Time Parameter Optimization for the Semiconductor Laser-Based Time-Delay Reservoir Computing System

Abstract

Time-delay reservoir computing (RC) systems, particularly those based on semiconductor lasers (SLs), have gained attention due to their low energy consumption, high response rates, and rich nonlinear dynamics. This work investigates the influence of key time parameters—virtual node interval (θ), delay feedback (τ), and data injection period (T) on the performance of SL-based time-delay RC systems. Using the Santa Fe time series prediction task and memory capacity evaluation task, we analyze how these parameters affect prediction accuracy and memory capability. The results reveal that θ = 0.2T_ro (where T_ro is the relaxation oscillation period of the SLs) optimizes prediction performance, while θ = 0.5T_ro maximizes memory capacity. Additionally, feedback delay τ significantly impacts system performance. Shorter τ values (e.g., τ = 0.54T) enhance prediction accuracy, whereas longer τ values (e.g., τ = 1.74T) improve memory capacity. These findings provide valuable insights for optimizing time-delay RC systems, enabling better task-specific performance and stability.

1. Introduction

Reservoir computing (RC) is a neural-inspired computational framework designed to address the challenges associated with training recurrent neural networks (RNNs). In 2001, Jaeger et al. and Maass et al. proposed the Echo State Network (ESN) [1] and the Liquid State Machine (LSM) [2], respectively, both of which share the common feature of employing a training-free network to simplify the training process. In 2007, Verstraeten et al. unified these approaches under the term RC [3]. Both RNNs and RC employ a recursive structure and nonlinear activation functions, which not only grant them memory capabilities but also enhance their ability to learn and model complex nonlinear systems. However, RNNs typically update all network weights using backpropagation algorithms, which often lead to issues such as gradient vanishing or gradient explosion. In contrast, RC processes information through a sparsely connected network with fixed weights, known as the reservoir [4,5]. Notably, in RC, only the output weights require training, while the input weights and internal reservoir weights remain unchanged. As a novel paradigm of artificial neural networks, RC has demonstrated exceptional performance in various tasks, including time series prediction, nonlinear channel equalization, handwritten digit recognition, and speech recognition [6]. Moreover, RC has garnered significant attention due to its inherent advantages, such as low energy consumption, high computational speed, and ease of hardware implementation [7,8].

In 2011, Appeltant et al. pioneered a time-delay RC system using a Mackey–Glass circuit [9]. This approach utilized a single physical node combined with a feedback loop, significantly simplifying the hardware implementation of RC. Since then, extensive research has been conducted on time-delay RC, leading to the development of RC systems based on electronic circuits, optoelectronic systems, and all-optical systems [10,11,12,13,14]. For instance, in 2012, Larger et al. proposed an optoelectronic time-delay RC system [15], which demonstrated excellent performance in tasks such as digital speech recognition and chaotic time series prediction. Similarly, Duport et al. developed an all-optical RC system by employing a semiconductor optical amplifier (SOA) as the nonlinear node of the reservoir [16]. Compared to electronic and optoelectronic devices, semiconductor lasers (SLs) have emerged as a superior choice for constructing time-delay RC systems due to their low energy consumption, high response rates, and rich nonlinear dynamics. In 2013, Brunner et al. implemented the first all-optical RC system using an SL as the nonlinear element and evaluated its performance through tasks such as time series prediction and speech recognition. The results demonstrated that the system could achieve high-speed operation with excellent performance [17]. In 2015, Romain et al. constructed a time-delay RC system based on a semiconductor ring laser (SRL) capable of handling two tasks simultaneously, significantly improving the computational efficiency of RC [18]. In 2018, Hou et al. proposed a time-delay optical RC system using an SL with dual optical feedback loops. This system exhibited enhanced prediction performance due to the increased complexity of nonlinear states compared to single-loop configurations [19]. In 2020, Guo et al. introduced an electrically modulated RC system based on a semiconductor nanolaser (SNL), which further improved the information processing rate of optical RC systems [20]. More recently, extensive research has focused on time-delay RC systems based on SLs, covering laser type selection, system architecture optimization, key parameter tuning, and task-specific applications [21,22,23,24,25]. For instance, in 2025, Li et al. successfully constructed a time-delayed RC system to process image recognition task based on mutually coupled multimode SLs [26]. By leveraging the multi-mode characteristics of SLs, they effectively reduced the feedback loop length and therefore enhanced the information processing rate.

Previous studies have shown that the performance of SL-based time-delay RC systems is influenced by several factors. Firstly, it depends on the system architecture and the type of SL used [27,28,29]. Secondly, it is significantly affected by parameters such as feedback strength, optical injection strength, and several time-related parameters [30,31,32,33,34,35,36]. These time parameters include the interval between virtual nodes (θ), the delay feedback (τ), and the data injection period (T), all of which play a crucial role in determining the performance of the RC system. Currently, the optimal values for these parameters are primarily determined through empirical testing, which makes the tuning process of SL-based time-delay RC systems highly complex and time-consuming.

In this work, we investigate the relationship between these time parameters in an SL-based time-delay RC system under different types of tasks. By employing the Santa Fe time series prediction task and the memory capacity evaluation task, we provide a detailed analysis of how these time parameters influence the prediction performance and memory capabilities of the system. Our findings offer valuable insights into optimizing the configuration of time parameters in time-delay RC systems, thereby enhancing their computational performance and stability.

2. System Model

Figure 1 illustrates the structure diagram of the SL-based time-delay RC system. In the system, the drive-SL (D-SL) is used as a light source, and the input signal converts the amplitude variation in the electrical signal into the phase variation in the optical signal through the phase modulator (PM). The optical signal, carrying the input information, is injected into the Response-SL (R-SL), where a variable attenuator (VA1) adjusts the injection intensity. R-SL acts as the physical node of the reservoir layer, receiving signals from both the Drive-SL and its own feedback loop to generate a nonlinear response. The optical circulator (OC), along with a second variable attenuator (VA2) and a fiber coupler (FC1), forms a feedback loop, where VA2 regulates the feedback intensity. Part of the output of FC1 is fed back to R-SL, and the other part is collected in time as the state of the node in the reservoir.

According to the information processing flow, this RC system can be divided into three main parts: the input layer, the reservoir layer, and the output layer, as illustrated in Figure 2. In the input layer, the information to be processed first undergoes mask preprocessing. Since the input data are typically in discrete form, denoted as u(n), they needs to be converted into a continuous signal u(t), where each data point remains constant over time T. The time parameter T usually ranges from a few nanoseconds to tens of nanoseconds, determining the data input period. Within a period T, the response state of the SL is sampled n times to obtain n reservoir states. It is noteworthy that the relaxation oscillation period of the SL is generally less than 1 nanosecond. Therefore, if u(t) is directly injected into the SL within each period T, the SL output will quickly reach a steady state after a brief transient response. This leads to a homogenization of subsequently collected n reservoir states, which is unfavorable for effective feature extraction from the data. To achieve richer dynamic responses, the input layer performs mask processing on u(t). The data input period T is divided into n intervals, each of length θ (θ = T/n). Within each interval, a random weight is assigned to construct the input weight signal, known as the mask. This process is known as mask operation, where u(t) is multiplied by the mask signal to produce the final input signal j(t). The parameter θ significantly impacts the diversity of reservoir states. If θ is too large, it leads to homogenization of reservoir states, On the other hand, if θ is too small, it may result in an insufficient laser response and the loss of input information features.

Furthermore, to enhance the system’s nonlinear characteristics and endow the reservoir with memory capabilities, the SL is subjected to delay feedback with a delay feedback of τ. Although previous studies have conventionally set the time parameter τ to a value comparable to the data injection period T, theoretically it can be set to any arbitrary value if the impact of τ on the system’s memory capacity and nonlinear characteristics is not taken into account.

In the output layer, the discrete reservoir states x(n) are obtained by collecting the output light intensity of the SL, which are then used to train the readout weights W_out. The W_out can be solved using the ridge regression method. After training, the output of the reservoir y(n) is obtained by multiplying x(n) with W_out.

In this RC system, the nonlinear dynamics of the R-SL under optical injection and time-delay feedback can be expressed as [37]:

$$\begin{array}{ll}{\displaystyle \frac{d{E}_r(t)}{dt}}=& {\displaystyle \frac{1+i\mathsf{\alpha }}{2}}\left\{{\displaystyle \frac{G_N(N_r(t)-N_0)}{1+\mathsf{\epsilon }|E_r(t){|}^2}}-{\displaystyle \frac{1}{{\mathsf{\tau }}_p}}\right\}{E}_r(t)+{\mathsf{\kappa }}_{inj}{E}_d(t)\exp (i∆\mathsf{\omega }t)\\ & +\mathsf{\kappa }{E}_r(t-\mathsf{\tau })\exp (-\mathrm{i}{\mathsf{\omega }}_{\mathrm{r}}\mathsf{\tau })\end{array}$$

(1)

$${\displaystyle \frac{d{N}_r(t)}{dt}}=J_r-{\displaystyle \frac{N_r(t)}{{\mathsf{\tau }}_s}}-{\displaystyle \frac{G_N(N_r(t)-N_0)}{1+\mathsf{\epsilon }|E_r(t){|}^2}}|E_r(t){|}^2$$

(2)

where N_r is the carrier density of the response SL, $\mathsf{\alpha }$ is the linewidth enhancement factor, G_N is the gain coefficient, N₀ is the transparency carrier density, ε is the saturation coefficient, and τ_p and τ_s are the photon and carrier lifetimes, respectively. κ_inj is the optical injection strength, κ is the feedback strength, ω_d, and ω_r, are the optical angular frequencies of the D-SL and R-SL, respectively, Δω is the angular frequency detuning (Δω = 2πΔf = 2π(f_d − f_r)), and τ is the delay time of the feedback loop. J_r is the injection current of the response laser, defined as J_r = j × Jth, where Jth is the threshold current and j is the normalized pump current. E_d and E_r are the slowly varying electric fields of the D-SL and R-SL, respectively. E_d can be expressed as:

$$E_d(t)=\sqrt{I_d}\exp (i\mathsf{\pi }j(t))$$

(3)

where I_d is the light intensity of the D-SL, and j(t) is the information injected into the reservoir.

The delay differential equations are solved using the fourth-order Runge–Kutta method with an integration step of 2ps. The parameter values used are as follows: α = 3.0, N₀ = 1.4 × 10²⁴, G_N = 8.4 × 10⁻¹³, ε = 2.0 × 10⁻²³, τ_p = 1.927 × 10⁻¹², τ_s = 2.04 × 10⁻⁹, ω_d = 1.23 × 10¹⁵, Δf = −4 × 10⁹.

3. Results and Discussion

3.1. Santa Fe Time Series Prediction

We first investigate the optimization method for the time parameters of the RC system using the Santa Fe time series prediction task. The Santa Fe dataset consists of a chaotic time series with 9000 sample points. The objective of this task is to predict the next data point based on historical data. The prediction performance is evaluated using the normalized mean square error (NMSE) which can be expressed as:

$$NMSE={\displaystyle \frac{1}{L}}{\displaystyle \sum _{n=1}^L[y(n)}-y_d(n){]}^2/\mathrm{var}(y_d)$$

(4)

In Equation (4), L is the length of the test data, y(n) is the system output, y_d is the desired output, and var denotes the variance. It is generally considered that the RC system exhibits good prediction performance when NMSE ≤ 0.1. In this work, 2000 data points are used as the training set and another 1000 data points are used as the testing set.

Among the three time parameters T, θ, and τ, T is denoted as T = n × θ, where n represents the number of virtual nodes. In this work, we set n = 50. Therefore, our focus is on discussing the impact of θ and τ on the system’s performance.

In previous studies, θ has often been measured by the relaxation oscillation period T_ro of the SL under free-running. It is suggested that θ should take values within the range of [0.1T_ro, T_ro], where T_ro can be expressed as:

$$T_{ro}={\displaystyle \frac{2\mathsf{\pi }}{\sqrt{G_N{J}_{th}(j-1)}}}$$

(5)

In Equation (5), Jth = Nth/τ_s, where Nth is the threshold carrier density (Nth = N₀ + 1/(G_N τ_p))). Throughout the testing, j is set to 1.05. Therefore, we know that the relaxation oscillation period T_ro is 0.97 ns.

Next, we discuss the impact of θ on the system’s prediction performance. As θ changes, both the injection strength κ_inj and feedback strength κ affect the system’s prediction performance, we analyze the evolution of NMSE in the κ_inj-κ parameter space for different θ values.

Figure 3 illustrates the evolution maps of the prediction error NMSE in κ_inj-κ parameter space as θ varies from 0.1T_ro to 0.6T_ro under τ = T. Different colors in the figure indicate different NMSE values. To enhance color recognition, only regions with good prediction performance (NMSE < 0.1) are displayed. The white dashed lines in the figure outline areas where NMSE ≤ 0.02. In the six figures, the regions with good prediction performance are predominantly located in the lower-right portion of the scanning interval. This indicates that a certain injection strength κ_inj is required to achieve good prediction performance, ensuring that the SL responds sufficiently to input signals. Additionally, under a given injection strength, the feedback strength k should not be excessively high, as this could induce system oscillations or even drive the system into a chaotic state. From Figure 3a–f, it is evident that as θ increases, the regions with good prediction performance gradually shrink. The minimum NMSE values (NMSE_min) in the six figures are 0.011, 0.009, 0.009, 0.011, 0.017, and 0.016, respectively. Notably, when θ is 0.2T_ro and 0.3T_ro, the system achieves the lowest prediction error of 0.009 under τ = T.

Figure 4 illustrates the minimum prediction error NMSE_min obtained by scanning the κ_inj-κ parameter space as θ increases from 0.1T_ro to T_ro. The subplot in the figure shows a magnified view as θ increases from 0.1T_ro to 0.8T_ro. The NMSE_min values in Figure 4 is defined as the minimum NMSE obtained in evaluation map of k_inj-k parameter space. For instance, at θ = 0.1T_ro we scanned a parameter space of 0 ns⁻¹ ≤ k ≤ 5 ns⁻¹ and 0 ns⁻¹ ≤ k_inj ≤ 10 ns⁻¹ (shown in Figure 3a). Within this space, the minimum NMSE is 0.011, which is defined as the NMSE_min for θ = 0.1T_ro. It can be observed that as θ increases, NMSE_min first decreases and then increases. The minimum value of 0.009 is achieved at 0.2T_ro and 0.3T_ro. When θ exceeds 0.8T_ro, the system fails to achieve good prediction performance (NMSE > 0.1), indicating that larger θ values lead to reduced prediction accuracy. It is evident that determining θ requires a comprehensive consideration of the results from Figure 3 and Figure 4. On one hand, a larger interval in the evolution map that yields good prediction performance indicates relatively simpler parameter adjustment in hardware implementation. On the other hand, a smaller NMSE_min signifies a stronger capability of the system to make accurate predictions. Based on the regions of good prediction performance and the NMSE_min values, we conclude that θ = 0.2T_ro is the optimal choice for maximizing prediction accuracy in SL-based time-delay RC systems.

Next, we investigate the impact of delay feedback τ on the prediction performance, here we set θ = 0.2T_ro, κ_inj = 4.2 ns⁻¹, κ = 0.8 ns⁻¹. In previous studies, τ was conventionally set either synchronized with T (τ = T) or not synchronized with T (τ ≠ T, with a tendency for τ = T–θ), but there was no in-depth analysis of more specific scenarios. Figure 5 depicts the variation in NMSE as τ increases from 0.1T to 2T. Throughout the change in τ, the NMSE values remain below 0.04, fluctuating mainly around 0.02. When τ is 0.4T, 0.54T, 0.8T, 0.9T, 0.98T, and T, NMSE is below 0.01, and the minimum NMSE of 0.0055 is achieved at 0.54T. When τ > T, it is almost impossible to achieve a prediction error of NMSE < 0.01. It is evident that when adopting the non-synchronous scheme, apart from the traditional setting of τ = T−θ (τ = 0.98T), there are more options available. Particularly, setting τ equal to 0.54T is more conducive to achieving relatively fewer prediction errors.

3.2. Memory Capacity Evaluation

Subsequently, we investigate the influence of time parameters on the system’s memory capacity based on the memory capacity evaluation task. This task assesses the system’s ability to store and recall the previously input information. In this task, the data to be processed is a set of random numbers uniformly distributed within the interval of [−0.5, 0.5], and the system is required to output the data from j steps earlier. The memory performance of the system is evaluated using MC, which can be expressed as follows [38,39]:

$$MC={\displaystyle \sum _{\mathrm{j}=0}^{+\infty }m(j)}$$

(6)

In Equation (6), m(j) represents the correlation coefficient between the system output sequence and the expected values when the delay step is j. m(j) can be expressed as:

$$m(j)={\displaystyle \frac{<[v(i-j)-<v(i-j)>][y_j(i)-<y_j(i)>])}{\sqrt{<|v(i-j)-<v(i-j)>{|}^2><|y_j(i)-<y_j(i)>{|}^2>}}}$$

(7)

where i represents the current input step, y is the estimated value, and v is the target value. The symbol <.> denotes the time average. In this test, we use 2000 data points for training and 1000 data points for testing.

Figure 6 illustrates the evolution of MC within the parameter space of injection strength κ_inj and feedback strength κ under different θ. Different colors in the figure represent varying MC values, with the black dashed lines outlining the regions where MC ≥ 10. The parameter intervals displayed here are 15 ns⁻¹ ≤ κ_inj ≤ 40 ns⁻¹ and 4 ns⁻¹ ≤ κ ≤ 18 ns⁻¹. Compared to the Santa Fe time series prediction task, achieving a high MC requires greater injection strength and feedback strength. We hypothesize that higher injection strengths induce more significant response states in the SL, which persist across multiple feedback cycles but gradually diminish over time. The regions outlined with high MC values (MC ≥ 10) are located at the edge of chaos, indicating that, under certain injection strengths, larger feedback can enhance the system’s memory capacity, but excessive feedback that causes system oscillations can reduce memory capacity. From Figure 6a–f we can see that as the θ increases from 0.1T_ro to 0.6T_ro, the maximum MC values (MC_max) obtained in each evolution map are 13, 11, 12, 12, 16, and 13, respectively. It is evident that the system’s maximum MC of 16 occurs in Figure 6e at θ = 0.5T_ro under τ = T. Additionally, we can observe that when θ = 0.5T_ro, the region with MC ≥ 10 is more extensive, which is beneficial for parameter adjustment in hardware implementation.

Figure 7 depicts the variation in MC_max as θ increases from 0.1T_ro to T_ro, where each value represents the maximum MC obtained by scanning the κ_inj-κ parameter space. When 0.1T_ro ≤ θ ≤ 0.4T_ro, MC_max fluctuates around 12. When θ = 0.5T_ro, MC_max reaches its peak value of 16. When θ > 0.5T_ro, MC_max gradually decreases to around 11. Combining the insights from Figure 6 and Figure 7, we find that θ = 0.5T_ro is optimal for achieving high memory capacity, as it provides a balance between parameter tuning simplicity and system performance. We find that 0.5T_ro represents an intermediate phase of the system’s relaxation oscillations, during which the system state is characterized by a combination of nonlinearity, rapid dynamics, energy release, and transient instability. Compared to the early relaxation oscillation phase at 0.2T_ro, the system exhibits a more comprehensive response and tighter coupling between input signals. We hypothesize that this is the reason for the emergence of the MC peak at 0.5T_ro.

Next, we investigate the influence of τ on memory capacity. Here, we set the parameters as θ = 0.5T_ro, κ_inj = 19 ns⁻¹, κ = 16 ns⁻¹. Figure 8 illustrates the variation in MC as τ increases from 0.1T to 4T. When τ < 1.74T, MC fluctuates with an upward trend, and MC is 16.5 at τ = T. When τ = 1.74T, MC reaches its maximum value of 21.7. When 1.74T < τ≤ 3.72T, MC oscillates between 14 and 21, and the value of 21 are observed at τ = 2.48T, 2.92T, and 3.6T. When τ > 3.72T, MC oscillates and decreases to around 13. Comparing Figure 5 and Figure 8, it can be observed that in the Santa Fe time series prediction task, NMSE fluctuates around 0.02 as τ increases from 0.1T to 2T, with NMSE below 0.01 occurring at several specific points where τ ≤ T. In contrast, in the memory capacity test task, a relatively large τ tends to favor higher MC values, but MC decreases when τ > 3.72T. We can infer that a relatively large delay feedback is conducive to achieve high memory capacity, as the current data injection is superimposed with the reservoir state from several cycles earlier and injected into the reservoir together. As τ becomes excessively large, the system’s memory capacity gradually declines, likely due to the reservoir’s struggle to forge a linkage between the current input and the reservoir state from a too distantly delayed cycle. The difference in the trend of the curves in Figure 5 and Figure 8 can enlighten us that when dealing with different tasks, it is necessary to adjust the delay feedback according to the task’s requirements for memory capacity and nonlinear transformation capability.

3.3. Discussion

In the above tests, the optimal time parameters θ and τ for the Santa Fe time series prediction task and the memory capacity evaluation task were identified. We recognize that during hardware implementation, certain parameters of the SL such as the linewidth enhancement factor (α) and the gain coefficient (G_N), may deviate from their nominal values. Therefore, the robustness of the identified optimal parameters needs further discussion.

Figure 9a shows the NMSE variation with respect to α and G_N for Santa Fe prediction task under κ_inj = 4.2 ns⁻¹, κ = 0.8 ns⁻¹, θ = 0.2T_ro, τ = 0.54T. It shows that the NMSE remains below 0.01 when α varies within −2% to +3% of its nominal value, and when G_N fluctuates within ±6% of its nominal value. Figure 9b shows the MC variation versus α and G_N for memory evaluation task under κ_inj =19 ns⁻¹, κ =16 ns⁻¹, θ = 0.5T_ro, τ = 1.74T. Compared to Figure 9a, the MC exhibits significantly stronger robustness against parameter drift in both α and G_N. Specifically, the MC maintains values above 16 when α varies from −10% to +6% or G_N varies from −5% to +8% of their nominal values. These findings indicate that the RC system’s performance can tolerate moderate parameter variations during hardware implementation, which is advantageous for maintaining stable long-term operation.

It should be noted that this work does not directly address the influence of injection strength (κ_inj) and feedback strength (κ) on the time parameters θ and τ. However, previous studies have demonstrated that a higher injection strength can enhance the reservoir’s response bandwidth [40], which facilitates a reduction in the virtual node interval and consequently improves the information processing rate while maintaining comparable reservoir performance. In this study, our investigation consistently focused on optimizing reservoir performance. Therefore, in Figure 3 and Figure 6, when evaluating system performance across the κ_inj-κ parameter space for different θ values, we selected θ based on configurations yielding optimal performance. Under this research framework, we observe that although the input data from both Santa Fe prediction task and MC evaluation task were normalized before being injected into the system, the parameter ranges (k_inj, k) for achieving good performance are distinct. For prediction tasks, achieving good performance requires relatively small values of κ and κ_inj (0 ns⁻¹ ≤ κ ≤ 5 ns⁻¹, 2.5 ns⁻¹ ≤ κ_inj ≤ 10 ns⁻¹). In contrast, memory evaluation tasks demand significantly larger κ and κ_inj to attain a high MC (4 ns⁻¹ ≤ k ≤ 18 ns⁻¹, 15 ns⁻¹ ≤ k_inj ≤ 40 ns⁻¹). According to previous studies on the nonlinear dynamical states of SLs, increasing feedback strength enriches the nonlinear states of SLs and can even drive them toward chaotic regimes. Numerous studies on SL-based reservoir computing have demonstrated that optimal reservoir performance is typically achieved near the edge of chaos [17]. Furthermore, we recognize that optical feedback is crucial for enabling memory capacity in RC. Comprehensive analysis of these factors reveals that achieving high MC generally requires strong feedback, while maintaining the system at the edge of chaos under high feedback conditions necessitates correspondingly high injection strength.

This work achieves a minimum NMSE of 0.0055 for the Santa Fe time series prediction task under optimized parameters θ = 0.2T_ro and τ = 0.54T, while the memory capacity evaluation task reaches its peak performance of 21.7 at θ = 0.5T_ro and τ = 1.74T. The obtained results outperform previous approaches using τ = T or τ = T ± θ configurations. Specifically, for the Santa Fe prediction task, our NMSE of 0.0055 represents an improvement over previously reported values of 0.0293 [20], 0.008 [41], and 0.0075 [33]. Similarly, in the memory evaluation task, the achieved MC of 21.7 exceeds the prior results of 10 [42] and 17.6 [33]. The research results demonstrate that adaptively adjusting the virtual node spacing θ according to specific task requirements can enhance RC performance. Furthermore, the feedback delay τ need not be synchronized with the data injection period T. Compared to conventional approaches that limited τ to a few predetermined values, our findings provide new insights for implementing high performance RC systems.

4. Conclusions

This work systematically explores the impact of time parameters of θ, τ, and T on the performance of SL-based time-delay RC systems. For the Santa Fe time series prediction task, we found that θ = 0.2T_ro and τ = 0.54T yield the lowest prediction error of NMSE = 0.0055, highlighting the importance of balancing virtual node interval and delay feedback for accurate forecasting. In contrast, for memory capacity evaluation task, θ = 0.5T_ro and τ = 1.74T maximize memory performance (MC = 21.7), as longer delay feedback allow the system to retain information over multiple cycles. However, excessively large τ (τ > 3.72T) values degrade memory capacity, likely due to the reservoir’s inability to link current inputs with distant past states.

With the progress in research on SL-based RC, the processing of diverse tasks based on RC has attracted extensive attention. Unlike previous studies that primarily focused on using a single RC system to handle diverse tasks to demonstrate its universality, our findings reveal that different types of tasks actually require different nonlinear transformation characteristics and memory capacities from the system. The results indicate that employing task-specific time parameters (θ and τ) can enhance system performance. Alternatively, one could design systems with tunable feedback delays, adaptively adjusting the feedback delay according to task requirements while employing different θ values at the input layer for data preprocessing to achieve optimal RC performance. In the future, we will conduct in-depth research on time parameter optimization of RC, including a theoretical analysis of the system and its hardware implementation.