IEEE Workshop on Nonlinear Circuit Networks
December 8-9, 2023

Performance of Two-Layer Reservoir Computing
with van der Pol Oscillators
Yasufumi Kajino, Kazuki Yasufuku, Yoko Uwate, Yoshifumi Nishio
Dept. Electrical and Electronic Eng, Tokushima University
2-1 Minami-Josanjima, Tokushima 770-8506, Japan
Email: {kajino, yasufuku, uwate, nishio}@ee.tokushima-u.ac.jp
Abstract—We studied network reservoir computing using interconnected oscillators. One VDP (van der Pol) oscillator wasconsidered as one node, and it was interconnected to form areservoir layer. In this study, two-layer reservoir was used andits performance was evaluated. The input waveform generation
task was compared between a single-layer (100 nodes) and a twolayer (50 nodes in each layer) reservoir. The results showed thatthe two-layer reservoir generated input waveform more faithfully
than the single-layer reservoir when the input waveform weresine, triangle, and square waves. The accuracy was better with
two-layer, but it depends on the input waveform.

I. I NTRODUCTION
Deep learning is a machine learning method based on artificial neural networks and has excellent performance in image,speech recognition, and natural language processing[1]-[5].
However, training large models requires a lot of computationaltime and resources, and it is also difficult to predict thetime required for training. Since real-time time-series patternrecognition requires fast learning, machine learning models
with low computational complexity and high performance areimportant.
Reservoir computing has attracted attention as a methodwith high accuracy while reducing energy and computational
complexity compared to conventional AI models. Unlike traditional neural networks, reservoir computing mainly only
requires operations at the readout layer and has its own dynamics of nonlinear projection of input into a high-dimensional
space. This simplifies learning while still providing effective
performance. Furthermore, reservoir computing has been implemented using a variety of physical systems, and its potential
has been widely explored [6]-[8]. However, there are still gaps
in our understanding of optimal architecture and performancemaximization, and future research should focus on innovative
reservoir design, enhanced learning algorithms, and integration
with other AI techniques.
This research focuses on network reservoir computing using
oscillators as computational elements. The coupled oscillator
model is the basis for modeling and analyzing rhythmic
behavior in ecology and neuroscience as well as engineering[9]-[12]. In particular, the synchronization phenomena of vander Pol oscillators, a simple yet versatile coupled system ofnonlinear oscillators, has attracted a great deal of attention,
which has helped to approximate various natural phenomena[13], [14].

Our research will improve the efficiency and accuracy ofreservoir computing systems by increasing the number ofreservoir layers to two, paving the way for a revolution in AI
applications. We focus on exploring different configurations
to develop energy-efficient, high-performance systems suitable
for resource-limited environments.
II. TWO - LAYER RESERVOIR WITH O SCILLATOR CIRCUITIn this study, two-layer reservoir using interconnected oscillators network is presented. Proposed reservoir computing
is used the unique dynamics of oscillators. Fig. 1 shows an
overview of the structure of two layers networked reservoir
computing system.

Fig. 1. Composition of the proposed two-layer reservoir

The input signal is input by adjusting the coupling strength
between the input node and the reservoir layer. It is then input
into the oscillator network as shown Fig. 1 and propagatedby mutual coupling between oscillators. As a result, output
signal is obtained from a number of output terminals locatedin the network. The output signal is obtained by measuring thevoltage difference between each oscillator and the reference
oscillator. It is important to note that the coupling strength
in the reservoir layers is fixed throughout the calculation. In
the readout layer, liner regression is performed. To evaluate theeffectiveness of the proposed two-layer reservoir, we compared
it with a single-layer reservoir.
van der Pol oscillators were used in this study. Fig. 2 showsits circuit model. van der Pol oscillator is a circuit consistingof a capacitance C, and inductance L, and a nonlinear resistor
G connected in parallel.

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III. SIMULATION METHODS

Fig. 2. Circuit model of van der Pol oscillator

In this experiment, the oscillators are connected with aresistor. The circuit equation of the VDP oscillator is expressed
as follows:

N

X
1
dv


 C n = −iL − iG −(vk − vn )
dt
Rnk
n,k=1


din


= vn .
Ldt

Ŵout = DX T (XX T + βI)−1(1)

In this experiment, nonlinear resistors exhibiting third-ordercharacteristics were used. The current-voltage characteristic of
the nonlinear resistor is expressed as follows:ig = −g1 v + g3 v 3 .

(2)

where g1 , g3 > 0.
The circuit equation is normalized as follows using thenormalization parameters. The normalization parameters andthe normalization equations are shown in Eqs. (3) and (4).

s
r

√
g1
g1 C



 v = g x, i = g L y, t = LCτ
3







3

r
ε = g1

L
1
, γ=
C
R

r

(3)

L
C


N

 dxn = εx (1 − x2 ) − y − X K (x − x )


n
n
nk k
n
n
dτ
n,k=1


dyn


= xn
dτ

RM SE
T
1X

T n=1
(4)

(5)

Here, Enk represents the adjacency matrix of the network.
It indicates whether the kth oscillator is connected to the nthoscillator. If Enk = 1, it means they are connected, Enk = 0
indicates that they are not connected.

(6)

To evaluate the proposed model, the error between the
output waveform using the optimal output weights and target waveform was measured. Since the simulation using theRunge-Kutta method showed a transient response in the initial
10,000τ , the error evaluation evaluation was performed on
the results after 10,000τ . Normalized root mean square error(NRMSE) was used for error evaluation. In general, theroot mean square error (RMSE) is used for error evaluation.
However, RMSE is susceptible to outliers in data or predicts
because it calculates the time-averaged square error of thesquared error between the model output and the target output.
NRMSE prevents this effect. NRMSE normalizes RMSE by
the variance of the target output. RMSE and NRMSE are
shown in Eqs. (7) and (8), where y(n) is the input waveformand d(n) is the target time series data.
v
u
T
u1 X
||d(n) − ŷ(n)||(7)
RM SE = t
T n=1
N RM SE =N is the number of oscillators in each reservoir layer. Knk
is the coupling strength and represented by as follows:
Knk = Enk γnk .

For the input signal waveform, a data set of 60,000 sine,
triangle, and square waves with amplitudes ranging from -1 to
1 was used. Each data point was input at 1τ intervals according
to the Runge-Kutta method. The first 28,000 data sets wereused for training and 32,000 for test. The Runge-Kutta method
was used to simulate the oscillators by solving the normalized
equations of the van der Pol oscillator. A moving average ofthe oscillators in the reservoir layer was taken, which wasused as the output of the reservoir layer. Ridge regression
was performed on the output waveform of the reservoir layerusing the target waveform to find the optimal output weights.
Ridge regression can be used to determine the output weights
to achieve the desired performance.
Ridge regression is represented by the following equation
where Ŵout is the output weight and D is the matrix of theteacher signal.

(8)

ˆ||d(n) − d(n)||The values of each parameter are as follows. Coupling
probability p is p = 0.8. Coupling strength s is s = 0.05.
Number of oscillators in two-layer reservoir N1 and N2 are
N1 = N2 = 50. Number of oscillators in reservoir layer whensingle-layer N is N = 100.
The simulation was performed with different parameters foreach layer. However, the best accuracy was obtained when
the parameters were the same for both layers. Therefore, thesimulation results are shown for the same parameters for bothlayers.

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Fig. 3. Output during training of reservoir with sine Fig. 4. Output during training of reservoir with Fig. 5. Output during training of reservoir withwave input. (a) single-layer reservoir. (b) two-layer triangle wave input. (a) single-layer reservoir. (b) square wave input. (a) single-layer reservoir. (b) twotwo-layer reservoir.
layer reservoir.
reservoir.

Fig. 6. Results of the waveform generation task. (a) Fig. 7. Results of the waveform generation task. (a) Fig. 8. Results of the waveform generation task. (a)single-layer reservoir. (b) two-layer reservoir. Input single-layer reservoir. (b) two-layer reservoir. Input single-layer reservoir. (b) two-layer reservoir. Input
waveform is triangle wave.
waveform is square wave.
waveform is sine wave.

TABLE IE RROR EVALUATION OF INPUT WAVEFORM GENERATION TASK WITH SINGLE - LAYER AND TWO - LAYER RESERVOIR (NRMSE)
input waveform
train
test

sine wave
single-layer two-layer
0.04430.02810.04910.0288trianglewave
single-layer
two-layer
0.09380.0844
0.09350.0857- 12 -

square wave
single-layer
two-layer
0.17410.16640.17640.1686

IV. SIMULATION RESULTS

R EFERENCES

This section presents the results of the simulations conducted. The output waveform of the reservoir when the input
waveform is a sine wave is shown in Fig. 4. The output
waveform of the reservoir when the input waveform is atriangle wave is shown in Fig. 5. The output waveform of thereservoir when the input waveform is a square wave is shownin Fig. 6. As parameter values ε is ε = 0.1. The results of the
input waveform generation task are also shown in Figs. 7, 8,
and 9. The green line represents the target waveform, the blueline represents the training output, the red line represents thetest output. The training and verification errors are summarizedin Table I.
The graphs in Figs. 4, 5, and 6 show that a two-layer
reservoir with 50 nodes in each layer has a response closer
to the input waveform than a single-layer reservoir with 100
nodes. Referring to Table I, when comparing the single-layer
and two-layer when the input wave is sine wave, the twolayer has a learning error of 0.0162 and a test error of 0.0203
smaller. when the input wave was a triangle wave, comparing
a single-layer to a two-layer, the training error was 0.0094 andthe test error was 0.0078 smaller with two-layer. Furthermore,when the input wave was a square wave, comparing a singlelayer to a two-layer, the training error was 0.0077 and thetest error was 0.0078 smaller with two-layer. As can be seenfrom Fig. 8, the output waveform does not reproduce the input
waveform well when the amplitude takes a constant value withrespect to the passage of time.

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V. C ONCLUSIONS
We have increased the number of network reservoir layerswith interconnected oscillators and investigated their performance. The proposed reservoir computing approach is superior
to other types of reservoirs in terms of flexible reservoir
layer design. Specifically, we focused on the VDP oscillator
and performed the task of input waveform generation. The
accuracy of the input waveform generation task was particularly improved when the input waveform was a sine wave,
while Other waveforms did not show much improvement inaccuracy compared to the sine wave. However, for all three
waveforms, the two-layer reservoir was more faithful to the
input waveform than the single-layer reservoir. These results
highlight the importance of carefully selecting and setting thetotal number of reservoir layers and nodes. Flexibility in the
choice of materials comprising the reservoir and in structuraldesign offers the potential for further advances in this area.
In this study, it was not possible to clarify the role of eachlayer with respect to the input signal because the accuracy
was improved when both layers had the same parameters. As
a future prospect, we would like to evaluate the output andaccuracy of each reservoir layer by switching the input signal
in time and inputting it to reservoir layer 1 and reservoir layer2.

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