Ultrashort pulse propagation in optical fibers is known to contain a series of complex nonlinear dynamics, and artificial intelligence (AI) has been applied in various ways to analyze and optimize the spectral or temporal field distributions of the nonlinear evolution and final output^[1]. For example, evolutionary algorithms for feedback controls have been applied to optimize specific supercontinuum generation (SCG) in single-mode or multimode fibers^[2,3], as well as to classify mode-locked states and adjust mode-locked fiber lasers to specific regimes^[4,5]. In addition, nonlinear pulse evolution in optical fibers is generally described by the generalized nonlinear Schrödinger equation (GNLSE)^[6]. However, GNLSE-based simulations have great difficulty in performing time-efficient optimization through traditional numerical methods because of their high nonlinearity, complexity, and sensitivity to the governing parameters. Recently, machine learning has drawn much interest from the academic community and has also shown prominent potential to promote technological development and break through the bottleneck of nonlinear dynamics modeling for its robust nonlinear tolerance and strong computation ability^[7–10].

As an efficient prediction tool for time-series data, the recurrent neural network (RNN) has been applied to predict nonlinear dynamics in optical fiber with great performance in accuracy and generalization ability for different conditions^[11–14]. To improve the time efficiency, feed-forward neural networks (FNNs) have been used to predict the SCG with intensity and phase information in both temporal and spectral domains^[15]. Compared with RNN, the FNN model has a faster speed and simpler structure but lower accuracy. For both of the above AI models, there is a trade-off between computational speed and accuracy. However, fast and accurate full-field prediction of nonlinear dynamics could be achieved by a convolutional neural network (CNN) with feature separation modeling (FSM)^[16], where the linear features and nonlinear features are modeled by NLSE-derived numerical simulations and CNN, respectively. Very recently, by combining physical knowledge with loss function, an improved neural network (NN) termed physics-informed neural network (PINN) has been introduced to solve forward and inverse prediction problems involving nonlinear partial differential equations^[17]. In the issues of modeling GNLSE-based fiber systems, the generalization ability of PINN is extremely limited in spite of its outstanding accuracy for the model has to be retrained once the input pulse has changed^[18,19].

The mentioned studies focus on modeling complex nonlinear dynamics from a given input pulse. The trained NN with excellent generalization ability makes it possible to optimize and improve the performance of nonlinear fiber laser systems^[20]. Remarkably, modeling the forward pulse propagation by NNs is not the only solution for the potential inverse design and optimization applications. The inverse propagation prediction that evaluates the intensity distributions of initial pulses can also be used to solve design and optimization problems^[21]. Specifically, according to the initial pulse predicted by the AI model, the desired system output can be obtained through pulse or spectral shaping (including phase distributions). However, the exploration for inverse predicting ultrashort pulse nonlinear dynamics in optical fibers is far from sufficient.

In this paper, we aim to solve the inverse prediction problem of pulse nonlinear dynamics based on an FNN model with a simple structure. After being trained by full-field SCG data set, the AI model can predict the intensity and phase of initial pulses. We also provide a new perspective to analyze the prediction accuracy through the reconstruction testing, and further explore the influence of specific characteristics of predicted pulses on the reconstruction results. For the SCG with chirped input pulses, spectral and temporal features are also well restored. Our results are of practical importance for the parameter optimization of ultrafast laser systems.

The SCG field evolutionary process is governed by the GNLSE, as described in the following equation:$$\begin{gathered} \frac{\partial A(z,t)}{\partial z}+\frac{\alpha }{2}A(z,t)-\sum _{k⩾1}{j}^{k+1}\frac{{\beta }_k}{k!}\frac{{\partial }^{k}A(z,t)}{\partial t^k} \\ =j\gamma A(z,t){|A(z,t)|}^2 \\ -\frac{\gamma }{{\omega }_0}\frac{\partial }{\partial t}[A(z,t){\int }_{-\infty }^{t}R(t^{\prime }){|A(z,t-t^{\prime })|}^2\mathrm{d}{t}^{\prime }], \end{gathered}$$(1)where $A(z,t)$ is the pulse envelope, $z$ is the distance variable, $t$ is the time in the reference frame, $j$ is the imaginary unity, ($k=2,\dots ,5$) is the $k$-order dispersion, $\alpha$ is the attenuation, $\gamma$ is the nonlinear coefficient, and ${\omega }_0$ is the central angle frequency. $R(t)$ is the nonlinear response function related to the stimulated Raman scattering (SRS) and self-phase modulation (SPM), which are of great importance to the SCG. The initial pulse complex amplitude can be described by $$A(0,t)=\sqrt{P_0}\mathrm{sech}\left(\frac{t}{t_0}\right)\exp [-jC{\left(\frac{t}{t_0}\right)}^2],$$(2)where $P_0$ is the peak power, $t_0$ is the pulse duration, and $C$ is the chirp value of the injected hyperbolic secant pulse. The split-step Fourier method (SSFM) is a numerical method to solve the GNLSE by splitting the optical fiber into numerous steps and calculating the linear and nonlinear effects, respectively.

The transform-limited (TL) samples are generated from unchirped initial pulse ($C=0$) with 0.77–1.43 kW peak power, 70–130 fs pulse duration and central wavelength at 830 nm. There are 1500 TL simulations with 1400 for training and 100 for testing. The chirped samples are generated from a chirped initial pulse with −2-2 (without zero) chirp value, 0.88–1.43 kW peak power, 70–130 fs pulse duration, and central wavelength at 830 nm. There are 2000 chirped simulations, with 1900 for training and 100 for testing. The width of temporal window is 2 ps with 1024 grid points, and the length of the nonlinear fiber is 0.2 m with a step length of SSFM at 0.002 cm to ensure the accuracy of the numerical simulation. We then downsample the propagation developments along the fiber length from 10,000 steps to 200 steps and the step length after downsampling is 0.1 cm, 50 times larger than that used in the numerical simulation for data generation. The fiber dispersion parameters used in the numerical simulation are ${\beta }_2=-5.90\times {10}^{-27}{\mathrm{s}}^2{\mathrm{m}}^{-1}$, ${\beta }_3=4.21\times {10}^{-41}{\mathrm{s}}^3{\mathrm{m}}^{-1}$, ${\beta }_4=-1.25\times {10}^{-55}{\mathrm{s}}^4{\mathrm{m}}^{-1}$, and ${\beta }_5=2.45\times {10}^{-70}{\mathrm{s}}^5{\mathrm{m}}^{-1}$. The fiber nonlinear coefficient is $\gamma =0.1{\mathrm{W}}^{-1}{\mathrm{m}}^{-1}$.

The FNN model contains an input layer, several hidden layers, and an output layer, as shown in Fig. 1(a). We preprocess the full-field evolution map before feeding it to the AI model. The temporal complex amplitude of a particular evolution (indicated by subscript $n$) at a distance $z_i$ is characterized by vector $[I_n(z_i,T),{\mathrm{\Phi }}_n(z_i,T)]$, where $I_n$ and ${\mathrm{\Phi }}_n$ represent intensity and phase, respectively. Both intensity and phase profiles extracted from the complex amplitude vector consist of 1024 grid points and thereby form a 2048-dimension intensity-phase vector. For the training procedure, the input intensity-phase vectors are extracted from the simulated evolution in inverse sequence. Once trained, only the last intensity-phase vector from a simulated SCG evolution is input to the FNN model for frame-by-frame prediction. The procedure that the predicted output vector at distance $z_i$ is fed to the trained model as input for the prediction at distance $z_i-\mathrm{\Delta }z$ and is repeated until the entire evolution is predicted.

Figure 1(b) demonstrates more details about the data processing in the FNN-based model. Generally, for forward propagation issues, a quantitative comparison between the AI-predicted results and that simulated with the GNLSE can be performed using the average normalized root mean squared error (NRMSE)^[14]. However, the average error calculation might mask huge deviations at some specific positions, which is intolerable for nonlinear pulse evolution with strong position correlation. For the inverse prediction of pulse propagation, reconstruction testing is proposed to optimize the local error evaluation of the AI model, in which the initial pulses predicted by the AI model are recovered to a complex amplitude vector as the input of the GNLSE-based numerical model to reconstruct SCG evolution. Then, we record the reconstructed NRMSE, which has been commonly used as the core performance evaluation parameter in previous studies^[11,15], indicated in Eq. (3), $$\mathrm{NRMSE}=\frac{\sqrt{{\sum }_{d,i}{(x_{n,d,i}-{\tilde{x}}_{n,d,i})}^2}}{\sqrt{{\sum }_{d,i}{(x_{n,d,i})}^2}},$$(3)where $x_n$ denotes the raw data generated by numerical simulations, ${\tilde{x}}_n$ denotes the FNN predictions or numerical results reconstructed from predicted initial pulses, subscript $n$ indicates a particular realization, $d$ and $i$ represent the coordinates of data points over intensity-phase vectors and propagation steps, respectively. The error between the reconstructed evolution maps and the raw data generated by the numerical simulation provides a new insight into evaluating the performance of the AI model.

The FNN model consists of five dense layers, including an input layer, three hidden layers of 2000 nodes with ReLU activation, and an output layer of 2048 nodes with tanh activation. The FNN is trained for 50 epochs with a root mean squared (RMS) prop optimizer. The learning rate is initialized and decreases with the epochs. The FNN-predicted temporal evolution for input peak power of 1 kW and pulse duration of 125.9 fs is shown in Fig. 2. It should be emphasized that the arrow points towards the FNN-predicted direction. The visual agreement between the temporal intensity evolution of GNLSE simulation and FNN inverse prediction indicates the excellent ability of the FNN model to grasp the regular pattern of nonlinear effects in optical fibers, even with a chaotic input. For detailed comparison, the intensity and phase profiles at specific distances are also shown in Fig. 2(c). The NRMSE of the predicted SCG evolution is 0.06, while the average value computed over the whole testing set is 0.15. The minimum and maximum of NRMSE among the testing samples are 0.04 and 0.69, respectively.

To further illustrate the prediction performance of the FNN model, the predicted initial pulses are used as the input of the GNLSE simulation, and the reconstruction evolutions are compared with the raw data simulated by GNLSE (10,000 steps), as shown in Fig. 3. It is worth noting that only the temporal full-field initial signal predicted by FNN model is used for the reconstruction testing. The reconstructed spectral and temporal evolutions accurately reproduce the TL pulse nonlinear propagation in optical fibers. The intensity and phase profiles at $z=0\mathrm{m}$ and $z=0.2\mathrm{m}$ are, respectively, depicted in the right panel of Figs. 3(a) and 3(b). The corresponding phase profiles are not given because of the TL initial pulse. It turns out that the full-field SCG evolution is well modeled by one FNN model. Note that the phase deviation at the profile edge between GNLSE and reconstruction is meaningless for both the temporal and spectral domains because of the small intensity distribution. It is worth noting that the FNN model prediction will induce intensity noise at the near-zero zone in both temporal and spectral domains. The irregular noise is invisible in the temporal domain because the noise is minor numerically on linear scales. But the spectrum is shown on logarithmic (dBm) scales and the minor noise is more obvious. The NRMSE of reconstruction testing is 0.69, which is $\sim 5$ times larger than that of FNN prediction, indicating that the errors further accumulate in the numerical simulation with 10,000 steps. The FNN model accurately predicts both the intensity and phase profiles, demonstrating that AI has the ability to predict nonlinear dynamics and the potential to solve the inverse prediction problem of fiber laser systems.

The inverse prediction of nonlinear pulse propagation enables GNLSE-based numerical simulations to be treated as a “black box” similar to experimental systems. That is, the predicted effects can be directly evaluated through the reconstruction process, which paves the way for the inverse design of ultrashort pulse propagation in optical fibers. However, due to the accumulation of errors in the prediction process, there is an inevitable deviation between the predicted initial pulse and the data label. Therefore, it is necessary to figure out whether the deviation will be over-amplified during the reconstruction process, which is related to whether the AI prediction has guiding significance for actual systems. We calculated the similarity of normalized intensity profile (SNIP) and peak power difference (PPD) between all 100 testing samples and the corresponding label data to analyze their influence on the reconstruction results, as shown in Fig. 4. The SNIP is essentially the maximum of the cross-correlation function between the predicted and simulated initial pulse intensity signals. The intensity profiles are normalized in order to exclude the influence of peak power deviation. The SNIP for a particular realization is calculated by $$\mathrm{SNIP}=\max \{\frac{I(t)}{\max [I(t)]}*\frac{I^{\prime }(-t)}{\max [I^{\prime }(-t)]}\},$$(4)where $I(t)$ is the simulated initial pulse intensity and $I^{\prime }(t)$ is the predicted initial pulse intensity. The reconstructed NRMSEs of 100 testing samples are sorted in ascending order, as shown in Fig. 4(b). To quantify the correlation with the reconstructed NRMSE, Spearman correlation coefficients ($r_{\rho }$) are calculated to characterize the intensity of the monotonic relationship^[22]. As shown in Fig. 4(c), with the increase of reconstructed NRMSE, the predicted NRMSE exhibits the noise distribution, attributed to the random error accumulation from GNLSE-based 10,000 step simulations. The polynomial fitting (solid line) can effectively filter the error accumulation in the reconstruction process and has a clear upward trend ($r_{\rho }=0.94$), indicating a highly positive correlation. Note that when the correlation coefficient is 1 or $-1$, it indicates a strict monotonic increase or decrease. The scatter diagrams shown in Figs. 4(d) and 4(e) intuitively show the influence of SNIP and PPD on the reconstruction results. The $r_{\rho }$ of SNIP is $-0.88$, indicating an obvious negative correlation between SNIP and reconstruction results. However, there is no obvious correlation between PPD and reconstructed NRMSEs. The low $r_{\rho }$ of $-0.40$ is attributed to the disordered distribution of reconstructed NRMSEs with large power differences. Even subject to the limitation of sample size, the SNIP exhibits a stronger influence on the reconstruction results. Our results show that AI models could be optimized by combining the SNIP of the last predicted frame with the loss function for backpropagation during the training procedure, albeit at the expense of increasing the training time of the NN.

To figure out the applicability of our method over chirped stimulated data, another FNN model is trained with the propagation samples that are stimulated by chirped pulses. Compared with the model for TL simulations, this FNN model for chirped simulations consists of four hidden layers. The training set is trained for 80 epochs with the RMSprop optimizer and decay learning rate. The average predicted NRMSE over the chirped testing samples is 0.43, which is $\sim 2.9$ times that of the TL testing samples (0.15). As the parameter freedom of the initial pulse increases, it seems inevitable that the prediction performance of AI will worsen^[11,15]. The spectral and temporal comparisons between the raw data of a particular sample stimulated by a chirped initial pulse and reconstruction of FNN prediction are shown in Fig. 5. Adding a chirp to the initial hyperbolic secant pulse drives the transition of the preprocessed initial phase from an all-zero vector to an ordered real vector, which increases the difficulty of FNN model in predicting full-field nonlinear dynamics. The intensity and phase profiles at $z=0.2\mathrm{m}$ and $z=0\mathrm{m}$ are shown in the right panel of Fig. 5. The reconstructed NRMSE for this particular sample is 0.89, which is $\sim 1.5$ times the predicted NRMSE of 0.59. Although the predicted ability of AI for the chirped initial pulse scenario declined somewhat, the agreement between the reconstruction and raw data in both temporal and spectral domains manifests that our method has a certain generalization ability over chirped simulated data. In addition, the performance over a high-nonlinear situation could be improved with a more complex structure of the AI model, designed loss function, suitable learning rate decay strategy, and even additional physical equation operators^[13,16,18].

In conclusion, we propose an FNN model to inversely predict the full-field (intensity and phase) SCG evolution. It is proven that simple AI model can realize precise prediction of ultrashort pulse nonlinear propagation in optical fibers from greatly different input layer vectors with significant advantages in speed and accuracy. Benefiting from the inverse prediction, we perform reconstruction testing to verify the accuracy of predicted initial pulses. Although the NRMSE is further accumulated during the reconstruction process, the results are highly consistent with the label data. In order to further explore the influence of predicted initial pulses on the reconstruction results, we extracted the SNIP and PPD calculated by the predicted pulses and corresponding label data. The SNIP exhibits a strong correlation with the reconstructed NRMSE while the reconstructed results are not sensitive to PPD in limited testing samples. In addition, to demonstrate the generality of the proposed AI model, the reconstruction testing has also been applied to the SCG data set with chirped input pulses. We expect our work focused on inverse prediction problem would bring novel insight into optimization of ultrafast laser systems and control of nonlinear dynamics in a practical perspective.