Linear photonic reservoir computer based on

coherently driven passive cavity Quentin Vinckier1, François Duport1, Anteo Smerieri1, Kristof Vandoorne2, Peter Bienstman2, Marc Haelterman1 et Serge Massar3

1 Service OPERA-Photonique, CP 194/5, Université Libre de Bruxelles (U.L.B.), avenue F.D. Roosevelt 50,1050 Bruxelles, Belgique

2 Photonics Research Group, Dept. of Information Technology, Ghent University – IMEC, Sint-Pietersnieuwstraat 41, 9000 Gent, Belgium

3 Laboratoire d’Information Quantique, CP 225, Université Libre de Bruxelles (U.L.B.), Boulevard du Triomphe, 1050 Bruxelles, Belgique

quentin.vinckier@ulb.ac.be

We present the first experimental implementation of a passive linear reservoir computer working in coherent light for analogue

signal processing. By using an optical cavity, the neuron states are coded by temporal multiplexing in the amplitude and the

phase of a coherent electromagnetic field. We report the performances obtained with our experimental setup compared to

simulations on some benchmark tasks. In the future, our goal will be to realise a high-speed implementation of our system in

integrated photonics.

Abstract

Experimental setup

Acknowledgements: Fonds pour la formation à la Recherche dans l’Industrie et dans l’Agriculture (FRIA, Belgique), the Photonics@be project of

the Interuniversity Attraction Poles Photonics@be Program (Belgian Science Policy) and the Fond de la Recherche Scientifique FRS-FNRS.

- September 2014 -

Sequential processing of the neurons coded in A(t)

The nonlinearity, provided by a photodiode, is in the output layer

Computer post-processing (for the readout weights)

N+1 neurons

in 1 round-

trip time

2 steps:

1 Training: the readout weights Wi are

optimised by Ridge regression to

minimize (𝑦 𝑡 − 𝑦(𝑡))²𝑡2𝑡1 , where

y(t) is the desired output.

2 Testing: the readout weights Wi are

kept fixed. Evaluation of 𝑦 (𝑡) and

the error (NMSE).

- Traditional Recurrent Neural

Network: Mask Mi, aij and Wi are

optimised.

- Reservoir Computer: only the

readout weights Wi are optimised:

Mask Mi and aij are randomly

chosen.

Next step: analog readout

Goals:

No digital post-

processing

Rapidity

Goal: Differentiate digits spoken 10 times by 5 female speakers.

Inputs of the reservoir: pre-processed signals according to the Lyon

Ear Model.

We’ve presented the first experimental passive all-optical linear

reservoir computer working with coherent light for analogue

signal processing. The performances are state of the art for all

the benchmark tasks that we tested. The next step will be to

implement an analog readout layer. In the future, the challenge

will be to realise a high-speed implementation of our system on

a chip.

Conclusion

Memory Capacities Speech Recognition Task

Channel Equalization Task Radar Task

𝒒 𝒕 = 0.08𝑑 𝑡 + 2 − 0.12𝑑 𝑡 + 1 +d(t) +0.18d(t-1)-0.1d(t-2)+0.091d(t-3)-0.05d(t-4) +0.04d(t-5)+0.03d(t-6)+0.01d(t-7) e (t ) =q(t)+0.036q²(t)-0.011q³(t)+noise

Goal: Recover an input symbol

sequence d(t), at the output e(t) of a

standardised nonlinear multipath

RF channel defined as follows:

LMF(k):

P(e(t-k))=e(t-k)

CL= LMF(k)𝑘𝑚𝑎𝑥𝑘=0

CL=

Memory Function MF = 1-NMSE ∈ [0,1] : ability to recall P(e(t-k))

QMF(k):

P(e(t-k))=3e2(t-k)-1

CQ= QMF(k)𝑘𝑚𝑎𝑥𝑘=0

CQ=

XMF(k,k’):

P(e(t-k),e(t-k’))=e(t-k)e(t-k’)

CX= XMF(k,k′)𝑘′𝑚𝑎𝑥𝑘′=0𝑘′≠𝑘

𝑘𝑚𝑎𝑥𝑘=0

CX=

Input data from soma.ece.mcmaster.ca/ipix/

Goal: Predict the X-band (λ=3 cm)

radar signal reflected from the sea

surface.

2 inputs: in-phase and quadrature

components of the reflected signal

(VV polarization).

With 50 neurons and 10 data

samples of 6000 symbols

With 50 neurons

and 10 data samples

of 2200 inputs

Word Error Rate using 5 data samples of 100 digits

200 neurons, no noise

500 neurons, SNR = 3dB

Experimental performances compared to simulations:

Results are state of the art

Will be partially integrated on a chip

With 50 neurons and 10 data

samples of 1000 inputs

Recurrent Neural Network:

« The Reservoir »

Input Signal 𝑒(𝑡)

Input Mask

Mi 𝐸𝑖 𝑡 = 𝑀𝑖 . 𝑒 𝑡

Neuron

States

𝑥𝑖(𝑡)

Readout

Weights

𝑊𝑖 Output Signal

𝑦 𝑡 = 𝑊𝑖 . 𝑥𝑖(𝑡)

𝑖

𝑥𝑖(t)=FNL 𝛽𝐸𝑖 𝑡 + 𝑏𝑖𝑎𝑠 + 𝛼𝑎𝑖𝑗𝑥𝑗(𝑡 − 1)

𝑗

𝑎𝑖𝑗

𝑎𝑖𝑗

=0

=1

Experimental performances compared to simulations:

Results are state of the art

Reservoir computer principle:

a non traditional neural network

Simulation WER= 0% WER= 0.6(+0.9)%

Experiment WER=0% WER=0.8(+0.8)%

• Simulation

• Experiment

Linear memory Quadratic memory Cross memory Total Memory

22.61+ 0.07 13.25+ 0.23 33.90+ 0.82

21.14+ 0.34 12.07+ 0.10 30.20+ 0.46

C=CL+CQ+CX

C=

* With 10 data samples of 60000

inputs

49.99+ 0.13 *

48.37+ 0.47