A modeling method of chaotic optoelectronic oscillator based on deep learning

Abstract

The present invention discloses a chaotic optoelectronic oscillator modeling method based on deep learning, specifically: according to the one-to-one correspondence between the previous cycle chaotic carrier excitation of the next cycle chaotic carrier of the feedback loop of the optoelectronic oscillator, a deep learning architecture of a Fourier neural operator is applied to model the nonlinear dynamics of the OEO chaotic system with parameters such as different bandwidths, different gains and different delays, so as to obtain a multi-parameter optoelectronic oscillator model for generating chaotic carriers efficiently and at low cost. The present invention is not only hardware-independent, but also does not require any prior knowledge about the underlying process, and the constructed model can be easily switched to a system with different parameters, which essentially increases the key space of the system. Therefore, the present invention has potential application prospects in secure optical communications such as chaotic optical secure communications and random number generation.

Description

Technical Field

[0001] The present invention belongs to the field of secure optical communication transmission and deep learning, and in particular relates to a chaotic optoelectronic oscillator modeling method based on deep learning. Background Art

[0002] In the past decade, optical chaos has been widely used in secure optical communications, high-speed random number generation and chaotic radar due to its advantages of broadband, noise-like and unpredictability. At present, there are two main methods to generate optical chaotic carriers, one is a chaotic system based on the nonlinearity inside semiconductor lasers, and the other is based on optoelectronic oscillators (OEOs). The former is extremely sensitive to the physical parameters of the laser, and the bandwidth is limited by relaxation oscillations. However, due to the significant advantages of OEO in generating high-dimensional, broadband and flat power spectrum chaotic carriers, it has become a promising method in high-speed secure optical communications. Lorenz et al. discovered the first chaotic attractor in 1963, which opened the prelude to chaos research. In recent decades, the nonlinear dynamics of chaotic attractors have been intensively studied, and nonlinear dynamic modeling of chaotic systems has become an important research hotspot. In 1987, Crutchfield and McNamara first proposed estimating the parameter family and spatial mode evolution of dynamical systems from bifurcation data. The method of constructing nonlinear mapping of chaotic systems from time series data to extract mathematical models reproduces the deterministic part of the dynamics, which usually does not require any prior knowledge about the underlying process. On the other hand, differential equations provide a good mathematical model for nonlinear dynamics. As we know, OEO chaotic systems belong to time-delay dynamic systems and can be modeled by differential delay equations. However, such systems are particularly sensitive to initial conditions, so the combination of delay and high gain makes it possible to realize infinite dimensions and complex dynamics.

[0003] Deep learning is an algorithm based on data representation learning, which uses neural networks as parameter structures to learn the internal mapping relationship of a specific system. Deep learning has strong generalization ability and excellent fault tolerance, so its application in chaotic systems has been reported for a long time. However, most of the early studies were based on the analysis of low-dimensional chaotic systems, and the accuracy of chaos prediction was relatively low. In recent years, people have conducted a lot of research on the application of deep learning in OEO chaotic systems. In 2019, J. Ke et al. proposed the use of neural networks to improve the OEO chaotic synchronization performance, and successfully demonstrated 32Gbit / s message transmission within 20 kilometers with the help of deep learning in the same year. In 2021, X. Gao et al. proposed a modeling scheme for OEO chaotic systems based on long and short-term memory (LSTM) neural networks to achieve delay feature extraction. In 2022, L. Jiang et al. proposed to apply LSTM to learn the nonlinear dynamics of OEO with different gains and delays, and use the training model to encrypt and decrypt information at the transmitter and receiver. Compared with hardware-based methods, encryption and decryption can be implemented more conveniently and at a lower cost by modeling the actual device. Summary of the invention

[0004] In order to learn the nonlinear dynamics of OEO chaotic systems with different wavelengths, different bandwidths, different gains, and different feedback delays in a data-driven manner without any prior knowledge, a multi-parameter optoelectronic oscillator model for generating chaotic carrier waves with high efficiency and low cost is obtained. The present invention provides a chaotic optoelectronic oscillator modeling method based on deep learning.

[0005] The present invention provides a chaotic OEO modeling method based on deep learning. According to the one-to-one correspondence between the previous cycle chaotic carrier and the next cycle chaotic carrier in the feedback loop of the OEO, a deep learning architecture of a Fourier neural operator is used to model the nonlinear dynamics of the OEO chaotic system with different bandwidths, different gains, and different delays, thereby obtaining a multi-parameter optoelectronic oscillator model for generating chaotic carriers efficiently and at low cost. Specifically, the following steps are included:

[0006] Step 1: The traditional chaotic OEO is characterized by a differential delay equation, and its mathematical model is as follows:

[0007]

[0008] Where x(t) = πV(t) / (2V π ) is a dimensionless variable, β=παPRG / (2V π ) is the gain of the feedback loop, V πis the half-wave voltage of the Mach-Zehnder modulator MZM, V(t) is the RF voltage of the MZM, φ0 is the bias phase of the MZM, P is the output power of the continuous wave laser source, G is the gain driven by the RF amplifier, τ is the time delay of the delay line, α and R are the total attenuation of the feedback loop and the responsiveness of the local discharge respectively; θ1 and θ2 are the response time of the feedback loop.

[0009] Formula (1) is solved by MATLAB using the fourth-order Runge-Kutta algorithm with an iteration step of 2.5 ps and a sampling rate of 400 GSa / s. The simulation output sequence x(t) is downsampled to a 100 GSa / s sampling sequence y(t). The corresponding parameters are set as θ1=15.9 ps, θ2=5.3 μs, φ0=π / 4, where τ and β are used as adjustable parameters.

[0010] Step 2: Calculate the autocorrelation of the simulation data with different parameters to obtain the time delay of a feedback loop cycle; the autocorrelation function is defined as follows:

[0011]

[0012] Where <·> represents the average operation; According to formula (1), the input and output pairs of the neural network need to satisfy the one-to-one correspondence between the previous cycle chaotic carrier and the next cycle chaotic carrier of the feedback loop of the optoelectronic oscillator, so it is assumed that the time delay is known; Here, a large number of input-output training data sets are constructed for the simulation data, that is, f: x i →x i +n, where n is the number of symbols corresponding to the feedback delay τ. Subsequently, each input-output data set is cut using a sliding window of length L to fit the input dimension of the Fourier neural operator.

[0013] Step 3: The Fourier neural operator is considered as a deep learning iterative architecture that can learn the mapping between two infinite-dimensional spaces from finite-dimensional input-output pairs. Its iterative update v t →v t+1 , t = 1, ..., t-1 is composed of the Fourier neural kernel operator, the linear transformation R, the weight matrix W and the nonlinear activation function σ, and its iteration rule is as follows:

[0014] v0(x)=Pa(x)+p (3)

[0015]

[0016] u(x)=Qv T (x)+q (5)

[0017] Where p and q are bias values, which are first lifted to a high-dimensional representation by a local transformation P parameterized by a shallow fully connected neural network; the lifting operation is considered as the first layer of the global neural network, and then v0(x) = P(a(x)) is obtained as The initial assumption is: then, after iterative updates, the output is u(x) = Q(v T (x)), which is obtained by transforming Q to v T The transformation Q is parameterized by a shallow fully connected neural network and is used to project back to the target dimension.

[0018] Step 4: Use python-pytorch to build the Fourier neural operator model; stack four Fourier integral operator layers, ReLU activation function and batch normalization layer to construct the Fourier neural operator; then, simulate the data of each delay and gain combination according to f:x i →x i +n to form a training set, of which 90% is used for training and 10% is used for testing; the Pearson correlation coefficient PCC is introduced to evaluate the consistency of the original sequence and the predicted sequence; the definition of PCC is as follows:

[0019]

[0020] Where x and y are the original sequence and predicted sequence respectively.

[0021] The beneficial technical effects of the present invention are:

[0022] 1) The present invention learns the nonlinear dynamics of optoelectronic oscillators in a data-driven manner without requiring any prior knowledge.

[0023] 2) Since the Fourier transform converts complex differentiation and convolution operations into simple product operations, the Fourier neural operator can speed up the operation compared to other deep learning architectures (such as LSTM).

[0024] 3) Compared with the traditional hardware-based optical chaotic carrier generation method, the multi-parameter chaotic OEO model constructed in the present invention is independent of hardware and can model OEO chaotic systems with different wavelengths, different bandwidths, different gains, and different feedback delays, thereby achieving efficient and low-cost chaotic carrier generation. Therefore, it has broad application prospects in the fields of chaotic optical secure communication and random number generation. BRIEF DESCRIPTION OF THE DRAWINGS

[0025] Figure 1 This is a structural block diagram of the chaotic optoelectronic oscillator modeling method based on deep learning of the present invention.

[0026] Figure 2Block diagram of chaotic data sequence preprocessing for the chaotic optoelectronic oscillator modeling method based on deep learning.

[0027] Figure 3 This is a structural block diagram of the Fourier neural operator model for the chaotic optoelectronic oscillator modeling method based on deep learning.

[0028] Figure 4 For different feedback gains ( Figure 4 a) and delay ( Figure 4 b) Comparison of PCC results from Fourier neural operator and LSTM prediction sequences under the same conditions.

[0029] Figure 5 The PCC results of the Fourier neural operator and LSTM prediction sequence under different oscillation frequency conditions are shown in Figure 2. Figure 5 (a)-(c) The oscillation frequencies are 5 GHz, 10 GHz and 15 GHz respectively). DETAILED DESCRIPTION

[0030] The present invention is further described in detail below with reference to the accompanying drawings and specific implementation methods.

[0031] The present invention discloses a chaotic optoelectronic oscillator modeling method based on deep learning. According to the one-to-one correspondence between the chaotic carrier of the previous cycle and the chaotic carrier of the next cycle in the feedback loop of the optoelectronic oscillator, the deep learning architecture of the Fourier neural operator is used to model the nonlinear dynamics of the optoelectronic oscillator characterized by the differential delay equation from data of different delays and gains, thereby obtaining a multi-parameter optoelectronic oscillator model for generating chaotic carriers efficiently and at low cost. A chaotic optoelectronic oscillator modeling scheme based on deep learning is mainly divided into the following three steps: 1) Constructing a mathematical model of the optoelectronic oscillator, and solving the mathematical model of the optoelectronic oscillator by applying the fourth-order Runge-Kutta algorithm through MATLAB simulation software, thereby obtaining chaotic sequence data of different parameters. 2) Solving the autocorrelation function of the chaotic sequence data of different parameters to obtain the feedback delay τ. Subsequently, a large number of input-output (i.e., f:x i →x i +n) training data set, where n is the number of symbols corresponding to the feedback delay τ. Next, a sliding window of length L is used to cut each input-output data set to adapt to the input dimension of the Fourier neural operator. 3) The Fourier neural operator model is built using python-pytorch, where 90% of the chaotic sequence data is used for training and 10% for testing, and the Pearson correlation coefficient is introduced to evaluate the consistency of the original sequence and the predicted sequence.

[0032] Figure 1This is a structural diagram of the chaotic optoelectronic oscillator modeling method based on deep learning of the present invention, which is mainly composed of three modules: (a) OEO chaotic system model, (b) Fourier neural operator structure and (c) multi-parameter chaotic OEO model. The OEO chaotic system consists of a continuous light source module with multiple wavelengths (1011~101 N )、MZM modulator 102、delay module (1031~103 N )、Feedback power module (1041~104 N ), photodetectors with different bandwidths (1051~105 N ) and a radio frequency amplifier module 106. The MZM modulator is seeded by a continuous wave laser source, and then the MZM modulator output is used to set the delay of the loop through a delay adjustment module. Next, an adjustable attenuator is used to implement a feedback power control module for adjusting the feedback gain of the OEO loop. The chaotic carrier is converted into an electrical signal by photodetectors of different bandwidths. The electrical signal is amplified by the radio frequency amplifier and used as a driving signal for the MZM to form a feedback loop to generate a chaotic carrier signal, which is digitized by a digital storage oscilloscope 107. Finally, the chaotic data is used to construct a multi-parameter chaotic OEO model.

[0033] Figure 2 The chaotic data preprocessing block diagram of the chaotic optoelectronic oscillator modeling method based on deep learning. In the chaotic data preprocessing module 108, the chaotic data generated by the OEO is used to calculate the feedback loop delay τ through the autocorrelation function, and then a large number of input-output (i.e., f:x) are constructed for the simulation data according to the one-to-one correspondence between the previous cycle chaotic carrier and the next cycle chaotic carrier of the feedback loop of the optoelectronic oscillator. i →x i+n ) training dataset, n is the number of symbols corresponding to the feedback delay τ. Subsequently, we slice each input-output dataset using a sliding window of length L to fit the input dimension of the Fourier neural operator.

[0034] Figure 3 The structure block diagram of the Fourier neural operator model of the chaotic optoelectronic oscillator modeling method based on deep learning is shown in FIG. In the Fourier neural operator model 109 , the Fourier neural operator is regarded as an iterative architecture, and its iterative update v t →v t+1 (t=1,…,t-1) is composed of Fourier neural kernel operator, linear transformation R, weight matrix W and nonlinear activation function σ. First, it is lifted to a high-dimensional representation (d dimension) through a local transformation P, which is usually parameterized by a shallow fully connected neural network. The lifting operation is regarded as the first layer of the global neural network, and then v0(x)=P(a(x)) is obtained as The initial assumption is . Then, after iterative updates, the output is u(x) = Q(vT(x)), which is obtained by the projection of the transformation Q on vT, where the transformation Q is usually parameterized by a shallow fully connected neural network projected back to the target dimension. The Fourier neural operator is constructed by stacking four Fourier integral operator layers with ReLU activation function and batch normalization layers using python-pytorch, and finally a multi-parameter chaotic OEO model is trained.

[0035] Figure 4 The PCC results of the Fourier neural operator and LSTM prediction sequence under different feedback gain and delay conditions are shown in Figure 2. Figure 4 It can be found that in terms of nonlinear mapping, the Fourier neural operator is more capable than LSTM. Figure 4 (a) It can be seen that as the gain increases, the chaotic time series becomes highly nonlinear and the learning ability of the Fourier neural operator gradually decreases. However, in the case of low feedback gain, the PCC can be greater than 0.99. In fact, the feedback gain is limited by the actual device and cannot be increased indefinitely in practice. On the other hand, Figure 4 (b) It can be seen that the nonlinear mapping ability of the Fourier neural operator is only affected by the feedback gain β and has nothing to do with the delay.

[0036] Figure 5 Under different oscillation frequency conditions ( Figure 5 (a)-(c), the oscillation frequencies are 5GHz, 10GHz and 15GHz respectively), PCC results from Fourier neural operator and LSTM prediction sequence. Figure 5 It is obvious that the learning effect of the Fourier neural operator deteriorates with the increase of the oscillation frequency, which indicates that the higher the oscillation frequency of the chaotic system, the faster the mutation of adjacent points in the discrete chaotic time series. Therefore, it will be more difficult to learn the nonlinear mapping of OEO. However, the nonlinear mapping ability of the Fourier neural operator shows very good robustness in the low gain range, and the PCC is greater than 0.99 in the small feedback gain range.

Claims

1. A chaotic optoelectronic oscillator modeling method based on deep learning, characterized in that: According to the one-to-one correspondence between the previous cycle chaotic carrier and the next cycle chaotic carrier of the feedback loop of the optoelectronic oscillator, a deep learning architecture of a Fourier neural operator is used to model the nonlinear dynamics of the OEO chaotic system with different wavelengths, different bandwidths, different gains and different feedback delays, so as to obtain a multi-parameter optoelectronic oscillator model for generating chaotic carriers efficiently and at low cost; specifically, the following steps are included: Step 1: The traditional chaotic OEO is characterized by a differential delay equation, and its mathematical model is as follows: Where x(t) = πV(t) / (2V π ) is a dimensionless variable, β=παPRG / (2V π ) is the gain of the feedback loop, V π is the half-wave voltage of the Mach-Zehnder modulator MZM, V(t) is the RF voltage of the MZM, φ0 is the bias phase of the MZM, P is the output power of the continuous wave laser source, G is the gain driven by the RF amplifier, τ is the time delay of the delay line, α and R are the total attenuation of the feedback loop and the responsivity of the partial discharge respectively; θ1 and θ2 are the response time of the feedback loop; Step 2: Calculate the autocorrelation of the simulation data with different parameters to obtain the time delay of a feedback loop cycle; the autocorrelation function is defined as follows: Where <·> represents the average operation; According to formula (1), the input and output pairs of the neural network need to satisfy the one-to-one correspondence between the previous cycle chaotic carrier and the next cycle chaotic carrier of the feedback loop of the optoelectronic oscillator, so it is assumed that the time delay is known; Here, a large number of input-output training data sets are constructed for the simulation data, that is, f: x i →x i +n, n is the number of symbols corresponding to the feedback delay τ. Subsequently, each input-output data set is cut using a sliding window of length L to fit the input dimension of the Fourier neural operator; Step 3: Iterative update v of the Fourier neural operator t →v t+1 , t=1,…,t-1 is represented by the Fourier neural operator It consists of a linear transformation R, a weight matrix W, and a nonlinear activation function σ, and its iteration rules are as follows: v0(x)=Pa(x)+p(3) u(x)=Qv T (x)+q(5) Where p and q are bias values, which are first lifted to a high-dimensional representation by a local transformation P parameterized by a shallow fully connected neural network; the lifting operation is considered as the first layer of the global neural network, and then v0(x) = P(a(x)) is obtained as The initial assumption is: then, after iterative updates, the output of the Fourier neural operator u(x) = Q(v T (x)), which is obtained by transforming Q to v T The transformation Q is parameterized by a shallow fully connected neural network and is used to project back to the target dimension; Step 4: Use python-pytorch to build the Fourier neural operator model; stack four Fourier integral operator layers, ReLU activation function and batch normalization layer to construct the Fourier neural operator; then, simulate the data of each delay and gain combination according to f:x i →x i +n to form a training set, of which 90% is used for training and 10% is used for testing; the Pearson correlation coefficient PCC is introduced to evaluate the consistency of the original sequence and the predicted sequence; the definition of PCC is as follows: Where x and y are the original sequence and predicted sequence respectively.

2. The method for modeling a chaotic optoelectronic oscillator based on deep learning according to claim 1, characterized in that: Formula (1) is solved by MATLAB using the fourth-order Runge-Kutta algorithm, with an iteration step of 2.5 ps and a sampling rate of 400 GSa / s; the simulation output sequence x(t) is downsampled to a 100 GSa / s sampling sequence y(t); the corresponding parameters are set as θ1=15.9 ps, θ2=5.3 μs, φ0=π / 4, where τ and β are used as adjustable parameters.

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