# Inverse-designed multi-dimensional silicon photonic transmitters

K.Y.Yang1,∗, A.D.White1,∗, F.Ashtiani2,∗, C.Shirpurkar3,∗, S.V.Pericherla3,∗,
L.Chang4, H.Song5, K.Zou5, H.Zhou5, K.Pang5, J.Yang1, M.A.Guidry1, D.M.Lukin1,
H.Hao2, L.Trask3, G.H.Ahn1, A.Netherton4, T.C.Briles6, J.R.Stone6,
L.Rechtman7, J.S.Stone8, K.Van Gasse1, J.L.Skarda1, L.Su1, D.Vercruysse1,
J.P.W.MacLean1, S.Aghaeimeibodi1, M.-J.Li8, D.A.B.Miller1, D.Marom7,
S.B.Papp6,9, A.E.Willner5, J.E.Bowers4, P.J.Delfyett3, F.Aflatouni2,
and J.Vučković1,†
1E.L.Ginzton Laboratory, Stanford University, Stanford, CA, USA.
2Department of Electrical and Systems Engineering, University of Pennsylvania,
Philadelphia, PA, USA.
3The College of Optics and Photonics, University of Central Florida, Orlando,
FL 32816, USA.
4Department of Electrical and Computer Engineering, University of California,
Santa Barbara, CA, USA.
5Department of Electrical Engineering, University of Southern California, Los
Angeles, CA, USA
6Time and Frequency Division, National Institute of Standards and Technology,
Boulder, CO, USA
7Applied Physics Department, The Hebrew University of Jerusalem, Jerusalem,
Israel
8Corning Incorporated, Corning, NY, USA
9Department of Physics, University of Colorado, Boulder, CO, USA
†Corresponding author<EMAIL_ADDRESS>*These authors contributed equally
to this work.

###### Abstract

Modern microelectronic processors have migrated towards parallel computing
architectures with many-core processors. However, such expansion comes with
diminishing returns exacted by the high cost of data movement between
individual processors. The use of optical interconnectsMiller:2009:IEEE ;
Miller:2017:JLT  has burgeoned as a promising technology that can address the
limits of this data transfer. While recent pushes to enhance optical
communication have focused on developing wavelength-division multiplexing
technology, this approach will eventually saturate the usable bandwidth, and
new dimensions of data transfer will be paramount to fulfill the ever-growing
need for speedWillner:2012:NaturePhotonics ; Siddharth:2013:Science ;
Richardson:2013:NaturePhotonics ; Kahn:2017:NaturePhotonics . Here we
demonstrate an integrated intra- and inter-chip multi-dimensional
communication scheme enabled by photonic inverse design. Using inverse-
designed mode-division multiplexers, we combine wavelength- and mode-
multiplexing and send massively parallel data through nano-photonic waveguides
and optical fibres. Crucially, as we take advantage of an orthogonal optical
basis, our approach is inherently scalable to a multiplicative enhancement
over the current state of the art.

Figure 1: An optical transmitter using multi-dimensional multiplexing (a)
Principle of chip-to-chip WDM-MDM data transmission. A frequency comb laser is
evenly distributed to individual WDM transmitters (ring modulators) using
cascaded Y-junction splitters. The multi-frequency data from each WDM
transmitter are routed into spatial modes of a multi-mode waveguide using an
MDM multiplexer. The optical data can then be transmitted through a multimode
optical link to the receiver, where the mode and wavelength channels are
separated by MDM-WDM demultiplexers and detected using photodiodes. (b)
Microscope image of integrated four channel MDM transceiver (Inset 1: ring
modulator, Inset 2: MDM multiplexer, Inset 3: Si/Ge photodetector, Inset 4:
wire-bonded chip). Figure 2: Photonic inverse design of broadband, low-
crosstalk, all-passive MDM multiplexers. (a) SEM images of back-to-back MDM
multiplexer devices with simulated mode conversion (right). (b) Measured
S-parameters of the back-to-back MDM multiplexer structure. The structure is
optimized at multiple wavelengths over the entire C band. The broadband
structure is utilized in the WDM-MDM data transmission experiments using
soliton microcombs (13-wavelength channels, 300 GHz channel spacing over 3.6
THz spectral span; see Fig.3).

Rapid advances in computing and networking have led to an explosion of new
technology in areas ranging from genomicsAlipanahi:2015:NatureBioTech  and
medicineTopol:2019:NatureMedicine  to multimedia streaming to artificial
intelligenceWeiss:1999 ; Haddadi:2019:IEEE ; Subhasish:2021:NatureElectronics
. Now, as computation has become increasingly parallelised and offloaded to
the cloudBerkeley:2010 ; IBM:2017 , the limits of performance lie in the
interconnects between computational nodes. In data centers, massive amounts of
information need to be constantly transferred between many individual
processors and memory. While this can be done electrically, the higher
frequency and lower loss of on-chip electro-optical interconnects allow for
wider bandwidth and greater energy efficiencyMiller:2009:IEEE ;
Miller:2017:JLT . Optical interconnects are further enhanced by wavelength-
division multiplexing (WDM), which enables parallel signal transmission by
independently encoding data on multiple frequencies of
lightStojanovic:2015:Nature ; Koos:2017:Nature ; Bowers:2019:Optica ;
Ballani:2020 . However, this approach is still fundamentally limited to the
bandwidth that optical fibres can support. To further increase the bandwidth
of links, other dimensions of signal encoding must be used.

One promising dimension that can be utilized for multiplexing is the spatial
domain; light can be decomposed into a set of optical beams with orthogonal
spatial cross-sections. These orthogonal spatial modes in multi-mode optical
waveguides or in free space can serve as independent communication
channelsMiller:2019:Tutorial ; Willner:2012:NaturePhotonics ;
Richardson:2013:NaturePhotonics ; Siddharth:2013:Science ; Ryf:2011:JLT ;
Van:2014:NaturePhotonics ; Kahn:2017:NaturePhotonics ; Puttnam:2020:JLT ;
Rademacher:2020:ECOC , each of which can support a full WDM link. This
independence gives mode-division multiplexing (MDM) a multiplicative effect on
the bandwidth of an optical link. Recently, significant progress has been made
towards integrating mode and wavelength division multiplexing together on
chipGabrielli:2012:NatureCommunications ; Luo:2014:NatureCommunications ;
Miller:2020:Optica ; Dai:2018:LPR ; Baumann:2018:OFC ; Bell:2020 .

In this work, we present a multi-wavelength, multi-mode communication scheme
for on-chip and chip-to-chip interconnects. Using photonic inverse
designPiggott:2015:NaturePhotonics ; Molesky:2018:NaturePhotonics , we
implement a low-crosstalk, all-passive silicon MDM that supports parallel WDM
channels over a 15-THz spectral bandwidth. With this device, we demonstrate
error-free parallel data transmission using an integrated frequency comb
laserPapp:2020:PhyRevAppl  and multi-dimensional multiplexing: 52 data
channels derived from 13 wavelength channels launched into four spatial mode
channels. The MDM device also enables spectrally efficient WDM due to its non-
resonant operation, allowing us to demonstrate a 1.28-Tb/s aggregated data
rate using only 320 GHz of the available spectral bandwidth. Additionally, in
combination with inverse designed beam emitters and a multimode-matched
fibreMarom:2019:JSTQE ; Marom:2017:IPC , we demonstrate chip-to-chip data
transmission across four spatial mode channels at three different wavelengths
between separate silicon chips.

A silicon multi-dimensional transmitter

We use inverse design to implement broadband multi-dimensional multiplexing on
an integrated silicon photonics platform. We show that inverse-designed MDM
devices can operate over a large, uninterrupted spectral bandwidth, and are
thus able to be combined with silicon photonic WDM circuits to achieve
orthogonal data encoding. In addition, the multiplexed data can be mapped
using silicon photonics to orthogonal spatial channels of free space or
optical fibres for chip-to-chip data transmission. The principles of this
massively parallel optical communications based on WDM-MDM are illustrated in
Fig.1a. A common multi-frequency laser sourcekallfass:2013:NaturePhotonics ;
Koos:2017:Nature ; Oxenlowe:2018:NaturePhotonics ; Loncar:2019:Nature ;
Moss:2020:NatureCommunications ; Bowers:2020:Nature ; Papp:2020:PhyRevAppl  is
evenly distributed into multiple WDM transmitter circuits, and each WDM
circuit independently encodes data onto different frequencies of light. An
inverse-designed MDM multiplexer takes the overlapping modes from multiple WDM
transmitters and transforms them into co-propagating spatially orthogonal
modes of a multi-mode optical waveguide. An efficient chip-to-chip optical
link (e.g. surface normal beam emitter) enables the transmission of the WDM-
MDM data between separate silicon chips throughout free space or optical
fibre. Finally, a receiver circuit de-multiplexes the spatially and spectrally
mapped signals and converts the data to electronic circuits via
photodetectors.

Fig.1b shows a simplified implementation of the proposed communication scheme
– four MDM channel transceiver with one ring modulator per spatial mode
(4-mode, 1-wavelength/mode). The chip was fabricated using a commercial
silicon foundry process, and the devices were designed to comply with the
process design rulesPiggott:2015:NaturePhotonics ; Su:2020 ;
Piggott:2020:ACSPhotonics  (details of the design and process are described in
the Methods). The integrated system consists of four ring modulators, MDM
multiplexers, and Si/Ge photodetectors. The MDM multiplexer, optimized for
operation in the communication wavelength band (Inset 2 of Fig.1b), routes
data from each ring modulator into orthogonal spatial modes (TE00, TE10, TE20,
TE30). It is important to note that the MDM multiplexer doesn’t require post-
processing/trimming or temperature tuning. To verify the operation of the MDM
link, we measured an open eye diagram of the integrated MDM transceiver and
the modal cross-talk of the all-passive MDM structure which ranges from -10 to
-20 dB (see Fig.S1). Our demonstration can be in principle scaled up to
implement a massively parallel WDM-MDM data transmission in the next section.

Figure 3: On-chip interconnect: WDM - MDM data transmission with a soliton
microcomb laser. (a) Parallel WDM-MDM data transmission scheme using soliton
microcombs as a multi-wavelength source. The transmitter optical source is
generated by pumping the Si3N4 microresonator (inset: optical microscope image
of the device) with a CW laser. WDM de-multiplxers (W-DEMUX) separate the comb
lines and intensity modulators (EOM) encode independent data (NRZ at a symbol
rate of 10 GBd/channel). The WDM data are recombined using WDM multiplexers
(W-MUX), and are coupled to on-chip MDM inputs simultaneously using a fibre
array. At the receiver, all the channels are separated by mode and wavelength
de-multiplexers and detected using a photodiode (PD). In our experiments, we
independently modulate even and odd WDM channels using two EOMs for emulating
WDM transmission (see Method section for more details). (b) Optical spectrum
of the soliton microcomb. For the data transmission measurements, 13 comb
lines at the C band (red shaded region) were modulated with 10-Gb/s NRZ
signals. (c) 10-Gb/s eye diagrams of all data channels directly detected using
PD. (d) Measured BERs (1012 bits compared) of the transmitted data channels,
and (e) BER sensitivity versus received optical power (or electrical voltage
amplitude). Figure 4: Chip-to-chip interconnect using inverse-designed beam
emitters and a rectangular core multimode fibre. (a) MDM chip-to-chip data
transmission scheme using a multimode fibre. A CW laser is modulated using an
amplitude modulator, and the data signal is coupled to a Tx input waveguide
via a lensed fibre. The Tx chip sends multi-channel MDM beams to a multimode
fibre using an inverse-designed coupler. The MDM data are sent to the Rx chip
that separates each mode into a single-mode output. All data channels are
directly detected using photodiodes at the receiver side. (b) Infrared images
of mode profiles at Tx-out measured at 1540 nm are overlaid on device
microscope images (bar: 10 $\mu$m). (c) Measured chip-to-chip MDM channel
cross-talks and insertion losses at 1540 nm (Left: multimode fibre chip-to-
chip link, Right: free-space). (d) 10-Gb/s eye diagrams for a multimode fibre
chip-to-chip link at three different frequencies. The signal degradation of
the mode channel 4 is attributed to the mode-dependent insertion loss.

On-chip interconnect

In this section, we show error-free data transmission using a multi-
dimensional multiplexing and integrated laser sources – demonstrating for the
first time a silicon photonic MDM interconnect with a chip-scale frequency
comb source. The MDM structure supports a broadband operation as well as an
arbitrary channel spacing for high-speed dense WDM (DWDM) communications. We
use external modulators and a 100-GHz-grid DWDM in this experiment; however
the MDM technology is fully compatible with silicon photonic WDM
transmittersStojanovic:2015:Nature ; Wade:2021:OFC ; Bergman:2021:arXiv .

The inverse-designed MDM devices were fabricated in a standard silicon-on-
insulator platform (silicon thickness: 220 nm). Fig.2a shows a scanning
electron micrograph (SEM) image of the MDM muliplexer, whose design area is
6.5 $\times$ 6.5 $\mu m^{2}$ with a foundry-compatible minimum feature size
(80 nm)GF:2020:OFC . The MDM multiplexer routes the fundamental transverse-
electric mode (TE00) of the 500-nm-wide input waveguides to the TE00 (channel
1), TE10 (channel 2), TE20 (channel 3), and TE30 (channel 4) modes of the
1800-nm-wide output waveguide. The device structure is optimized for operation
over multiple wavelengths within the communication wavelength band, as
broadband optimization has previously been shown to be an effective heuristic
for fabrication robustnessPiggott:2020:ACSPhotonics ;
Yang:2020:NaturePhotonics . Fig.2b shows measured S-parameters of the back-to-
back MDM multiplexers. We find that the peak insertion loss is less than 0.8
dB and 3-dB bandwidth is wider than 150 nm for all mode channels.
Additionally, the modal cross-talk, which must be suppressed to avoid MDM
signal degradation Luo:2014:NatureCommunications , is less than -18 dB for all
mode channels. Notably, the cross-talk of the broadband design is
significantly lower compared to that of the narrow band design (see Fig.S2),
which can be attributed to fabrication-error robustness.

The conceptual diagram of WDM-MDM data transmission using Si3N4 soliton
microcombs is depicted in Fig.3a. The soliton microcombs are generated as
circulating pulses in a chip-scale optical cavity pumped with a CW laser,
leading to low-noise, broadband optical frequency combsKoos:2017:Nature ;
Papp:2020:PhyRevAppl . In this experiment, a soliton source with 300-GHz
frequency spacing (Fig.3b) is the optical source of the transmitter and all
data channels are directly detected using photodiodes. We use non-return-to-
zero (NRZ) coding at a symbol rate of 10 GBd, and simultaneously launch 13 WDM
channels into 4 MDM channels. It is important to note that we use optical
fibre amplifiers (see Fig.S3) to compensate for the insertion loss of external
optical components (e.g. modulators, DWDM) but this extra loss can be
mitigated once all components are integrated.

The eye diagrams and bit error rates (BERs) of the soliton data transmission
experiment are presented in Fig.3c-d. Out of the 52 carriers derived from the
four spatial mode channels and all comb lines in the C band, 42 data channels
show no error occurrences when a total of 1012 bits are compared (BER $<$
10-12). The data channels also show BERs better than 10-10 with 3 dBm input
optical power at 10 Gb/s/channel (see Fig.3e and Fig.S3c). The total data rate
in this experiment amounts to 520 Gb/s. Additionally, we show that the optical
link can also be driven at 20 Gb/s/channel with open eye diagrams to achieve
an aggregate nano-photonic waveguide bandwidth of 1.04 Tb/s (see Fig.S4). In
this experiment, data channels at specific WDM channels (e.g. C46 and C49)
show larger error rates than other wavelength channels, owing to lower comb
line power at the WDM channel and non-uniformity of the soliton optical
spectrum (see Fig.S3b).

In a single MDM device, which can achieve a spectral bandwidth of over 150 nm
(see Fig.2), further improvement in the data transmission capacity is
straightforward through the use of adjacent S and L bands. We can also
decrease the frequency spacing of the comb source to the standard 100-GHz
frequency gridKoos:2017:Nature  or even finerBowers:2020:Nature . As a proof
of concept, we demonstrate WDM-MDM data transmission (4-mode,
16-wavelength/mode) using a mode-locked laser with a 20-GHz channel spacing –
we achieve a spectral efficiency of 4 bit/s/Hz and BER of all channels lower
than the threshold of 20 % hard-decision error correction (see Fig.S5-S6).
This setup showed a 1.28 Tb/s aggregated data rate using only 320 GHz of the
spectral bandwidth.

Chip-to-chip interconnect

The WDM-MDM communication link can be implemented for chip-to-chip
interconnects onto which multiple chips and computing nodes are connected. In
this section, we demonstrate a chip-to-chip MDM link using inverse-designed
beam emitters and a rectangular core multimode fibreMarom:2017:IPC , shown
schematically in Fig.4a.

The chip-to-chip interconnect requires an efficient means of coupling light
from on-chip transceivers to a communication link. In both free-space and
fibre systems, this can be achieved using a beam emitterYoo:2012:OE ;
Cai:2012:Science ; Baumann:2018:OFC ; Tsang:2019:JQE  that preserves and
launches all spatial modes perpendicular to the chip. Here, the beam emitter
structure is designed for an operation bandwidth of 40 nm with an 80-nm
minimum feature size using photonic inverse design. The output of the MDM
multiplexer is adiabatically tapered into the inverse-designed beam emitter to
send data into orthogonal spatial modes and maintain the mode profiles from a
silicon waveguide to free space or fibres. The dimensions of the beam emitter
are flexible and can be varied over a wide range to maximize chip-to-fibre
coupling efficiency.

A modulated CW laser (NRZ at a symbol rate of 10 GBd) is first coupled to the
Tx chip through one of four single-mode input waveguides via a lensed fibre.
The Tx chip then sends light into a unique spatial mode of the multimode fibre
using the MDM multiplexer (Fig.2-3) and inverse-designed grating (see Fig.4b).
The five-meter-long multimode fibre, which has a rectangular core (dimension:
32 $\times$ 8 $\mu$m2; see the inset of Fig.4a) constrained to support a
single mode in the narrow transverse direction and four modes in the
orthogonal direction, transmits the signal from the Tx chip to the Rx chip
while maintaining low modal crosstalk. Finally, the Rx chip de-multiplexes the
multimode signals and allows us to individually characterize the transmitted
signals at single-mode output waveguides using a second lensed fibre.

Fig.4c presents the measured channel cross-talks of the chip-to-chip MDM link
using a multimode fibre. To benchmark the multimode fibre link, we also
conduct a free-space experiment where the orthogonal spatial mode channels are
transmitted between chips (see Fig.S7 for experimental setup and detailed
information). The cross-talk of the multimode fibre link ranges from -4.2 dB
to -13.9 dB, and a 3-dB-bandwidth of chip-to-fibre coupling is approximately
60 nm for all mode channels (see Fig.S8). The mode-dependent insertion loss
difference is approximately 8 dB at 1540 nm throughout the entire link. The
variations are attributed to fibre-to-chip alignment and mode-dependent
transmission loss of the fibre, which has higher loss in higher-order modes –
in fact, mode channel 1 - 3 show less than 2.5 dB insertion loss. Given that
the cross-talk is dominated by the fibre-to-chip coupling alignment, the
communication link quality can be greatly improved with appropriate photonic
packagingIMEC:2017:OFC . Fig.4d shows eye diagrams of the chip-to-chip link
using a multimode fibre link, driven with NRZ at a symbol rate of 10 Gb/s.

Conclusion

In summary, we have demonstrated a multi-dimesional optical communication
scheme using silicon photonic circuits. With an inverse-designed silicon MDM
device and chip-scale soliton microcombs, we multiplex four spatial modes and
comb channels covering the entire C-band to achieve natively error-free ($<$
10-12) data transmission in 42 out of 52 channels through a nanophotonic
waveguide. Additionally, we show chip-to-chip multi-mode communications
enabled by inverse-designed beam emitters and a rectangular core multimode
fibre. The MDM technology is fully compatible with silicon photonic WDM
transmitters and semiconductor foundry processes. Here, photonic inverse
design ensures device compactness and broadband operation. This enhances
robustness to fabrication errors and temperature fluctuations, enabling
practical applications in data centers and high performance computing.

Scaling of this technology for ultra-wide bandwidth high-fidelity
communication will require the maintenance of low mode cross-talk and
insertion loss. In particular, we observe signal degradation in chip-to-chip
data transmission experiments (both via multimode fibre and free-space). This
signal degradation can be compensated by using MZI meshesMiller:2017:LSA  or
digital MIMO signal processingKhan:2016:OL ; Willner:2015:ScientificReport ;
Bell:2020 . Future scaled-up implementations of our communication scheme can
employ such error-correction optical circuits on the receiver chip, and their
performance can be improved in terms of footprint, insertion loss and
operation bandwidth using photonic inverse design. Furthermore, efficient edge
couplersLipson:2020:CLEO ; Shen:2020:OE , foundry-compatible bi-level
gratingsVuckovic:2018:OE ; Popovic:2016:OFC  as well as advanced fibre
technologiesPuttnam:2021:Optica  can further mitigate mode cross-talk and
insertion loss.

As our MDM device features non-resonant, low-loss, low-crosstalk operation
over the entire C-band, it is compatible with a large number of novel light
sources, not limited to a Si3N4-based soliton microcomb (Fig.3) and a table-
top mode-locked laser (Fig.S5) demonstrated in this work. For example,
different types of chip-scale lasers and frequency comb sources such as mode-
locked quantum dot lasersBowers:2019:Optica , electro-optical frequency
combsLoncar:2019:Nature ; Idjadi:2020:CLEO , and vertical cavity surface
emitting laser (VCSEL)Bell:2021:ECOC  can be used. Co-design of the laser
source, optical link architecture, photonic devices and signal processing will
facilitate the next generation of optical interconnects in data-centre
networks, wireless communications, and hardware accelerators.

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Methods
Photonic inverse design Stanford Photonics Inverse Design Software
(SPINS)SPINs ; Su:2020  was used to design mode-division multiplexers and
surface-normal grating couplers for TE-polarized light. The inverse design
method can provide a variety of photonic circuit component designsWu:2019:LPR
; Liu:2019:NatureCommunications ; Gabrielli:2020:PTL ;
Piggott:2020:ACSPhotonics  (see Fig.S9) as well as a potential compact 12
channel multiplexer design (see Fig.S10).

Optoelectronic transceiver The AMF photonic process is a 180 nm silicon-on-
insulator fabrication process with 2 $\mu$m buried oxide thickness. The
thickness of the silicon layer is 220 nm and the single-mode (500 nm width)
photonic waveguides have propagation loss of about 2 dB/cm. The grating
coupler is designed by partially etching silicon and has a measured loss of
about 4 dBAshtiani:2019:Optica ; Idjadi:2020:NaturePhotonics . Also, the
Y-junction has a measured excess loss of 0.5 dBZhang:2013:OE . The implemented
PN ring modulator has a free spectral range of about 11 nm and a 3-dB
bandwidth of higher than 30 GHzAshtiani:2019:Optica . The photodiode has a
responsivity of higher than 0.75 A/W and a 3-dB bandwidth of more than 30
GHzAshtiani:2019:Optica ; Idjadi:2020:NaturePhotonics . Fig.S1 shows measured
results of data transmission from electro-optic ring modulator to MDM
components and Si/Ge photodiode. In addition, we show the channel cross-talk
of the all-passive MDM device.

Soliton-WDM-MDM data transmission The soliton microcombs are generated by
pumping the silicon nitride microresonator with a CW laser, IQ modulator in a
single-sideband suppressed-carrier configuration driven by a voltage-
controlled oscillator, and a subsequent amplifier; see Fig. S3 and
RefPapp:2020:PhyRevAppl  for a more detailed description of the comb
generation setup and silicon nitride device designs. The output of the comb is
then passed through a WDM filter that serves to suppress the pump power, and
the transmitted comb power is amplified and de-multiplexed using a commercial
100 GHz ITU grid based DWDM with an insertion loss of 3.5 dB per channel. The
‘even’ and ‘odd’ carriers are recombined using another multiplexer. Each
channel is separately amplified and passed through two intensity modulators
(20 GHz 3-dB bandwidth and 6-dB insertion loss, EOSpace) which are driven by
PRBS generators using non-return to zero (at a data rate of 10 Gb/s). The data
channels are de-correlated with a delay of approximately 5000 symbols. We
recombine the odd and even sets of carriers, amplify them using an EDFA and
split the power into four different channels. Mode channel 2 - 4 have a delay
of 50, 100 and 200 symbols with respect to mode channel 1. Output coupling is
performed with a lensed fibre aligned to one MDM output waveguide at a time.
The light at the output is then amplified and sent through another de-
multiplexer where the signal is detected using a 14 GHz bandwidth photodiode
(DSC40S) with a receiver responsivity of 0.8 A/W. The photodetected signal is
then sent to an error analyzer which we use to measure the bit-error rate.
Fig. S3 shows a detailed experimental setup.

MLL-WDM-MDM data transmission A mode-locked laser (Lumentum, repetition rate:
10-GHz) generates a frequency comb. The frequency comb is passed through a
delay line interferometer (free-spectral-range: 20-GHz) to increase the
optical comb spacing. Then the 20-GHz comb is passed through a highly
nonlinear fibre (HNLF) for spectral flattening and broadeningWillner:2014:OE .
The frequency lines of the optical comb are selectively filtered and de-
interleaved into ‘odd’ and ‘even’ carriers by a programmable filter (Finisar
WaveShaper). The ‘odd’ and ‘even’ carriers are separately amplified using
C-band EDFAs, and routed to in-phase/quadrature modulators which are driven by
arbitrary-waveform generators using quadrature phase-shift keying (at a symbol
rate of 10-Gbd). The data channels are decorrelated by 50 symbols delays and
combined into a standard single-mode fibre. Next we boost the power of all of
the data channels using an EDFA and simultaneously launch them into the four
input ports of the MDM chip using a fibre array (single-mode fibres with 127
$\mu$m spacing). Output coupling is performed with a tapered lensed fibre
aligned to one MDM output waveguide at a time, and the received signal is
characterized using intradyne coherent detection with electrical filtering and
a tunable external cavity laser as a local oscillator. Fig.S6 shows the
experimental setup and optical spectra of MLL sources.

Four spatial mode rectangular core fibre An optical fibre with a rectangular
core geometry was provided by Corning Incorporated. The germanium-doped core
material was machined to rectangular shape and with an annular silica cladding
material was drawn down to final dimensions (125 $\mu$m cladding diameter,
nearly perfectly circular, with core of 32 $\times$ 8 $\mu$m2 and index
modulation $\sim$ 0.005). The fiber supports four polarization-degenerate
spatial modes with intermode effective index differences $\geq$ 0.0005. A
5-meter-long fibre sample was used to image the four guided modes using low-
coherence interferogram image analysis (beating against a plane wave, see
Fig.S11). Measured differential modal group delay was $\sim$ 8 ns/km, which
can lead to signal skew in long mode multiplexed links and MIMO processing
complexity if mode mixing occurs.

Free-space communication experiments Optical signals were generated using a CW
tunable laser, passed through a circulator, and modulated with an amplitude
electro-optic modulator. The modulator was driven by a data generated by
micro-controller and electronic amplifier. To recover from modulator losses
and to compensate for extraneous losses in the free space system, the signal
was amplified with a fibre amplifier before coupling to the transmitter chip.
The amplified beam passes through a polarization controller, free space
coupler, and 50/50 beamsplitter and is focused onto input grating couplers
through a 100$\times$ nIR objective lens (Mitutoyo). Light that is coupled out
of the multimode output grating passes through the beamsplitter a second time
and is collimated with a 200mm focal length lens. This beam is sent
approximately 0.5 m to another beamsplitter, 200 mm lens, and 100$\times$
objective that focus the multimode beam onto the multimode grating of the Rx
chip. The output of the Rx chip is coupled out once more from the 100$\times$
objective and coupled to a single-mode fibre. 10% of the light from the laser
is picked off with a directional coupler, passed through a polarization
controller, and 1% of this light is coupled to the output fibre. The homodyne
signal is read using a photodiode and a custom electric amplifier chain fed to
an oscilloscope. Fig.S7 shows the experimental setup, free-space data
transmission, and measured optical mode profiles at multiple wavelengths.

Data availability The data that support the plots within this paper and other
findings of this study are available from the corresponding author upon
reasonable request.

Code availability An open source of the photonic optimization software used in
this paper is available at https://github.com/stanfordnqp/spins-b.

Acknowledgment We acknowledge insightful discussion with J.M.Kahn, S.Fan,
A.Dutt, and H.Kwon, and are also grateful for technical advice from S.K.Pai,
M.H.Idjadi, and D.Huang. The chip-scale devices were fabricated in the
Stanford Nanofabrication Facility, the Stanford Nano Shared Facilities,
Advanced Micro Foundry Pte Ltd, AIM photonics, and LIGENTEC. This work is
funded by the DARPA under the PIPES program, and the AFOSR under the MURI
program (Award No. FA9550-17-1-0002). A.E.W. acknowledges the support from
from AFRL (FA8650-20-C-1105). We thank G.Keeler, G.Pomrenke, and the programme
management teams for discussions throughout the project.

Competing interests. The authors declare they have no competing financial
interests.

Figure S1: Integrated MDM transceiver. (a) Data transmission measurement. A CW
laser is coupled to an integrated MDM transceiver (GC: grating coupler, MOD:
ring modulator, PG: pattern generator, MUX/DEMUX: MDM
multiplexer/demultiplexer, PD: photodiode). The cascaded Y-junctions evenly
distribute optical power and simultaneously send a light to four ring
modulators – one of ring modulators is driven by an external electric driver.
We measured transmitted data using integrated Si/Ge photodiodes (PD) and an
external scope. (b) Measured eye diagram of MDM channel 3 (TE20) obtained from
an integrated Si/Ge photodiode (PD) and external signal scope. (c) Measured
S-parameters of the back-to-back MDM multiplexer structure. We sent the
multiplexer design in (a) to two semiconductor foundries (AMF and AIM
photonics), and channel cross-talks at 1550 - 1560 nm (ring modulator
operation wavelength) of all channels consistently ranges from -10 dB to -20
dB on chips from two foundries.  Figure S2: Optimization of broadband, low-
crosstalk MDM multiplexer. Measured S-parameters of the back-to-back MDM
multiplexer structures with different design bandwidths. The structure in (b)
is optimized at multiple wavelengths over the entire C band, while the
structure in (a) is optimized only at a single wavelength (1560 nm). The
narrow band structure is implemented in the WDM-MDM communications using a
mode-locked laser (16 wavelength channels over 320 GHz spectral span; see
Fig.3), and the broadband structure is utilized in the WDM-MDM communications
using soliton microcombs (13 wavelength channels over 3.6 THz spectral span;
see Fig.3). Figure S3: Soliton microcomb data transmission. (a) A silicon
nitride microresonator generates a frequency comb with a 300 GHz repetition
rate (optical spectrum: top panel of (b)). The comb lines are de-multiplexed
using a standard ITU grid DWDM (W-DEMUX), and de-interleaved into ‘odd’ and
‘even’ carriers using another multiplexer (W-MUX). The ‘odd’ and ‘even’
channels are amplified and passed through intensity modulators (EOM) which are
driven by PRBS generator using a NRZ encoding. The data channels are
decorrelated by a fibre delay and re-combined into a standard single-mode
fibre (optical spectrum: bottom panel of (b)). The combined sets of carriers
are amplified and simultaneously coupled to four single-mode waveguide inputs
of the silicon MDM chip (see Fig.2c), and the transmitted light through the
MDM chip is coupled back to single-mode fibre using a lensed fibre. The
collected light is amplified and sent through another demultiplexer (W-DEMUX)
where the received signal is analyzed using a photodiode (PD) and an error
analyzer. (b) Frequency comb spectra before (top panel) and after (bottom
panel) optical amplification. (c) 10-Gb/s bathtub curves obtained from BERT by
sweeping the delay between the clocks for transmitter and receiver. We
measured a bit-error rate better than 10-10 with -3 dBm received optical
power. Left: Measured curves of the channel C37 (193.7 THz)/ M4 (TE30), Right:
Bathtub curves of the channel C34 (193.4 THz)/ M4 (TE30) at various received
optical powers. Figure S4: Eye diagrams of all data channels (13 wavelength
channels $\times$ 4 mode channels) at 20 Gb/s NRZ. Figure S5: On-chip
interconnect: WDM - MDM data transmission with a frequency comb laser. (a) A
mode-locked laser is employed as a multi-wavelength laser source and evenly
distributed to four WDM transmitters. WDM de-multiplxers (W-DEMUX) separate
the comb lines (16 WDM channels) and in-phase/quadrature modulators (IQ-mod)
encode independent data (QPSK at a symbol rate of 10 GBd per WDM channel). The
WDM data are recombined into single-mode fibres using WDM multiplexers
(W-MUX), and four fibres are coupled to MDM input ports simultaneously using a
fibre array. The on-chip mode multiplexer recombines all data channels
(aggregated channel numbers: 4 $\times$ 16) into a multimode waveguide, and
sends WDM-MDM data to the receiver. At the receiver, the mode and wavelength
channels are separated by mode and wavelength de-multiplexers and detected
using digital coherent receivers (Coh. Rx). In our experiments, we
independently modulate even and odd WDM channels using two IQ-modulators for
emulating WDM transmission (see Fig.S6 for more details). (b) Optical spectrum
of the frequency comb laser (see method section for more details). (c) Optical
microscope image of MDM multiplexer back-to-back structure. (d) Top: measured
BERs of the transmitted data channels along with BER threshold when applying
forward error correction with 20 % overheads (dashed black line), Bottom:
Optical signal-to-noise ratio (OSNR) of the transmitted channels.  Figure S6:
Data transmission using flattened mode-locked laser for parallel WDM and MDM.
(a) A mode-locked laser generates a frequency comb. The frequency comb is
passed through a delay line interferometer (DLI) to increase the optical comb
spacing. Then the 20-GHz comb is passed through a highly nonlinear fibre
(HNLF) for spectral flattening and broadening. Frequency combs (optical
spectrum: top panel of (b)) are selectively filtered and de-interleaved into
‘odd’ and ‘even’ carriers using a waveshaper (WS). Each carriers are
separately amplified (optical spectrum: bottom panel of (b)) and passed
through IQ modulators (IQ-M) which are driven by arbitrary-waveform generators
using QPSK. The data channels are decorrelated by a fibre delay and combined
into a single-mode fibre. All carriers are amplified and simultaneously
coupled to four single-mode waveguide inputs of the silicon MDM chip (see
Fig.2b). We collect light at the waveguide output using a lensed fibre, and
the collected light is analyzed using a coherent receiver and local oscillator
(LO). (b) Measured optical spectra of flattened 20-GHz frequency comb (top)
and selectively filtered comb tooth after interleaving and amplification
(bottom). (c) Measured BER versus optical signal-to-noise ratio of all 64
channels as well as back-to-back test case.  Figure S7: Chip-to-chip
interconnect: Wireless data transmission using inverse-designed beam emitter.
(a) A CW laser is modulated with on-off-keying data using an amplitude
modulator, and this data signal is coupled to the Tx input grating couplers.
The Tx chip emits multi-channel MDM data to free-space using an inverse-
designed coupler (Inset 1: measured optical mode profile at Tx grating
coupler, Inset 2: mode profile in free-space between Tx and Rx chips). The
free-space modes are projected onto an MDM demultiplexing device in the Rx
chip that separates each mode into a single-mode output (Inset 3). The
transmitted data are detected using a coherent receiver with the CW laser as
the local oscillator (LO). Infrared images are overlaid on device SEM images
(bar: 10 $\mu$m), and measured at 1525 nm. (b) Received data traces at each Rx
output grating coupler (Top: 1525 nm, Bottom: 1540 nm). The signal cross-talk
at Rx CH3 trace is attributed to the effects of channel-dependent optical link
insertion loss, the limited size aperture, and misalignment. Figure S8: Mode-
dependent loss measurement. (a) A CW laser is coupled to the MDM chip, and we
measured transmitted optical power through MDM multiplexer, grating coupler,
and a ractangular core multimode fibre using a photodetector. (b) Measured
insertion loss of mode channels exhibits a 3-dB-bandwidth of 60 nm for all
channels, and the mode-dependent loss difference is approximately 5 dB. The
mode-dependent loss is attributed to fibre transmission loss and chip-to-fibre
misalignment.  Figure S9: Arbitrarily routed multimode photonic circuit
element. Inverse design can provide a multimode photonics toolkit. We combine
functionalities of a multimode crossing and waveguide bendWu:2019:LPR  in a
compact design area. (a) Top: SEM image of a multimode photonic circuit which
consists of an MDM multiplexer, multimode splitter, and MDM demultiplexer. The
multimode splitter (white rectangular box, design area: 6 $\mu$m $\times$ 6
$\mu$m) can selectively route multimode signals to different waveguides.
Channels 1-3 are sent to the demultiplexer while the channel 4 is routed to
the other waveguide port. Bottom: Zoomed-in SEM image (bar: 5 $\mu$m) and
simulated multimode routing. (b) Measured transmission for the multiplexer-
splitter-demultiplexer structure.  Figure S10: Optimization of 12 channel
mode-division multiplexer. For computational simplicity, we restrict our study
to a two-dimensional structure with transverse-electric polarization. We model
the structure with infinite extent in the third dimension and an effective
relative permittivity of a silicon slab (220-nm-thick silicon at 1550 nm
wavelength). (a) Structure of optimized device (18 $\mu$m $\times$ 18 $\mu$m),
which consists of twelve single-mode input waveguides and one multi-mode
output waveguide (b-d) Mode 1-, 2- and 12- transmission through device. (e)
S-parameters of simulated device (note that it is a single multiplexer device
and it is not a simulation of back-to-back structure). The channel cross-talk
ranges from -53 dB to -25 dB, and insertion loss ranges from from -0.23 dB to
-0.47 dB. Figure S11: Four spatial mode rectangular core fiber. Experimental
setup for imaging fiber modes using low-coherence interferogram analysis.
Short optical pulse from a mode-locked laser at center wavelength of 1.55
$\mu$m is focused with an objective (Obj) into the rectangular core fiber and
excites all its spatial modes. (An optional band-pass filter (BPF) is used to
increase the coherence lengths). At the fibre’s distal end, the modes acquire
differential group delays (DGD). A copy of the excitation pulse is timed for
co-incidence onto an infrared camera, obtaining an interferogram of one mode
at a time, inset (a). The mode profiles are extracted following a Fourier
transform and selection of the off-axis diffracted signal (inset (b)), and
inverse Fourier transform to reveal the mode profile, inset (c).


# DropKAN: Regularizing KANs by masking post-activations

Mohammed Ghaith Altarabichi
Center for Applied Intelligent Systems Research
Halmstad University, Sweden
<EMAIL_ADDRESS>
###### Abstract

We propose DropKAN (Dropout Kolmogorov-Arnold Networks) a regularization
method that prevents co-adaptation of activation function weights in
Kolmogorov-Arnold Networks (KANs). DropKAN operates by randomly masking some
of the post-activations within the KANs computation graph, while scaling-up
the retained post-activations. We show that this simple procedure that require
minimal coding effort has a regularizing effect and consistently lead to
better generalization of KANs.

We analyze the adaptation of the standard Dropout with KANs and demonstrate
that Dropout applied to KANs’ neurons can lead to unpredictable behavior in
the feedforward pass. We carry an empirical study with real world Machine
Learning datasets to validate our findings. Our results suggest that DropKAN
is consistently a better alternative to using standard Dropout with KANs, and
improves the generalization performance of KANs. Our implementation of DropKAN
is available at: https://github.com/Ghaith81/dropkan

##  1 Introduction

Kolmogorov-Arnold Networks (KANs) [1] are recently proposed as an alternative
to Multi-Layer Perceptrons (MLPs). The computation graph of Kolmogorov-Arnold
Networks (KANs) is different form the standrad Multi-Layer Perceptrons (MLPs)
in two fundamental ways: 1) On edges: KANs use trainable activation functions,
unlike MLPs that rely on linear weights. 2) On neurons ("nodes"): KANs sum the
incoming signals, different to MLPs that apply non-linear activation functions
e.g., ReLU. These changes to the computation graph indicates that many of the
techniques used in MLPs might not directly transfer to KANs, or at least may
not necessarily give the same desired effect. In this work we explore whether
KANs could benefit from using Dropout [2] for regularization, and propose
DropKAN based on our analysis as a method to regularize KANs by randomly
masking post-activations. DropKAN is efficient at regularizing KANs and is
easy to incorporate within any implementation of KANs.

Our contributions in this paper can be categorized in two folds. First, we
analyze the behavior of Dropout applied to KANs and show that it behaves
differently from how it was originally designed to operate with MLPs. Our
second and main contribution is proposing DropKAN, an alternative to Dropout
applied to KANs, as we show that DropKAN consistently outperforms Dropout with
KANs using a number of real world datasets.

##  2 Motivation

We start by formalizing the definition of a KAN layer and the adaptation of
Dropout to KANs. We denote the input dimension of the $l^{th}$ layer of a KAN
model consisting of $L$ layers by $n_{l}$. The activation functions connecting
the layer $l$ to the following layer $l+1$ can be expressed as a 1D matrix of
functions:

$\Phi_{l}=\\{\phi_{l,j,i}\\},\\\ l=0,2,\ldots,L-1,\\\ i=1,2,\ldots,n_{l},\\\
j=1,2,\ldots,n_{l+1}$ (1)

where $\phi_{l,j,i}$ is an arbitrary function represented as a spline with
trainable parameters. We define $x_{l,i}$ as the pre-activation (input) of the
function $\phi_{l,j,i}$; the post-activation of $\phi_{l,j,i}$ is denoted by
$\tilde{x}_{l,j,i}=\phi_{l,j,i}(x_{l,i})$. The neurons in the KAN layer
performs a sum of all incoming post-activations to output $\tilde{x}_{l,j,i}$.

$x_{l+1,j}=\sum_{i=1}^{n_{in}}\tilde{x}_{l,j,i}=\sum_{i=1}^{n_{in}}\phi_{l,j,i}(x_{l,i}),\\\
j=1,2,\ldots,n_{l+1}$ (2)

By adding a Dropout layer between the KAN layers $l$ and $l+1$ as in Figure 1,
a binary dropout mask $m_{j}$ is applied to the $l$ layer outputs’ $x_{l+1,j}$
in training time. The mask is usually sampled independently from a Bernoulli
distribution with a probability $p$ of being 1 (indicating that the neuron $j$
is retained) and $1-p$ of being 0 (indicating that the neuron $j$ is dropped).
The output of the KAN layer with the inclusion of Dropout becomes:

$x^{\prime}_{l+1,j}=\frac{m_{j}x_{l+1,j}}{1-p},\quad j=1,2,\ldots,n_{l+1}$ (3)

The output of the Dropout layer in Equation 3 is usually scaled-up by a factor
of $\frac{1}{1-p}$, we denote this factor by $s$. This procedure is done to
compensate the effect of dropping out some nodes, with the rational of
ensuring that a similar level of signal will continue to propagate through the
network with the presence of Dropout in training time.

(a)

(b)

(c)

Figure 1: A standard KAN network (on the left), a KAN network regularized
using Dropout applied between the KAN layers by masking neurons/nodes, with
dashed lines indicating zero inputs to the connected activations (middle), a
KAN network regularized using DropKAN by masking post-activations, where
dashed lines indicate zero inputs to the connected neurons (right).

###  2.1 Why is Dropout problematic with KANs?

The key motivation of Dropout is to prevent co-adaptations [3] of weights by
sampling a thinned network through dropping out nodes. If a node in a MLP is
dropped by a Dropout layer then all the weights connected to it will take no
part in the feedforward and backward passes during the training step. The
weights of the dropped neuron in a MLP have no impact on feedforward because
they are multiplied by an input of zero from the dropped node, while on the
backward pass they have no influence on the loss calculation, and will
consequently get a gradient of zero in that training step.

However, applying Dropout to the outputs of a KAN layer is not enough to
exclude the masked nodes from actively participating in the feedforward and
backward passes. The zero outputs from the masked nodes will be used as inputs
to the activation functions $\Phi_{l+1}$ in the following layer $l+1$, given
that for an arbitrary activation function $\phi^{\prime}$ from $\Phi_{l+1}$,
the output of $\phi^{\prime}(x=0)$ is not necessarily zero, the nodes will
still propagate the corresponding value of $\phi^{\prime}(x=0)$ into the
network during feedforward. Consequently, the weights of activation function
(the spline coefficients) $\phi^{\prime}$ in the layer $l+1$ following the
dropout will also be updated in the backward pass.

Another key issue with applying Dropout to KANs is the compensation of
dropping out some nodes by scaling-up. As we have observed in Equation 3, the
outputs of the kept nodes are scaled by a factor of $s$. This procedure is
ineffective with KANs as the arbitrary activation function from $\Phi_{l+1}$
are not necessarily homogeneous function of degree 1 (linear),
$\phi^{\prime}(sx)\neq s\phi^{\prime}(x)$. The behavior of
$\phi^{\prime}(sx_{l+1,j})$ is unpredictable at training time, and it is not
trivial to identify $s$ that could lead to the proper scaling-up effect, as
this procedure must be done for each Dropout layer in the network.

##  3 Methods

We design DropKAN to address the previous issues of Dropout by applying a
binary dropout mask $M_{i}$ to the post-activations $\tilde{x}_{l,j,i}$ as
observed in Figure 1 in training time. This is different to Dropout where the
mask is applied to the output of the neurons $x_{l+1,j}$. Similar to Dropout,
we sample the mask independently from a Bernoulli distribution with a
probability $p$ of being 1 (indicating that the post-activation is retained)
and $1-p$ of being 0 (indicating that the post-activation is dropped). The
output of the KAN layer when DropKAN is applied becomes:

$x^{\prime}_{l+1,j}=\frac{1}{1-p}\sum_{i=1}^{n_{in}}M_{i}\tilde{x}_{j,i}=\frac{1}{1-p}\sum_{i=1}^{n_{in}}M_{i}\phi_{j,i}(x_{i}),\\\
j=1,2,\ldots,n_{out}$ (4)

In DropKAN the output is scaled-up by a factor of $\frac{1}{1-p}$ to
compensate for the dropped post-activations. In test time DropKAN behaves
exactly as a regular KAN layer as in Equation 2, effectively serving as an
identity function.

Applying the mask on the post-activations in DropKAN combined with scaling-up
ensures that the expected values of the sums in KAN nodes during training are
approximately the same with and without DropKAN:

$\mathbb{E}[\frac{1}{1-p}\sum_{i=1}^{n_{in}}M_{i}\tilde{x}_{j,i}]\approx\mathbb{E}[\sum_{i=1}^{n_{in}}\tilde{x}_{j,i}]$
(5)

Unlike with Dropout, DropKAN doesn’t mask the nodes after the sum, instead
some of the post-activations are masked while the kept ones are scaled-up,
which ensure the expected value of the summation taking place in the node with
the presence of DropKAN is about the same without DropKAN as observed in
Equation 5.

##  4 Results and Discussion

This section describes the experimental design we have used to evaluate
DropKAN, along with the results obtained in our experiments. The first
experiment aimed to evaluate the expected value of a KAN function using
DropKAN layers in training mode, and compare to a network with Dropout enabled
between the KAN layers. In the second experiment, we compare the performance
of a network using DropKAN layers against a regular KANs and KAN regularized
with Dropout using a number of classification problems.

###  4.1 Experimental Setup

Our experiments involve 10 popular [4, 5] data sets from the UCI Machine
Learning Database Repository222http://archive.ics.uci.edu/ml. We have included
datasets of varying sizes from different domains. Table 1 provides a summary
of the number of instances, features and classes of all data sets used in our
experiments.

Table 1: UCI Data sets used for evaluation. Data set | No. of Instances | No. of Features | No. of Classes
---|---|---|---
dermatology | 366 | 34 | 6
german | 1 000 | 24 | 2
semeion | 1 592 | 265 | 2
car | 1 728 | 6 | 4
abalone | 4 177 | 8 | 28
adult | 32 561 | 14 | 2
bank-full | 45 211 | 16 | 2
connect-4 | 67 556 | 42 | 3
diabetes | 101 766 | 49 | 3
census-income | 199 523 | 41 | 2

Each data set is divided into training (60%), validation (20%), and testing
(20%) splits. A unified approach of prepossessing is adopted for all data
sets, including categorical features encoding, imputation of missing values,
and shuffling of instances. Accuracy of the model is the metric we used for
the evaluation in all experiments. All reported accuracies are the ones
realized on the testing set.

As for the KANs hyperparameters we have adopted the default values.
Additionally, we set the grid to 3 and turned off the trainable parameters of
the base activation function. The KAN architecture [nin, 10, 1] is used across
all experiments (unless explicitly mentioned otherwise) for all datasets,
where nin is the number of input features in the dataset. The networks are
trained for 2000 steps using Adam optimizer with a learning rate of 0.01, and
a batch size of 32.

In describing the experiments, we will use KANs with five different settings:
using Dropout, DropKAN and neither (No-Drop). For Dropout and DropKAN we will
report results in two modes with scale-up (Dropout w/ scale) and without
(Dropout w/o scale).

###  4.2 Experiment I - Forward Pass Evaluation

In this experiment we evaluate the impact of using DropKAN and Dropout on the
forward pass of backpropagation for a KAN network. We train the KAN network
[6, 2, 2, 1] five times for a total of 100 steps. We carry five forward passes
using the validation split on every 10 steps using the five settings of: No-
Drop, Dropout w/ scale, Dropout w/o scale, DropKAN w/ scale, and DropKAN w/o
scale. We fix the rate of drop to 0.5 for all drop modes on each layer. We
average the value at the output neuron ($x_{3,1}$) to estimate the expected
value of the output using the validation split.

The plots in Figure 2 show the results of this experiment, and while it is
clear that the propagated signals for both Dropout w/o scale and DropKAN w/o
scale become weaker due to the process of zeroing-out, the scaling is allowing
DropKAN w/ scale to pretty accurately approximate the strength of the signal
comparing to the No-Drop mode consistent with Equation 5. On the other hand,
the procedure of scaling-up the input for Dropout is consistently off, and in
this case is causing to over-scale the signal. As we have explained earlier,
scaling up the input of the non-linear activation function $\phi^{\prime}$
would cause an unpredictable behaviour at training time since
$\phi^{\prime}(sx)\neq s\phi^{\prime}(x)$.

(a)  (b)

Figure 2: The expected value of the output neuron ($x_{3,1}$) of the [6, 2, 2
,1] KAN with No-Drop using the car dataset compared to DropKAN w/ scale and
DropKAN w/o scale (on the left), and to Dropout w/ scale and Dropout w/o scale
(right) at different stages of the training.

###  4.3 Experiment II - Classification Problems

In this experiment, we compare KANs equipped with DropKAN layers against KANs
with standard KAN layers and KANs regularized using Dropout. The goal of the
experiment is to validate the regularizing effect of DropKAN and to compare it
against using standard Dropout with KANs.

We use a random search to optimize the rates of drop for the DropKAN and
Dropout settings, we ran the search for 50 evaluation per setting, and choose
the setting with the highest accuracy performance on the validation split. For
evaluation we train every setting five times and report the average of the
runs.

The results in Table 2 suggest that regularization methods improved the
standard KAN performance (No-Drop) in 8 scenarios apart from two datasets
semeion, and car. DropKAN w/ scale is clearly the most successful setting
recording the highest test accuracy in six of the ten datasets, while the
standard Dropout w/ scale was only the best choice in one dataset.

The results suggest that scaling is showing a positive effect with DopKAN as
the scaled settings performed better than w/o scaling in seven scenarios,
consistent with the results of the previous experiment and Equation 5.
However, scaling is not as successful in the case of Dropout showing better
performance comparing to no scaling in only 4 scenarios.

Table 2: Results of KAN networks trained with Dropout, DropKAN (with and without scaling), and no Drop, each result is the mean of five independent runs. Dataset | KANs Accuracy
---|---
| No-Drop | Dropout w/o scale | Dropout w/ scale | DropKAN w/o scale | DropKAN w/ scale
dermatology | 74.84% | 87.75%$\pm 3.18$ | 90.47%$\pm 2.12$ | 84.37%$\pm 1.99$ | 91.84%$\pm 1.03$
german | 60.25% | 74.75%$\pm 0.50$ | 73.25%$\pm 0.94$ | 71.75%$\pm 1.58$ | 76.00%$\pm 0.51$
semeion | 98.27% | 97.65%$\pm 0.20$ | 97.33%$\pm 0.41$ | 97.96%$\pm 0.26$ | 98.27%$\pm 0.07$
car | 92.89% | 85.83%$\pm 4.90$ | 64.31%$\pm 3.93$ | 87.28%$\pm 2.48$ | 88.15%$\pm 1.91$
abalone | 27.11% | 27.29%$\pm 0.60$ | 27.53%$\pm 0.72$ | 27.65%$\pm 0.61$ | 28.07%$\pm 0.48$
adult | 84.07% | 83.76%$\pm 0.31$ | 83.54%$\pm 0.51$ | 83.98%$\pm 0.12$ | 84.84%$\pm 0.09$
bank-full | 86.93% | 89.18%$\pm 0.19$ | 87.69%$\pm 0.87$ | 89.11%$\pm 0.16$ | 88.79%$\pm 0.08$
connect-4 | 64.43% | 61.43%$\pm 1.07$ | 70.71%$\pm 1.37$ | 70.46%$\pm 0.82$ | 60.57%$\pm 1.92$
diabetic | 55.47% | 55.56%$\pm 0.35$ | 55.72%$\pm 0.82$ | 57.19%$\pm 0.24$ | 54.28%$\pm 0.71$
census-income | 94.19% | 94.55%$\pm 0.19$ | 94.49%$\pm 0.14$ | 94.67%$\pm 0.12$ | 94.82%$\pm 0.04$

##  5 DropKAN Implementation

DropKAN can not be implemented as a standalone layer, but instead it must be
applied to the post-activations within the KAN layer. Luckily, the
implementation is straightforward, and could be realized with a single line of
code, allowing DropKAN to be easily incorporated into any custom KAN
implementation. We demonstrate here the line of code to be added assuming
torch library is used for the KAN layer implementation:
postspline = postspline * torch.empty(postspline.shape,
device=postspline.device).bernoulli_(1 - rate)/ (1 - rate)

Where postspline is the post-activations (spline) variable, and the variable
rate is the dropping probability of DropKAN, the masking procedure is only
active in training time, and must be disabled for inference.

It would be interesting to implement and test DropKAN for the newly-emergeing
KAN-based architectures such as graph [6, 7, 8], convolutional [9], and
transformer [10]. For instance in FourierKAN-GCF (Graph Collaborative
Filtering ) [11], regularization strategies like node and message dropout were
tested, we believe DropKAN could be directly extended to such architectures.

##  6 Related Work

Expanding upon the Dropout technique, various methodologies [12, 13, 14] have
been proposed to refine the training regularization of Deep Neural Networks
(DNNs) for supervised learning. The fundamental idea is to introduce noise
into intermediate layers during training. Recent advancements have focused on
enhancing regularization in Convolutional Neural Networks (CNNs) through the
introduction of structured noise. For instance, the SpatialDropout method [15]
selectively removes entire channels from activation maps, the DropPath scheme
[16] opts to discard entire layers, and the DropBlock algorithm [14] zeros out
multiple contiguous regions within activation maps.

Techniques inspired by Dropout are not limited to ones during the feedforward
stage of backpropagation, as in Gradient Dropout proposed in the context of
meta-learning by [17], and later generalized by [18] to the supervised
training setting, [18] showed that masking the gradient during the backward
pass prevents the network from memorizing the data and learning overly simple
patterns early in the training, similar ideas could potentially be extended to
KANs.

##  7 Conclusions

We present DropKAN an effective method to regularize KANs leading to reliable
generalization improvement over the standard KANs, and KANs regularized with
Dropout. Our analysis of adapting Dropout to KANs indicates that using it
between KAN layers is problematic due to the speciality of the computation
graph of KANs. We design DropKAN to mask the post-activations instead of
masking the neurons/nodes. We show that a KAN network constructed using
DropKAN layers consistently outperforms a KAN of the same architecture using
standard KAN layers, even when the later is regularized with Dropout between
its layers.

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