85e+02     108325     109452     0.72     0.24     –     –     –     –     –     –     –     –

Ar III     $^3P\! -\! ^3P$ 2 - 1     8.9914     -1     3.10e-02     0     1112     1466.00     1.67     795.10     1.49     489.49     0.92     413.35     0.68     333.44     0.42

H I     9-21     9.0470     4     4.95e+02     108325     109430     0.76     0.65     0.40     0.63     –     –     –     –     –     –

H I     9-20     9.2605     1     6.45e+02     108325     109404     0.81     0.20     0.58     0.07     0.47     0.11     0.43     0.06     0.30     0.08

H I     8-13     9.3920     4     7.80e+03     107965     109030     7.22     0.29     5.24     0.08     2.31     0.14     2.09     0.15     1.64     0.13

H I     9-19     9.5217     4     8.56e+02     108325     109375     0.83     0.56     0.68     0.41     –     –     –     –     –     –

H$_2$     0-0 $S(3)$     9.6649     8     9.84e-09     705     1740     38.46     0.21     22.03     0.47     63.15     0.32     199.96     0.16     148.41     0.17

H I     9-18     9.8470     9     1.16e+03     108325     109340     1.42     0.31     0.99     0.29     –     –     –     –     –     –

H I     9-17     10.2613     5     1.60e+03     108325     109299     2.60     0.25     1.91     0.33     0.84     0.18     0.67     0.12     0.59     0.13

UID     –     10.3845     0     –     –     –     1.40     0.16     0.46     0.19     0.07     0.20     0.29     0.08     0.08     0.11

S IV     $^2P^o\! -\! ^2P^o$ 1/2 - 3/2     10.5105     2     7.30e-03     0     951     492.10     0.56     413.84     0.78     271.81     0.56     251.25     0.39     190.24     0.30

Ni II     $^4F\! -\! ^4F$ 9/2 - 7/2     10.6822     8     2.71e-02     8394     9330     2.20     0.40     3.97     0.32     0.58     0.27     0.54     0.11     0.47     0.17

H I     9-16     10.8036     4     2.27e+03     108325     109250     2.79     0.20     1.98     0.28     0.57     0.20     0.50     0.11     0.54     0.05

He I     $^3S\! -\! ^3P^o$ multiplet     10.8820     5     –     –     –     0.62     0.22     0.51     0.59     –     –     –     –     –     –

Ni III     $^3F\! -\! ^3F$ 3 - 2     11.0023     14     2.70e-02     1361     2270     2.17     0.80     1.96     2.93     0.89     1.96     0.55     0.94     0.52     0.48

H I     7-9     11.3087     6     8.24e+04     107440     108325     29.05     1.69     22.42     2.75     9.41     2.17     7.86     1.15     6.27     1.06

H I     9-15     11.5395     6     3.32e+03     108325     109191     2.76     0.29     1.15     0.81     0.14     0.40     0.15     0.27     0.17     0.20

H I     10-20     12.1568     2     6.17e+02     108582     109404     0.99     0.19     –     –     –     –     –     –     –     –

H$_2$     0-0 $S(2)$     12.2786     7     2.76e-09     354     1169     24.67     0.24     20.48     0.26     56.56     0.15     107.33     0.12     89.38     0.05

H I     6-7     12.3719     9     4.56e+05     106632     107440     92.29     0.42     68.00     0.38     27.00     0.40     23.62     0.31     19.23     0.23

H I     8-11     12.3872     9     2.30e+04     107965     108772     10.66     0.47     7.55     0.72     3.22     0.59     3.10     0.40     2.25     0.29

H I     9-14     12.5871     8     5.08e+03     108325     109119     4.29     0.06     2.92     0.27     0.75     0.08     0.99     0.15     0.82     0.02

H I     10-19     12.6110     6     8.25e+02     108582     109375     1.35     0.26     –     –     –     –     –     –     –     –

Ne II     $^2P^o\! -\! ^2P^o$ 3/2 - 1/2     12.8135     7     8.32e-03     0     780     6156.40     0.79     4612.99     1.96     1422.69     1.40     1188.46     1.42     1013.34     0.65

H I     10-18     13.1880     5     1.13e+03     108582     109340     1.47     0.23     –     –     –     –     –     –     –     –

H I     10-17     13.9418     -1     1.58e+03     108582     109299     1.48     0.20     –     –     –     –     –     –     –     –

H I     9-13     14.1831     4     8.19e+03     108325     109030     5.02     0.31     3.14     0.82     0.95     0.50     0.87     0.29     0.70     0.18

UID     –     14.3330     0     –     –     –     0.93     0.49     0.90     0.59     0.50     0.32     0.56     0.36     0.30     0.12

Cl II     $^3P\! -\! ^3P$ 2 - 1     14.3678     3     7.62e-03     0     696     20.66     0.61     82.39     0.68     36.86     0.35     30.19     0.30     10.67     0.22

H I     10-16     14.9623     5     2.28e+03     108582     109250     2.11     0.22     1.51     0.40     –     –     –     –     –     –

Ne III     $^3P\! -\! ^3P$ 2 - 1     15.5550     10     5.85e-03     0     643     2039.97     0.74     1468.13     1.15     874.50     0.72     759.39     0.57     582.45     0.43

H I     8-10     16.2091     0     4.68e+04     107965     108582     16.53     0.55     10.92     0.89     4.44     0.73     4.50     0.53     3.33     0.41

H I     10-15     16.4117     -2     3.42e+03     108582     109191     4.28     1.01     6.84     5.23     –     –     –     –     –     –

H I     9-12     16.8806     12     1.43e+04     108325     108917     6.16     1.09     4.15     1.25     1.73     0.59     1.53     0.64     1.16     0.49

H$_2$     0-0 $S(1)$     17.0348     19     4.76e-10     118     706     24.97     1.00     19.89     1.60     46.92     1.29     114.43     1.05     119.29     0.70

P III     $^2P^o\! -\! ^2P^o$ 1/2 - 3/2     17.8846     4     1.47e-03     0     559     39.80     0.22     26.08     1.44     14.54     1.15     13.53     0.90     10.63     0.87

Fe II     $a^4F\! -\! a^4F$ 9/2 - 7/2     17.9360     -0     5.84e-03     1873     2430     15.17     0.93     30.99     0.22     6.09     0.25     5.05     0.35     3.02     0.16

UID     –     18.6510     0     –     –     –     –     –     –     –     7.69     0.98     9.40     0.73     7.54     0.66

S III     $^3P\! -\! ^3P$ 1 - 2     18.7130     -1     1.43e-03     299     833     5996.73     1.28     4127.26     1.75     1884.99     1.39     1681.10     0.73     1297.20     0.96

H I     7-8     19.0619     4     2.27e+05     107440     107965     46.39     2.04     35.43     3.73     14.22     1.88     12.01     1.30     9.97     1.11

Ar III     $^3P\! -\! ^3P$ 1 - 0     21.8291     -2     5.19e-03     1112     1570     94.13     1.38     48.27     2.02     27.58     0.68     24.18     0.60     19.10     0.70

H I     9-11     22.3405     -26     2.81e+04     108325     108772     13.28     0.68     –     –     –     –     –     –     –     –

Fe III     $^5D\! -\! ^5D$ 4 - 3     22.9250     15     2.80e-03     0     436     141.05     3.71     88.91     3.20     39.08     2.11     29.57     1.23     25.94     1.21

S I     $^3P\! -\! ^3P$ 2 - 1     25.2490     44     1.40e-03     0     396     –     –     –     –     6.64     4.04     7.57     3.49     16.38     1.94

Fe II     $a^6D\! -\! a^6D$ 9/2 - 7/2     25.9884     70     2.13e-03     0     385     55.80     10.65     298.31     33.29     158.19     14.68     152.99     11.69     37.46     8.31

aIntensities omitted where line is not present or too weak for  reliable measurement.
bReported uncertainties $\sigma$ are based on local noise measurement only (Fig. <ref>), suitable to determine S/N.
We suggest a 3% fractional uncertainty to represent systematic calibration uncertainties. See Sect. <ref> for details.


#  Large scale traffic forecasting with gradient boosting
Traffic4cast 2022 challenge
Team Bolt

Martin Lumiste
Bolt Technology
Tallinn, Estonia
<EMAIL_ADDRESS>
Andrei Ilie
Bolt Technology, University of Bucharest
Bucharest, Romania
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Accurate traffic forecasting is of the utmost importance for optimal travel
planning and for efficient city mobility. IARAI 111The Institute of Advanced
Research in Artificial Intelligence organizes Traffic4cast, a yearly traffic
prediction competition based on real-life data
222https://www.iarai.ac.at/traffic4cast/, aiming to leverage artificial
intelligence advances for producing accurate traffic estimates. We present our
solution to the IARAI Traffic4cast 2022 competition, in which the goal is to
develop algorithms for predicting road graph edge congestion classes and
supersegment-level travel times. In contrast to the previous years, this
year’s competition focuses on modelling graph edge level behaviour, rather
than more coarse aggregated grid-based traffic movies. Due to this, we
leverage a method familiar from tabular data modelling - gradient-boosted
decision tree ensembles. We reduce the dimensionality of the input data
representing traffic counters with the help of the classic PCA method and feed
it as input to a LightGBM model. This simple, fast, and scalable technique
allowed us to win second place in the core competition. The source code and
references to trained model files and submissions are available at
https://github.com/skandium/t4c22.

##  1 Introduction

The Institute of Advanced Research in Artificial Intelligence proposes a
yearly traffic forecasting challenge at NeurIPS, Traffic4cast. This
competition has the goal of uncovering and deepening the application of
artificial intelligence methods in the traffic forecasting domain.

This year, the goal of the competition was to leverage public vehicle counter
data 333Counts of vehicles passing through map nodes, measured automatically
with the aid of electronic devices. for producing city-wide traffic estimates.
There is wide applicability to this, as traffic counters are easily accessible
sensors that many cities already use. Efficient and accurate methods to infer
city-wide segment-level traffic patterns on top of live counter data would be
highly useful for optimising city mobility and travel planning. These would
also represent an affordable, easy-to-scale approach, as opposed to having
traffic cameras in most city intersections or having live data streams from
many probes covering the city continuously.

The core and extended Traffic4cast tracks featured somewhat different
prediction problems. The core track contains a multi-class classification task
at a segment level, where the goal is to predict one out of three predefined
congestion classes (green, yellow, or red), which represent traffic
intensities. The training labels are sparse relative to the entity we are
predicting: in any given training temporal slice, we might observe labels for
only about 5% of the edges that we will need to make predictions for. The
evaluation metric used is weighted cross entropy, placing more emphasis on the
classes that appear more rarely, with the highest weight for red (congested
segments), then for yellow, and then for green classes.

The extended track proposes a regression task at a supersegment
444Supersegments are sequences of segments created according to some given
methodology, connecting main intersections. level, optimizing for mean
absolute error. The training data for this task consists of dense labels (we
observe all supersegment labels at all times).

We reduce both problems to a tabular format, capturing intrinsic features of
the road entities we want to create predictions for (e.g: latitude, longitude,
target encoding of historical speeds) and features representing the local and
city-wide traffic states. Most interestingly, we capture the city-wide traffic
state by performing dimensionality reduction on top of the available counter
data at any temporal slice. This, together with the gradient-boosted modelling
approach we used, constitute the key elements of our solution. The methodology
remains similar for both tracks, but notes are provided below for where slight
task-specific modifications were required.

##  2 Solution

###  2.1 Motivation

It is well known in the literature and applied machine learning competitions
that gradient-boosted tree ensembles generally outperform neural networks on
tabular data Shwartz-Ziv and Armon [2021] Grinsztajn et al. [2022] Borisov et
al. [2021] Lundberg et al. [2020], especially when one considers the accuracy
and training time trade off. Recently, this has also been shown for large
scale multivariate time series forecasting problems. Makridakis et al. [2022]
On the other hand, fields like computer vision and natural language processing
are dominated by deep learning approaches due to their unstructured data which
must first be projected into a more easily separable space by the first layers
of the networks. It is then a natural question whether spatio-temporal graph
problems such as traffic prediction are close enough to tabular that they can
benefit from gradient boosting. Our main hypothesis was that this is the case.

Another core motivation for our solution was that traffic is highly seasonal.
Therefore, features like the day of the week and the time of the day would be
greatly informational. Unfortunately, such temporal features are not available
at test time, so we tried to recover them implicitly from the city traffic
state. Our hypothesis was that the global city traffic context would be
beneficial for modelling as it not only acts as a proxy for temporal features,
but is even more powerful due to being robust against distribution shifts that
may occur between train and test sets.

Figure 1: The first two principal components of the last traffic counter
measurements are highly separable for periods reflecting different traffic
conditions.

###  2.2 Data preprocessing

The dataset provided by the competition consisted of three main sources: a)
the vehicle counter time series with roughly four thousand counters per city
at a granularity of 15 minutes, b) task-specific labels for road segments,
also in 15-minute intervals (shifted one step ahead of the counter
observation) and c) static road graph attribute metadata (e.g the coordinates
of nodes or the road types of edges). Aggregated traffic movies similar to
previous years are also provided, but we do not use these in our solution.

As the counter volumes are provided as a time series, the first step is to
transform those into numeric features. Traffic4cast 2021 showed that using
only the most recent 1 hour of data carries enough signal (Eichenberger et al.
[2022]). Due to this, we calculate only the sum of the past hour and the very
last measurement as numeric features. To reduce the relatively high dimensions
of the counter data (Melbourne has 3980 counters and 10260 measurement
intervals) and to embed the global city traffic context, we compute the
principal components of the $k\times t$ counter matrix $\mathbf{V}$ where $k$
is the number of counters in the city and $t$ the number of observations. We
leverage the $k\times k$ covariance matrix, meaning our resulting principal
components are time series. In practice, we use the first 8 principal
components of the last volume and the first 5 principal components of the sums
- confirmed on the validation set to be the optimal strategy. We observe that
the first few principal components explain well the data variance. For
example, in Melbourne’s case, the top 5 principal components of the last
hour’s sum of counters features explain 93.12% of the variance.

For embedding the spatial context of edges and supersegments, we mostly use
raw $x$ and $y$ coordinates. For edges, we use the start node as its
representative point. However, as supersegments are defined as a list of nodes
which can be multiple kilometres long, we use its 2-dimensional medoid as the
representative point. Later experiments showed that also using the start and
end points of supersegments together with its medoid and distance can be
beneficial. This is because the supersegments start and end by definition at
popular intersections. As the set of these intersections has low cardinality
(roughly 500), there is value in trying to learn to profile them.

On the other hand, the cardinality of the modelled entity itself is very large
(roughly $100.000$ segments in core, thousands of supersegments in extended).
We save model training bandwidth by precomputing segment characteristic target
encoding features based on training set labels. Moreover, we calculate segment
characteristic features conditional on the traffic state of the whole city.
This is equivalent to the IARAI baseline provided in the extended track -
overall ETAs per supersegment, conditional on city traffic cluster - but used
as a feature, rather than prediction. As decision tree ensembles are high-
capacity learners, we additionally guard against the risk of data leakage by
only calculating target encoded features based on the train set if an edge had
a significant enough amount of observations within that cluster bucket,
otherwise overwriting the values with fallback constants. We also leverage the
raw edge-level speed labels (data source for both core and extended track
labels) and apply this methodology to calculate the median speed and free flow
speed per edge.

Finally, to embed the graph nature of the problem, we implement a simple
spatial weighting method inspired by graph message passing. Namely, given the
counter matrix $\mathbf{V}$ from above, let us define a symmetric weight
matrix $\mathbf{B}\in\mathbb{R}^{k\times k}$ where each row element is a
normalized score summing up to one. Each element $b_{i,j}$ gives the spatial
relevance of counter $i$ with respect to $j$. To find $\mathbf{B}$, one could
apply any monotonically decreasing function in distance row-wise on the
pairwise distance matrix of counters. For example, we experiment with applying
softmax on the inverse distances. By multiplying
$\mathbf{W}=\mathbf{V}^{T}\mathbf{B}\in\mathbb{R}^{t\times k}$ we have that
$w_{t,i}=\sum_{j}^{K}v_{t,j}\times b_{i,j}$ so each row gives the spatially
weighted context around each counter. Note that this is a fairly generic
approach, as $\mathbf{B}$ could be based on any distance metric such as
Euclidean or graph hop distances between the counters. By making $\mathbf{B}$
a sparse matrix, we could enforce nearest neighbour features. We experiment
with multiple Euclidean and nearest neighbour weighting methods and join the
resulting matrixes $\mathbf{W}$ to our dataset based on the nearest counter to
the road.

We give a high level overview of the data preprocessing we do in Appendix A,
Figure 2.

###  2.3 Training

Gradient boosting can be thought of as performing gradient descent in
functional space Friedman [2001] by
$\hat{y}_{i}=\sum_{m=1}^{n}h_{m}(x_{i})+h_{0}$ where the weak estimators
$h_{m}$ use the residuals of the previous ensemble prediction $h_{m-1}$ as
labels and $h_{0}$ is usually a constant. But this is not a requirement, we
can use any predictor as $h_{0}$ to provide a robust basis for the model to
improve upon. To speed up model convergence, we initialize it from a smart
baseline - the target encoding features described above. In practice, this
means that for the core track, our first iteration consists of the weighted
logits coming from the train set average class distributions. For the extended
part, we use traffic conditional median ETAs, similar to the IARAI baseline.
This helps the model become not only more accurate than the baseline already
in a few iterations, but also converge to a better end result, likely due to
the avoided complexity that making splits for each road entity would entail.

We use the highly efficient gradient boosting library LightGBM Ke et al.
[2017] for training. Hyperparameters are tuned once according to a predefined
stepwise Optuna strategy Ozaki [2020]. We only tune the hyperparameters on a
small subset of extended track data for Melbourne and use these as a sane
baseline for every model. Thereafter, the only hyperparameters that we choose
are the number of leaves in individual decision trees ($num\\_leaves$) and the
number of gradient boosting iterations ($num\\_iters$). Surprisingly, for the
core track, we found that the $num\\_leaves$ parameter which controls the tree
complexity by the number of unique terminal leaf nodes could be increased
almost arbitrarily, speeding up convergence significantly and not hurting the
holdout test set accuracy. We therefore use 5k leaves for Melbourne and 10k
for Madrid and London.

We simulate interleaved validation week selection, keeping in line with how
test set is separated. For example, Melbourne has the following weeks of the
year available in the training data:
$[23,25,27,29,31,33,35,37,39,41,43,45,47,49,51,53]$. We use weeks
$[25,33,41,49]$ for validation, and the rest for training.

We generally prototyped all city models locally on a random subsample of the
data before scaling them to a cloud instance. Table 2 shows the summary of our
training strategy 555Extended models were trained on a laptop with swap
enabled, actual memory usage is higher. We think that the quick training times
with no GPU requirements are what make LightGBM an attractive solution to
traffic forecasting, lowering the technical and financial barriers of entry in
this field. For even larger problems, there exists a distributed
implementation of LightGBM which has been shown to scale up to a 1.7 billion
row dataset Microsoft [2017].

Table 1: Results Track | Model | PCA | init_score | Target encoding | Tuned | Score
---|---|---|---|---|---|---
Core |  |  |  |  |  |  Weighted cross-entropy
| IARAI baseline 666Vanilla GNN |  |  |  |  | 0.8978
| Bolt | X |  |  |  | 0.8877
| Bolt | X | X |  |  | 0.8692
| Bolt | X | X | X | X | 0.8497
Extended |  |  |  |  |  |  Mean absolute error
| IARAI baseline 777Average ETA per supersegment, conditional on city traffic cluster |  |  |  |  | 67.21
| Bolt |  | X |  |  | 65.03
| Bolt | X | X |  |  | 63.56
| Bolt | X | X | X |  | 61.58
| Bolt | X | X | X | X | 61.25

###  2.4 Results

Our approach ranked second in the core track and fourth in the extended one.
The results are presented in Table 1, where we also introduce the ablation
study of our models. We perform ablation on four of the key components of our
solutions, which are shared across both core and extended models: using city-
wide PCA features (PCA), feeding (traffic conditional) target encodings as
initialization score to LightGBM (init_score), using (traffic conditional)
target encodings (at multiple granularities) as features to the model (Target
encoding), and performing hyperparameter tuning using Optuna and early
stopping (Tuned).

##  3 Discussion

The biggest surprise to the authors was that message passing did not work at
all. We found no meaningful improvement from adding locally weighted counter
features, especially when compared to the city global principal component
features. We experimented with different weighting methods and the most
successful ones tended to be ones that still used the entire city’s counter
data and applied only a marginal penalty for high distances, essentially
converging to city global features. We think that this can be partly explained
by the heuristic nature of our feature engineering, while a weighting schema
learned by a GNN should work better. However, it should not be discounted that
decision trees are natural interaction learners - indeed, the original
motivation for the tree method came from analyzing interactions in survey data
Morgan and Sonquist [1963]. It seems that passing a combination of city global
context features and the spatial coordinates of roads is enough for learning
their interaction and therefore also the local context, conditional on the
overall traffic state.

Traffic is a spatio-temporal process and hiding the temporal data by
competition design definitely limits any solution’s potential accuracy.
However, we find that our reduced form city context features are enough to
recover the temporal features and act as powerful proxies for them. In a real-
life setting, including both the time features and counter data should give
the best result.

Unsurprisingly, target encodings were a critical component of making the
models competitive. However, we additionally found that the degree to which
one can leverage this signal is proportional to the density of the labels
used. Due to this, we managed to develop very intricate target encoded
features in the extended track, calculating multiple granularities of traffic
quantiles for observations and using all of them as features, without any
visible lack of generalization. On the core track, we are conditioning the
target encoded features only on a binary "low" or "high" traffic regime,
experiments with more granular partitions started to suffer due to leakage.

Although we found gradient-boosted decision tree ensembles to be competitive
for this competition, it is worth pointing out some of this architecture’s
shortcomings as well. Firstly, there is no natural embedding method for
decision trees. One could pass entity tokens as categorical features and
LightGBM would try to apply Fisher grouping Fisher [1958], but this is
essentially useless for a high cardinality feature such as $edge\\_id$. In
practice, we circumvent this by calculating target encoded features for each
token we think is relevant. However, this is suboptimal, as we need to
explicitly handle tokens with few values and there is an increased risk of
data leakage because of using the labels directly. Closely related to the
first problem is that it is difficult to embed sequential features with our
tabular method. In the extended track, explicit node sequences provide an
obvious sequential signal for predicting ETAs. An architecture such as an RNN
would elegantly aggregate the sequential node embeddings into vector space.
Therefore a neural network with a sequential input component should be
competitive in the extended track, but an alternative hybrid approach, as
shown by Leontjeva and Kuzovkin [2016] would be to extract the sequential
activations and use them as features in our tabular model. We did not
experiment with these approaches in the competition and these could remain as
future avenues of traffic research.

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##  Appendix A Data preprocessing overview

Figure 2: Summary of the preprocessed data used for training. We are using
three main sources of data: counter data, training set labels and static map
data.

##  Appendix B Training summary

Table 2 gives the LightGBM training summary.

Table 2: Training summary Track | City | Rows (Mil) | Features | RAM (GB) | Leaves | Iters | Time (h)
---|---|---|---|---|---|---|---
Core | Melbourne | 98 | 44 | 256 | 5k | 250 | <1
Core | London | 337 | 44 | 512 | 10k | 450 | 4
Core | Madrid | 482 | 44 | 768 | 10k | 600 | 5
Extended | Melbourne | 33 | 31 | 32 | 350 | 3200 | 1.5
Extended | London | 42 | 31 | 32 | 400 | 5700 | 2
Extended | Madrid | 41 | 31 | 32 | 350 | 4900 | 2

##  Appendix C Feature importance

Figure 3: Exact Shapley decision tree feature importances - core, Melbourne
Lundberg et al. [2020]

Figure 3 visualizes the most important features (out of 44) of a fully trained
model for core track, including their contributions to each congestion class.
We see how city global features such as PCs and $volumes$ (simple city wide
time based medians) dominate the predictions, with the spatial context in
$x$/$y$ close behind. Everything else comes from edge specific target
encodings. Edge attributes do not make it into the most relevant features.