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Abstract
Reservoir computing is a machine learning approach that enables us to use recurrent neural networks without involving the complexity of training algorithms and make hardware implementation possible. We present a novel photonic architecture of a reservoir computer that employs a nonlinear node and a resonator to implement a virtual recurrent neural network. This resonator behaves as an echo generator component that substitutes the delay line in delay-based reservoir computers available in the literature. The virtual neural network formed in our implementation is fundamentally different from the delay-based reservoir computers. Different virtual architectures based on the FSR and the Finesse of the resonator are investigated to provide higher performance depending on the task. We test the performance of our proposed reservoir by 3 benchmark tasks, signal classification, nonlinear channel equalization, and memory capacity. Our system enhances the memory capacity of the reservoir compared to the delay-based systems and shows reliable results in signal classification and nonlinear channel equalization.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Artificial neural networks (ANNs) are inspired from the brain’s operation and are able to process highly complex computational tasks such as speech recognition, image recognition, and time series prediction which are difficult and time-consuming for von Neumann computers. The two main categories of neural networks are feedforward and recurrent neural networks (RNNs). In feedforward ANNs, connections (equivalent to axons and synapses) between neurons steer the signal from input to output in one direction, while in RNNs, feedback connections are also present. Due to the complexity of training RNNs (finding the correct connection weights between neurons) [1], reservoir computers (RCs) have been introduced in recent years as an attractive solution [2,3]. A reservoir computer consists of three layers: the input layer, the reservoir, and the output layer. The reservoir itself is typically a fixed RNN which does not need training. The enticing feature is that only the output layer is trained, thus the reservoir computing approach has become particularly attractive for hardware realization of RNNs as most of the connections of neurons are fixed in the design stage and does not need reconfiguration (training).
Optical implementations of RCs have gained considerable research attention in recent years and various architectures have been introduced thanks to low power consumption and crosstalk, high bandwidth, and high-speed computing in optics [4–7]. The fixed reservoir layer in an optical RC may be implemented with actual (physical) neurons or virtual neurons. In the physical implementation, nodes (neurons) are spatially distributed and physically connected e.g. for instance by using waveguides on an integrated chip. Different optical elements such as semiconductor optical amplifiers [8,9], ring resonators [10], passive optical structures [11], and photonic crystal cavities [12] have been used as physical nodes in these physical implementation. For high computing performance, typically a high number of nodes is required in the reservoir layer ($10^2$ to $10^3$) which complicates the integration of such realizations significantly.
To circumvent these demanding issues regarding the high number of physical nodes required in an RC, the concept of virtual nodes by means of time multiplexing was introduced in [13]. In this approach by using a delay line, virtual nodes are connected successively in time which are then fed back into the nonlinear node, yielding a particular form of RNN topology [14,15]. The schematic of the virtual reservoir using delay line is shown in Fig. 1(a). The input signal ($u(t)$) containing bits of data is multiplied by a mask signal. This mask is of higher data rate (pulses of smaller width called chips). The masked signal is the input signal of the reservoir layer which is made of a single nonlinear node and a delay line. At each chip, the input signal is combined with the delayed version of itself which has travelled through the delay line. With the proper choice of the delay, one can successfully connect the virtual nodes. The equivalent network of a virtual RC for 5 virtual nodes is shown in Fig. 1(b). Different approaches have been adopted to realize delay-based virtual reservoir computers e.g. electronic circuits [15], opto-electronic systems [13,16], all-optical systems [17,18], and laser systems [19–22]. Although these delay-based implementations reduce the number of physical nodes, but unfortunately they are in need of considerable time delay whenever a large number of virtual nodes is required and thus are difficult to implement. The most straight forward approach to realize this time delay is using a transmission line, e.g. an optical fiber, which requires long lengths (in the orders of meters to kilometers) [17–19] and rendering them unfavorable if not impossible for integrated circuits. In more elegant implementations, a delay line is substituted by a resonator and thereby the physical length of the delay element is reduced from meters in [23] to millimeters and centimeters in [24–26]. Still the length of the structure which determines the delay required by the delay line is dictated by the number of virtual nodes.
Fig. 1. (a). Schematic of delay-based reservoir. (b). The virtual equivalent network of the delay-based reservoir for $N=5$. (c). Schematic of resonator-based reservoir. (d). The virtual equivalent network of the resonator-based reservoir for $N=5$. The strength of the connections is shown by their darkness and brightness. The weaker the connections, the lighter their color. The time delay of the connections is demonstrated using the number of triangles on each arrow.
This inspires us to seek alternative virtual node based solutions which can be more easily integrated. Our idea is to use an echo generator replacing the lengthy delay line in Fig. 1(c). This echo generator block is realized in our proposal with a simple yet properly designed multi-mode Fabry-Perot resonator, in which the signal goes back and forth and generates an echo at each round trip. In a single-mode resonator; on the other hand, the signal is trapped within the resonator before gradually coming out of it. The former acts as an echo generator block while the latter imitates a delay line. Aside from the significant size reduction if one employs a compact multi-mode resonator, the dynamics of the system also allows a more versatile topology as depicted in Fig. 1(d). The use of resonator causes fundamental differences in the architecture of the virtual neural network. This means that unlike the previous implementations in which merely two neurons were connected at any moment, more than two neurons are connected at the same time. This can improve performance in some tasks such as memory capacity and nonlinear channel equalization which is demonstrated throughout our paper.
The paper is structured as follows. The resonator is a fundamental element in our structure which determines the equivalent virtual network of the reservoir computer. The details related to the multi-mode resonator are presented in section 2. Also, the effect of resonator parameters on the virtual network is investigated in this section. Our proposed structure and the equivalent virtual network of the reservoir computer is presented in section 3 and the performance of this RC under some standard tasks are presented in section 4. Finally, the conclusions are given in section 5.
2. Resonator in the multi-mode regime
Resonators are most commonly operate in the single-mode regime such that the input signal bandwidth is smaller than the resonator bandwidth. In this regime, the output wave shape of a single pulse after propagation through the resonator will be unchanged and the output of the resonator is the delayed version of the input. If the signal bandwidth is larger than the single-mode resonator bandwidth, a distorted and slightly delayed version of the input pulse reaches the output. Neither of these behaviours are desired for our case. We seek an output for the resonator which contains one or more delayed copies of the input pulse, with sufficient delay between the input and the last copy or echo of the input pulse. This requires to use the resonator in a multi-mode fashion, and the input pulse bandwidth to span multiple resonances of the resonator, to create proper echoes and delay. Different types of resonators such as bandpass and stopband structures are candidates such an echo generator block. In this paper, without the loss of generality and for simplicity’s sake, we use Fabry-Perot resonators which can be used in transmission and reflection mode.
First, we assume a Fabry-Perot resonator constructed by two mirrors with reflection coefficients $r_1$, $r_2$ and transmission coefficients $t_1$ and $t_2$ which are spaced L apart from each other. Two dielectric slabs can be used as the mirrors of the Fabry-Perot. For a slab having a length equal to an odd multiples of quarter wavelength, it can be found that the reflection and transmission coefficients of mirror will be $r_i$ and $jt_i$ such that $t_i^2 + r_i ^2 =1$. The medium between the two mirrors is filled homogeneously with a material of $n_r$ refractive index. In this model of the resonator, we assume that the variations of the refractive index and the length of the resonator over the bandwidth of the input signal can be ignored. Also, diffraction losses at the mirrors due to finite dimensions are neglected.
The impulse response of this Fabry-Perot resonator can be written as
in which $t_{RT} = {2Ln_r}/{c}$ is the round-trip time of light propagating in the resonator and $c$ is the speed of light [27].The impulse response of the resonator shows that the output of the resonator can be echoes of the input pulse if the pulse width in the time domain is smaller than the round trip time of the resonator. This means that in the frequency domain, the pulse bandwidth should be wider than the resonator bandwidth which includes multiple modes of the resonator as depicted in Fig. 2(a). In this regime, if a modulated Gaussian pulse with temporal width $\tau _p$ is incident on the resonator, the output of the resonator will be that shown in Fig. 2(c). The output contains the echoes of the input pulse with decaying amplitudes.
Fig. 2. (a). Transmission coefficient of the resonator (blue curve) and optical Gaussian pulse (red curve) in frequency domain when the bandwidth of Gaussian pulse is wider than resonator bandwidth so that the bandwidth of the pulse contains several modes of the resonator. Both the curves are normalized. (b) Transmission phase of the resonator (blue curve) and optical Gaussian pulse (red curve) in frequency domain. (c) Input Gaussian pulse and the output of the resonator in time domain.
If one of the mirrors of the Fabry-Perot resonator is a perfect mirror, the reflection coefficient of the resonator can be written as:
In reflection mode, in order for the resonator to reflect echoes of the input signal, the input signal pulse width must include several resonant modes, as before. The response of this resonator to a Gaussian pulse and the resonator response is shown in Fig. 3. The comparison of two Figs. 2 and 3 shows that the reduction rate of echoes in reflection mode resonator is lower than transmission mode, as expected. The FSR for a Fabry-Perot is defined as: $FSR = c/2n_rL$ and the finesse is approximated by:
in which $R_1 = |r_1|^2$ [27]. A Fabry-Perot resonator is characterized by these two parameters which determine the length of the resonator and the reflection coefficients of its mirrors. The length between two mirrors in the resonator determines the time interval between echoes at its output. The longer the resonator, the longer the time interval between echoes. The amplitude of the echoes is controlled by the reflection coefficient of the mirrors. The higher the reflection coefficient of the mirrors, the longer the wave stays inside the resonator. The reduction rate of echoes at the resonator output is determined by the reflection coefficient of the mirrors. In order to find a resonator-based reservoir with good performance, it is sufficient to train and test the system with different values for these two parameters.Fig. 3. (a). Reflection coefficient of the resonator (blue curve) and optical Gaussian pulse (red curve) in frequency domain when the bandwidth of Gaussian pulse is wider than resonator bandwidth so that the bandwidth of the pulse contains several modes of the resonator. Both the curves are normalized. (b) Reflection phase of the resonator (blue curve) and optical Gaussian pulse (red curve) in frequency domain. (c) Input Gaussian pulse and the reflected signal from the resonator in time domain.
3. Proposed architecture of reservoir computer
Two possible architectures of resonator based RC is shown in Fig. 4 and 5 which use Fabry-Perot resonator in transmission and reflection mode respectively. Our proposed RC consists of input, reservoir, and output layers. In both architectures, a laser is an element in the input layer and the optical source that feeds the $LiNbO_3$ Mach-Zehnder modulator (MZM). The reservoir layer of both implementations consists of: a Mach-Zehnder Modulator (MZM) as the nonlinear node, a Fabry-Perot resonator, an attenuator, and an electronic feedback loop. The MZM has a $\sin$ functionality and the phase $\phi$ determines the working point of the modulator which has been set to $\pi /4$. The modulated optical signal is fed to the resonator and a photodetector (PD) is employed to detect the resonator output. The modulation signal is generated using an arbitrary wave generator (AWG) in the input layer. The delayed versions of the input signal are added to the modulated signal and are applied to the MZM. In the architecture with the resonator in transmission mode, a significant portion of the power is reflected by the resonator, which is not favorable. The isolator avoids back reflections of the resonator. But, in Fig. 5 the resonator reflects all power and the input and output paths are separated by a circulator. In both implementations, the output of the reservoir layer is detected using a PD in the output layer and the electronic signal is sampled using a 100 Giga Sample/s digital oscilloscope. The number of nodes is equal to 50 neurons and the reflectivity of two mirrors of Fabry-Perot resonator and its FSR are two parameters that are investigated to achieve good performance.
3.1 Input layer
In the input layer, to connect the original signal to the virtual nodes, a pre-processing step is needed. First, the original signal $u(t)$ undergoes a sample and hold procedure with a sampling time interval $T$. Then, the resulting signal $u[n]$ is multiplied with a periodic masking signal $m(t)$. The masking signal has $N$ different values in each interval $T=N\theta$ and is a periodic signal with period $T$. The time interval $T$ and $\theta$ are called bit time and chip time respectively and $N$ is the number of virtual neurons.
3.2 Reservoir layer
The reservoir consists of a MZM as the nonlinear node, a resonator, and an electronic feedback loop which connects the echoes from the resonator output to the nonlinear node.
3.2.1 Topology of the equivalent virtual network
In order to determine the difference between a delay-based reservoir and a resonator-based reservoir, we consider their equivalent virtual neural networks. For delay-based reservoir computing in the unsynchronized regime, the reservoir state is formulated as follows:
In the resonator-based reservoir with a multi-mode Fabry-Perot resonator, the input signal is combined with delayed versions of the reservoir node states. Every echo from the resonator acts as a delay line and therefore the reservoir states are calculated according to the following equation:
3.2.2 Dynamics of the reservoir
In order to simulate the dynamics of the reservoir layer, the time domain reservoir output should be determined. The laser signal with frequency $f_0$ and amplitude $E_0$ enters the input port of the modulator. The output signal of the modulator is written as follows:
After substituting Eq. (1) in (7), the low frequency part of reservoir state, $X(t)$, can be expressed as:
It should be noted that for the reflective implementation $T\left (t\right )$ should be replaced with $R\left (t\right )$. The reservoir output after photodiode is calculated as $x_{PD} = |x(t)|^2$ and is sampled in the middle of each chip time interval.
3.3 Output layer
The output of the reservoir $y[n]$ is a linear combination of reservoir states as: $y[n] = \textbf {W}_\mathrm {out}\: \textbf {x}[n]$, which $W_{out}$ is the weight matrix of the output layer and the output weights are determined by minimizing the mean square error (MSE) which is defined as follows:
4. Simulation results
We evaluated the performance of our proposed RC on three benchmark tasks: signal classification, memory capacity, and nonlinear channel equalization. The number of virtual nodes in our simulations is $N=50$ and two parameters input gain $\alpha$ and feedback gain $\beta$ are optimized for each reservoir.
4.1 Signal classification
In the signal classification task, the input is a combination of square and triangular waves and the target output is 0 while the input is triangular wave and 1 otherwise. For every triangle sample, 10 voltage levels are used. Each of the training and test datasets consists of 1000 samples randomly chosen between square and triangle samples. We test our proposed RC in transmission mode using this task. Figure 6(a) shows the NMSE by varying the mirror reflectivity of the resonator. The FSR of the resonator is equal to 12.5 GHz. This plot shows that the NMSE is low for the reflectivity values below 0.95 and increases as the reflectivity increases. This is because by increasing the reflectivity, the echo amplitudes damp slowly and results in a network with long delay connection which connects input samples of the reservoir to each other. However, in the signal classification task, the arrangement of triangular and square samples is random and does not correlate with each other and also long delay connections are not required in this task. This is while, the strength of the important first echoes is low. Therefore, the performance of the RC decreases as the reflectivity increases. It should be noted that the NMSE obtained is several orders of magnitude better than the NMSE reported in [13]. We also add noise to the input signal of our RC and train the output layer using the noisy data. The NMSE obtained in different values of reflectivity is shown in Fig. 6(b) which is below 0.01 for some values of mirror reflectivity. The target output and the output of our RC for mirror reflectivity 0.91 is demonstrated in Fig. 7. Figure 7(a) shows the results for noiseless case which reveals that the output and target signals perfectly match and Fig. 7(b) shows the noisy case on which the output follows the target with a slight difference.
Fig. 6. Signal classification Task for the RC in transmission mode. (a) NMSE as a function of mirror reflectivity for signal classification without considering the effect of noise.(b) NMSE as a function of mirror reflectivity for signal classification with a noisy signal, SNR is 30 dB.
Fig. 7. Signal classification Task for the RC in transmission mode. (a) The output of the reservoir and the target output for ideal signal classification. (b) The output of the reservoir and the target output for noisy signal classification.
The performance of our reservoir computer when the reflectivities of the two mirrors of the resonator are 0.91 and 0.95 is studied under noisy signal classification task and the obtained results are shown in Fig. 8. For low SNR values the classification becomes harder and the performance of the reservoir is severely affected.
Fig. 8. Noisy signal classification Task for the RC in transmission mode. NMSE as a function of SNR for two mirror reflectivities 0.91 and 0.95.
4.2 Nonlinear channel equalization
In this task, the reservoir computer is used as an equalizer of a nonlinear with memory channel. Assume we want to send 4 symbols $\{-3,-1,+1,+3\}$ on a wireless communication channel. These symbols will distort after propagating on the channel and the receiver, receives distorted symbols. In this task, the model of the communication channel is as follows, first a sequence of symbols $d[n]$ passes through a memory channel,
Then it will transform nonlinearly as,
in which $\nu [n]$ is zero-mean white Gaussian noise and its power is adjusted to the signal-to-noise (SNR) from 12-32 dB. This distorted signal is the input of the reservoir computer. In this task, the reservoir is trained to estimate the input of the channel which is $d[n]$. In order to evaluate the performance of our reservoir computer, the symbol error rate (SER) is calculated. Figure 9 and Fig. 10 show the SER for different values of mirror reflectivity of Fabry-Perot resonator in transmission and reflection modes respectively. The FSR of the resonators are 12.5 GHz, 6.25 GHz, and 3.125 GHz which provides the echo distance of $2\theta$, $4\theta$, and $6\theta$, respectively. The SER decreases when the echo interval increases. The reason for the increase in reservoir performance is that by increasing the echo interval, the stronger connections have longer latency. As a result, the reservoir gains richer dynamics. Also, this figure shows that when echo distance equals $2\theta$ and the reflectivity is in the range $[0.8, 0.94]$, the error is less than other values. For lower reflectivities, the amplitude of the output echoes damp more rapidly and the number of echoes with significant amplitude decreases. Therefore, it is possible that none of echoes result in delay larger than the period T. But as the reflectivity approaches 1, the damping rate of echoes decreases which leads to long-delay connections. A good reservoir should have fading memory to slowly reduce the effect of the past input samples. Therefore the output becomes independent of much earlier inputs. Due to the fact that increasing the distance of echoes, changes the virtual neural network and makes the connections with longer delays more stronger, the performance of the reservoir changes and the SER will be minimum in another reflectivity value. In other hand, as the echo distance increases to $6 \theta$, the SER is less dependant to the reflectivity values. Figure 9 and 10 show that the $\mathrm {SER} < 10^{-4}$ is possible which is the best values reported in papers yet. The length of an air-filled Fabry-Perot resonator which causes the echo distance equal to $6\theta$ will be 3.6 cm. It should be however noted that the air-filled Fabry-Perot resonator is merely an easy to design example that can be substituted with other types of electromagnetic resonators, e.g. micro ring arrays, point defects in photonic crystals, etc. that would have smaller foot print and can be much more easily integrated. Also, Fig. 11 and 12 demonstrate the SER as the SNR ranges from $12$ to $32$ dB. The value of the reflectivity of two mirrors is 0.91 and 0.55 in Fig. 11 and 12, respectively where the SER has its minimum value.Fig. 9. Nonlinear channel equalization for the RC in transmission mode. Symbol error rate as a function of reflectivity of mirrors in transmission mode. The results are for 5 different runs for each reflectivity value. SNR is set to 32 dB.
Fig. 10. Nonlinear channel equalization for the RC in reflection mode. Symbol error rate as a function of reflectivity of mirror in reflection mode. The results are for 5 different runs for each reflectivity value. SNR is set to 32 dB.
Fig. 11. Nonlinear channel equalization for the RC in transmission mode. Symbol error rate as a function of SNR in transmission mode with constant mirror reflectivity of 0.91. The results are for 5 different runs for each reflectivity value.
Fig. 12. Nonlinear channel equalization for the RC in reflection mode. Symbol error rate as a function of SNR in transmission mode with constant mirror reflectivity of 0.55. The results are for 5 different runs for each reflectivity value.
4.3 Memory capacity
This task is introduced to determine the network memory capacity [28]. In this task, the reservoir computer is trained to generate delayed input signal. The input signal $u[n]$ is a uniformly distributed random series in the interval $[-1,+1]$. The desired output $u[n-k]$ is the input signal shifted k time steps in the past. In order to evaluate the k-delay memory capacity, the reservoir is trained for each k so that the memory capacity related to delay k maximizes. The total linear memory capacity is obtained by summing the memory capacities over all delays as follow:
where the k-delay memory capacity $MC_{i}$ is defined as:Here, $d\left [W_k^{out}\right ]$ is the determination coefficient and is defined as the squared correlation coefficient of two signals. The k-delay memory capacity is in the range $[0,1]$ and the summation on $MC_k$ will be greater than 1. The memory capacity of our reservoir in transmission mode is shown in Fig. 13 for different values of mirror reflectivity and the echo distance. The memory capacity for echo distance equal to $2\theta$ ranges from 2 to 3 and for $R \in [0.9, 0.98]$ is larger than other values. Also, the memory capacity becomes larger by increasing the echo distance. As the echo distance increases the connections with longer delays are stronger. This results show that our reservoir has considerably higher memory than the one presented in [25].
5. Conclusion
We have proposed a novel photonic RC architecture which uses a multi-mode resonator instead of the lengthy delay line to realize virtual nodes required in the neural network. The resonator generates multiple echoes, each of them behaves like a delay line. Therefore, the virtual neural network is fundamentally different from the delay based reservoir computers. This new implementation causes richer dynamics that provides considerably higher memory capacity which is beneficial for tasks with high dynamics. Our proposed RC shows good performance in other standard tasks like signal classification and nonlinear channel equalization.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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