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Abstract
We experimentally investigate delay-based photonic reservoir computing using semiconductor lasers with optical feedback and injection. We apply different types of temporal mask signals, such as digital, chaos, and colored-noise mask signals, as the weights between the input signal and the virtual nodes in the reservoir. We evaluate the performance of reservoir computing by using a time-series prediction task for the different mask signals. The chaos mask signal shows superior performance than that of the digital mask signals. However, similar prediction errors can be achieved for the chaos and colored-noise mask signals. Mask signals with larger amplitudes result in better performance for all mask signals in the range of the amplitude accessible in our experiment. The performance of reservoir computing is strongly dependent on the cut-off frequency of the colored-noise mask signals, which is related to the resonance of the relaxation oscillation frequency of the laser used as the reservoir.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Artificial intelligence based on neural networks and deep learning has been widely used in information and computer technologies. A recurrent neural network is a type of neural network that uses self-feedback for memorizing past input signals. Recurrent neural networks process empirical data very effectively owing to the existence of internal fading memory. However, these networks require large amounts of calculation and complex algorithms for learning the connection weights. To overcome these problems, the concept of reservoir computing has been proposed [1,2], where the weights between the input and the network, as well as the weights among the nodes of the network, are fixed randomly, and only the output weights are trained by machine learning. Therefore, the learning algorithm used for reservoir computing is simpler than that for conventional recurrent neural networks, and low calculation power is required. Reservoir computing is based on mapping an input signal into a high-dimensional space to facilitate time-series prediction and classification.
A concept for delay-based reservoir computing using a single nonlinear element has been proposed [3], and many implementations based on optical and photonic systems have been reported [4–19]. For delay-based reservoir computing, a single nonlinear system with a time-delayed feedback loop is treated as a virtual network, and the internal states of virtual nodes are assigned by sampling the temporal waveform of the dynamical output within the feedback loop. Easy hardware implementation of the virtual network can be realized using a single nonlinear dynamical system with a time-delayed feedback loop, but without using a real complex network. Many studies have been reported on the implementation of delay-based reservoir computing using optical and photonic systems, such as optoelectronic systems [4–6], all-optical systems [7–9], and laser dynamical systems [10–15]. In particular, semiconductor lasers with time-delayed feedback are very promising for the high-speed implementation of reservoir computing over gighertz [10], owing to their fast relaxation oscillation frequencies [20].
In delay-based reservoir computing, a temporal mask is applied to each input data to introduce a complex transient response, and the masked input data is sent to the reservoir. The temporal mask is treated as the weight between the input signal and the virtual nodes in the reservoir. In most studies, a binary random signal is used as the input temporal mask, consisting of a piecewise constant function with a randomly modulated binary sequence. Several studies on the design of the input mask signal have been reported using a six-level digital mask [16], a binary mask with optimized combination [17], and a sinusoidal analog mask [18] for the reduction of noise. It has also been shown numerically that the performance of reservoir computing can be improved by using a chaos mask signal [19]. However, no comprehensive investigation of the comparison between the different types of input mask signals has been reported by experiment yet.
In this study, we experimentally investigate delay-based photonic reservoir computing using semiconductor lasers with optical feedback and injection. We apply different types of temporal masks, such as digital, chaos, and colored-noise mask signals to an input signal. We evaluate the performance of a time-series prediction task by using the different mask signals and clarify the condition of the temporal mask signal for performance improvement.
2. Scheme and experimental setup for delay-based photonic reservoir computing
We introduce an implementation of delay-based photonic reservoir computing with two semiconductor lasers, as shown in Fig. 1. The processing scheme consists of three stages: the input layer, reservoir, and output layer. In the input layer, an analog input signal (e.g., time series) is expanded for time duration T, and a temporal mask signal with the length of T is applied to each input signal. The expanded input signal with the mask signal is used as a modulation signal.
Fig. 1 Schematic of delay-based photonic reservoir computing with two semiconductor lasers. The processing scheme consists of the input layer, the reservoir, and the output layer.
In the reservoir, one laser (referred to as “drive laser”) is used as the input light source, and the other laser (referred to as “response laser”) with an optical feedback loop is used as the reservoir. The optical phase of the drive laser output is modulated by the modulation signal from the input layer using a phase modulator. The phase-modulated drive signal is injected into the response laser. The delay time τ of the feedback loop in the response laser is matched to the input mask length T. The temporal dynamics of the response laser are affected by the injection signal from the drive laser as well as the feedback signal in the response laser.
The temporal waveforms of the response laser in the feedback loop are sampled at the node interval θ for N data, where τ = Nθ is satisfied. The sampled outputs are treated as virtual node states and used to calculate the output signal. The output signal is obtained from the sum of the weighted values of all the virtual node states. Learning of the weight values is carried out using the linear least-squares method with training data [3,19].
Figure 2 shows our experimental setup for delay-based photonic reservoir computing with two semiconductor lasers. We used a commercial semiconductor laser (NTT Electronics, NLK1C5GAAA) as the drive laser. The laser is mounted on a butterfly package and the optical isolator is inserted in front of the laser to avoid back reflection of unwanted feedback light. We used another semiconductor laser (NTT Electronics, KELD1C5GAAA) as the response laser without an optical isolator to introduce optical feedback. The optical wavelengths of both lasers are approximately 1548 nm, used for standard optical communication. The injection currents of the drive and response lasers are set to 30.00 mA (2.84 Ith) and 12.30 mA (1.31 Ith), respectively, where Ith is the lasing threshold. The injection current of the response laser is selected near the lasing threshold to achieve consistency with respect to the drive input signal. The relaxation oscillation frequencies of the response laser without and with drive light injection are 1.7 and 6 GHz, respectively.
Fig. 2 Experimental setup for delay-based photonic reservoir computing with two semiconductor lasers.
The optical phase of the drive laser output is modulated by an input signal with a temporal mask signal using a phase modulator (Photline, MPZ-LN-20, 20 GHz bandwidth). The modulation signal is generated by an arbitrary waveform generator (Tektronix, AWG70002A, 25 GigaSamples/s). The output of the drive laser is injected into the response laser unidirectionally through an optical isolator and a variable attenuator. The wavelengths of the two lasers are precisely controlled by using a temperature controller, so that the two wavelengths can be matched by injection locking. The injection strength is set to 16.3 μW and it needs to be large enough to achieve injection locking between the drive and response lasers. Under the condition of injection locking, the response laser output shows consistency [21] with respect to the drive input signal, where a reproducible temporal waveform of the response laser output can be obtained when the response laser is driven repeatedly by the same input signal. The response laser has an external cavity with a fiber reflector to introduce an optical feedback loop used as the reservoir. The external cavity length of the optical fiber is set to 3.68 m (one-way), and the corresponding feedback delay time (roundtrip) is τ = 35.4 ns. The signal-to-noise ratio (SNR) of the response laser output is 8 dB in our experiment.
The optical output of the response laser is sent to a photodetector (New Forus, 1554-B, 12 GHz bandwidth) and converted into an electric signal. The electric signal is amplified by an electric amplifier (New Forus, 1422-LF, 20 GHz bandwidth) and detected by a digital oscilloscope (Tektronix, DPO72304DX, 100 GigaSamples/s, 23 GHz bandwidth). The temporal waveform of the electric signal is detected at the sampling time of 0.01 ns, and it is used as the virtual node state for reservoir computing. The interval of the virtual nodes is set to θ = 0.2 ns, and the number of the virtual nodes is N = 177 (N = τ /θ).
3. Experimental results of time-series prediction task
3.1 Digital and chaos mask signals
We use the Santa Fe time-series prediction task [22] to evaluate the performance of reservoir computing. We perform single-point prediction of the chaotic data generated from a far-infrared laser. In this task, the input signal corresponds to the chaotic waveform at the n-th sampling point, and reservoir computing intends to predict the chaotic data at the (n + 1)-th sampling point. We use 3000 steps for training and 1000 steps for testing.
We introduce three digital mask signals and an analog chaos mask signal to compare the performance of the time-series prediction task. We use a binary mask signal consisting of a binary sequence {−1, 1} that varies randomly at each interval θ, as shown in Fig. 3(a). The mask interval is set to be the same as the virtual node interval (θ = 0.2 ns). The second mask is a six-level mask signal, where the levels of the digital mask signal are changed randomly at the six values { ± 1.0, ± 0.6, ± 0.3} with the constant interval θ [not shown in Fig. 3]. In addition, we use a random-level mask signal in which the levels of the digital mask signal are changed randomly at continuous values between −1 and 1, as shown in Fig. 3(b). All digital masks are changed at the same interval θ = 0.2 ns, and the shortest period of the mask signal corresponds to 0.4 ns (2.5 GHz in frequency).
Fig. 3 Temporal input mask signals: (a) binary mask, (b) random-level mask, and (c) chaos mask signal. The digital mask interval θ = 0.2 ns corresponds to the virtual node interval. (d) The fast Fourier transform of the chaos mask signal of (c).
We also use an analog chaos mask signal generated experimentally from a semiconductor laser with optical feedback [20], as shown in Fig. 3(c). The chaos mask signal is generated from the response laser without drive injection. The fundamental frequency of the chaotic oscillation is determined by the relaxation oscillation frequency of 1.7 GHz [see Fig. 3(d)]. We generate a sequence of the digital and chaos mask signals for the time duration T ( = τ) of the mask, and this sequence is repeatedly used for each input data to create the modulation signal in the input layer.
Here, we introduce the standard deviation of the temporal mask signals as a measure of the amplitude of the mask signals. We match the standard deviation of the amplitude of the temporal waveforms of all the digital and chaos mask signals as closely as possible, for comparison. In other words, the performance of reservoir computing is compared when the input mask signals with the same standard deviation of the amplitudes are used.
Figure 4 shows an example of the temporal waveforms of the mask signals and the response laser outputs for the random-level and chaos mask signals. The standard deviations of the random-level and chaos mask signals are set to 0.36 and 0.34, respectively. For both cases, irregular temporal waveforms of the response laser outputs are observed, and various node states are obtained that are denoted as blue dots in Fig. 4. The amplitude of the response laser output for the chaos mask signal is larger than that for the random-level mask signal. On the contrary, the frequencies of the response laser outputs are similar for both cases.
Fig. 4 Temporal waveforms of the mask signals and the response laser outputs for (a) digital random-level mask signal and (b) analog chaos mask signal. Blue dots indicate node states.
Figure 5 shows the experimental results of the Santa Fe time-series prediction task using reservoir computing with the random-level and chaos mask signals. The temporal waveforms of the original signals and the prediction results are shown by black and red lines, respectively, in Fig. 5. The error signals are shown by blue lines in Fig. 5. It is found that the prediction results are similar to the original signals for both cases. However, smaller errors for the case of the chaos mask signal are obtained, compared with the case of the random-level mask signal.
Fig. 5 Experimental results of the Santa Fe time-series prediction task using reservoir computing with (a) digital random-level mask signal and (b) analog chaos mask signal. Original signals (black lines), prediction results (red lines), and error signals (blue lines) are shown. The normalized mean-square errors (NMSEs) are (a) 0.305 and (b) 0.154, respectively. The feedback strength of the response laser is set to zero.
The performance of the time-series prediction task is quantitatively evaluated by using the normalized mean-square error (NMSE) as follows [3,19]:
where n is the index of the input data and L is the total number of the data. y is the prediction result of reservoir computing that is compared to the original signal ȳ as a target. var represents the variance. A smaller NMSE indicates better performance of the time-series prediction task.We calculate the NMSEs of the prediction results shown in Fig. 5. The NMSEs for the cases of the random-level and chaos mask signals are 0.305 and 0.154, respectively. Therefore, better prediction results can be obtained using the chaos mask signal. The values of NMSEs is one order of magnitude larger than those in the numerical simulations [19], since the small SNR of 8 dB is found in our experiment due to the existence of relaxation oscillation of the response laser output. The NMSEs can be improved by averaging the temporal waveforms of the response laser output before reservoir computing is performed, and we confirmed that smaller values of the NMSEs can be obtained using averaged temporal waveforms. We also found that it is important to set the fundamental frequency of the chaos mask signal to the same order of magnitude as the relaxation oscillation frequency of the response laser under drive injection to obtain good performance of reservoir computing.
Figure 6 shows the NMSEs of the time-series prediction task as the feedback strength is changed for the analog chaos mask signal. The NMSEs slightly decrease with an increase in the feedback strength; however, the NMSEs do not change much within the range between 0.1 and 0.2 for all feedback strengths in Fig. 6. In fact, the phase fluctuation of the feedback light of the response laser becomes dominant as the feedback strength is increased, and large deviations of the NMSEs are observed for different trials of the prediction task even for the same parameter values. Therefore, we decided not to use the feedback light of the response laser for the comparison of different mask signals. This open-loop configuration can be considered as a similarity of extreme learning machines, as described in [23]. On the contrary, the phase stability of the drive laser is not sensitive to the NMSEs of the prediction task.
Fig. 6 NMSEs of the time-series prediction task as a function of the feedback strength of the response laser using reservoir computing with analog chaos mask signal.
We investigate the NMSEs of the time-series prediction task for the three digital mask signals and the chaos mask signal when the standard deviation of the mask signal is continuously changed. Figure 7 shows the NMSEs for the cases of the binary, six-level, random-level, and chaos mask signals as the standard deviation of the mask signal is varied. For all mask signals, the NMSEs decrease monotonically as the standard deviation of the mask signal is increased. This result indicates that a larger mask signal improves the performance of time-series prediction task for all mask signals. In addition, it is worth noting that the NMSE curves for the three digital (binary, six-level, and random-level) masks almost overlap each other. Therefore, similar performance of the time-series prediction task can be obtained for the three digital mask signals under similar standard deviations of the mask signals. More interestingly, the NMSEs for the case of the chaos mask signal are smaller than those for the three digital mask signals at the same standard deviation. We interpret that the chaos mask signal effectively induces large complex dynamics of the response laser output and a variety of virtual node states can be obtained, resulting in the performance improvement of the time-series prediction task.
Fig. 7 NMSEs of the time-series prediction task for the binary (black curve with circles), six-level (red curve with squares), random-level (blue curve with triangles), and chaos mask signals (green curve with diamonds) as the standard deviation of the mask signal is varied.
3.2 Colored-noise and chaos mask signals
Next, we introduce analog colored-noise mask signals for reservoir computing and compare their performance with the case of the chaos mask signal. The colored noise signals are generated from the Ornstein–Uhlenbeck process using white Gaussian noise and a first-order low-pass filter in numerical simulations [24]. The cut-off frequency fc of the colored noise signal is changed by using different correlation times in the numerical simulation. We generate four colored-noise signals with different cut-off frequencies of fc = 1.5, 3, 6, and 12 GHz. In our experiment, fc is limited at 12.5 GHz, which is the Nyquist frequency of the arbitrary waveform generator (25 GigaSamples/s).
Figure 8 shows the input mask signals of the colored-noise mask signals with the cut-off frequency of fc = 3 and 12 GHz. A faster oscillation is observed for a larger cut-off frequency. The frequency spectra of the colored-noise mask signals show flat spectrum components from 0 Hz to fc, which corresponds to −3 dB degradation of the power spectrum. For analog mask signals, we generate a sequence of the colored-noise signal for the time duration T of the mask, and this sequence is repeatedly used for each input data to create the modulation signal.
Fig. 8 (a), (c) Temporal input mask signals of colored-noise signals and (b), (d) corresponding fast Fourier transforms. (a), (b) Cut-off frequencies of fc = 3 GHz, and (c), (d) fc = 12 GHz.
Figure 9 shows the temporal waveforms of the colored-noise mask signals with the cut-off frequencies of fc = 3 and 12 GHz, and the corresponding response laser outputs. The standard deviations of the colored-noise mask signals are set to 0.34 for both cases. Complex oscillations of the response laser outputs are observed for both cases, and a variety of node states can be obtained. However, the oscillation frequency of the response laser output for the case of fc = 12 GHz [Fig. 9(b)] is much faster than that for the case of fc = 3 GHz [Fig. 9(a)]. In Fig. 9(b), the average frequency of the response laser output is approximately 6 GHz, corresponding to the relaxation oscillation frequency of the response laser under optical injection from the drive laser.
Fig. 9 Temporal waveforms of the colored-noise mask signals with the cut-off frequencies of (a) fc = 3 GHz and (b) fc = 12 GHz, and the corresponding response laser outputs. Blue dots indicate node states.
Figure 10 shows the experimental results of the time-series prediction task using the colored-noise mask signals with fc = 3 and 12 GHz. The time-series prediction is successful for both cases. However, the error signals for the case of fc = 12 GHz are smaller than those for fc = 3 GHz. The NMSEs are 0.243 and 0.146 for the cases of fc = 3 and 12 GHz, respectively. This result indicates that better performance can be achieved for the colored-noise mask signal with a larger cut-off frequency.
Fig. 10 Experimental results of the Santa Fe time-series prediction task using the colored-noise mask signals with the cut-off frequencies of (a) fc = 3 GHz and (b) fc = 12 GHz. Original signals (black lines), prediction results (red lines), and error signals (blue lines) are shown. The NMSEs are (a) 0.243 and (b) 0.146, respectively.
We investigate the NMSEs of the time-series prediction task for the different colored-noise mask signals when the standard deviation of the mask signal is continuously changed. Figure 11 shows the experimental results of the NMSEs for the four colored-noise mask signals with different cut-off frequencies (fc = 1.5, 3, 6, and 12 GHz) as the standard deviation of the mask signal is varied. It is found that the NMSEs decrease monotonically as the standard deviation of the mask signal is increased for all cases, which is similar to the cases of digital mask signals, as shown in Fig. 7. We cannot further increase the standard deviation of the mask signals due to the limitation of the gain of the amplifier connected to the phase modulator. In addition, better NMSEs are obtained for larger fc. The NMSE for the case of fc = 12 GHz (the blue curve) shows the best performance among the four colored-noise mask signals. The results of NMSEs for the case of the chaos mask signal are also plotted in Fig. 11 (the purple curve) for comparison. It is found that the NMSEs for the chaos mask signal are similar to the case of the colored-noise mask signal with fc = 6 GHz (the green curve). The chaos mask signal produces a similar resonance effect to the case of the colored-noise mask signal with fc = 6 GHz to induce the complex dynamics of the response laser output. Therefore, no significant difference is found between the chaos mask signal and the colored-noise mask signal with a certain fc on the performance of the time-series prediction task.
Fig. 11 NMSEs for the four colored-noise mask signals with the cut-off frequencies of fc = 1.5 GHz (black curve with circles), fc = 3 GHz (red curve with squares), fc = 6 GHz (green curve with triangles), and fc = 12 GHz (blue curve with diamonds) as the standard deviation of the mask signal is varied. The results of NMSEs for the chaos mask signal (purple curve with crosses) is also plotted for comparison. CNmask indicates the colored-noise mask.
From Fig. 11, we understand that the NMSEs are strongly dependent on the cut-off frequency of the colored-noise mask signals. We found that the performance can be improved by setting the cut-off frequency slightly higher than the relaxation oscillation frequency of the response laser under optical injection, to effectively induce a large amplitude of the laser dynamics by the input modulation signal [21]. We consider that strong resonance occurs when the relaxation oscillation frequency of the response laser is set to be in the same order of magnitude as the cut-off frequency of the colored-noise mask signal. Note that the relaxation oscillation frequency of the response laser is increased due to the external optical injection from the drive laser [25]. We cannot further increase the cut-off frequency of the colored noise signals due to the limitation of the sampling rate of the arbitrary waveform generator (25 GigaSamples/s). However, we experimentally confirm that the NMSEs are strongly dependent on the cut-off frequency of the colored-noise mask signals.
4. Discussion
From the comparison between the digital and analog mask signals [Figs. 7 and 11] for different standard deviations of the mask signals, we found that mask signals with larger amplitudes improve the NMSEs for all the digital and analog mask signals. No significant difference of the NMSEs exists among the binary, six-level, and random-level digital mask signals. However, the chaos mask signals result in better performance than those of these digital mask signals for the same standard deviation of the mask signals [see Fig. 7]. In the case of colored-noise signals, the NMESs are strongly dependent on the cut-off frequencies of the colored-noise mask signals [see Fig. 11]. However, no significant difference of the NMSEs exists between the chaos mask signal and the colored-noise mask signal with the cut-off frequency of 6 GHz, whose spectrum is close to the power spectrum of the chaos mask signal [see Fig. 3(d)]. Therefore, we consider that a larger and faster colored-noise mask is the best input mask in our study.
These experimental findings seem to contradict with the previously reported numerical results, obtained from the Lang–Kobayashi equation model [see Figs. 6 and 12(b) in [19] for the amplitude and frequency dependence]. In [19], the optimal amplitude and frequency of the input mask signal are found in numerical simulation, whereas this is not the case in our experiments. We speculate that there exists the limitation of the amplitude and frequency of the mask signals due to the experimental apparatus, such as the limited gain of the amplifier and the limited sampling rate of the arbitrary waveform generator used for the input mask signals. If the experimental results are compared with the numerical results in the regions of small amplitudes and low frequencies before reaching the optimal values, both results agree well with each other. (That is, prediction errors decrease monotonically as the amplitude or the frequency of the mask signal is increased.)
For the comparison between the digital and analog mask signals, the digital mask signal is easy to design and implement in experiment. In contrast, the analog mask signal is more preferable to induce a variety of complex dynamics of the reservoir’s laser outputs. The relaxation oscillation frequency and the frequency of the chaos or colored-noise mask signal need to be in the same order of magnitude to achieve resonance between them, and large-amplitude and high-frequency oscillations of the laser outputs can be obtained that are useful for the performance improvement of delay-based reservoir computing.
We found that the design of both the amplitude and the frequency of the mask signal is crucial to improve the performance of delay-based reservoir computing. We consider that a good input mask signal induces large resonant dynamics of the response laser. Therefore, large complex mask signals with the oscillation frequency close to the relaxation oscillation frequency of the response laser under optical injection could be useful to improve the performance of delay-based reservoir computing.
We have not investigated the influence of the complexity (e.g., Lyapunov exponents) of the chaos mask signal on the system performance. The complexity of chaos may be related to the performance of reservoir computing and this topic will be part of our future work. In addition, we expect that our experimental results would be valid for other benchmark tasks for reservoir computing, and the use of other benchmark tasks will also be part of our future work.
5. Conclusions
We experimentally investigated delay-based photonic reservoir computing using semiconductor lasers with optical feedback and injection. We applied different types of temporal mask signals, such as binary, six-level, random-level, chaos, and colored-noise mask signals as the weights between the input signal and the virtual nodes in the reservoir. We evaluated the performance of reservoir computing using the Santa Fe time-series prediction task and compared the normalized mean-square errors of the prediction task for the different mask signals. We found that the chaos mask signal results in better performance than the digital mask signals. However, there is no significant difference in performance between the chaos and colored-noise mask signals, if the cut-off frequency of the colored-noise signal is adequately selected. Mask signals with larger amplitudes result in better performance for all mask signals in the range of the amplitude accessible in our experiment. The frequency of the mask signal is crucial for improving the performance of reservoir computing, and the fundamental oscillation frequency of the mask signal needs to be in the same order of magnitude as the relaxation oscillation frequency of the response laser under optical injection.
A design of the temporal mask signal provides flexibility of the weights between the input signal and the reservoir in delay-based reservoir computing. A suitable design of the temporal mask signal could result in significant improvement of the performance of delay-based reservoir computing.
Funding
Grants-in-Aid for Scientific Research from Japan Society for the Promotion of Science (JSPS KAKENHI Grant Number JP16H03878); JST CREST (Grant Number JPMJCR17N2).
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