## 2 Materials and Methods  
### Data Generation via SSFM  
To verify and evaluate the performance of DeepONet in modeling the nonlinear dynamics of fiber propagation, we generate a series of high-quality simulation data. In this study, we utilize the Split-Step Fourier Method (SSFM) to generate these data [6]. SSFM is a widely used numerical method for solving the Nonlinear Schrodinger Equation (NLSE) by iteratively simulating the propagation of optical pulses in fibers through both the time and frequency domains [19]. The NLSE describes the evolution of the optical field within the cavity and can be written as follows:  
\[ \begin{split}\frac{\partial A}{\partial Z}+\frac{\alpha-g}{2}A+ \frac{i\beta_{2}}{2}\frac{\partial^{2}A}{\partial T^{2}}-\frac{\beta_{3}}{6} \frac{\partial^{3}A}{\partial T^{3}}-\frac{g}{2\Omega_{S}^{2}}\frac{\partial ^{2}A}{\partial T^{2}}=\\ i\gamma\bigg{(}|A|^{2}A+\frac{i}{\omega_{0}}\frac{\partial(|A|^{ 2}A)}{\partial T}\bigg{)}+i\gamma T_{R}\frac{\partial|A|^{2}}{\partial T}A \end{split}  \tag{1}\]  
where \(A\) denotes the pulse envelope, which describes the complex envelope of the optical pulse propagating in the fiber. Here, \(Z\) is the propagation distance, \(\alpha\) is the linear loss coefficient representing absorption or scattering loss, and \(g\) is the gain coefficient depicting the gain due to stimulated emission in a fiber amplifier. The term \(\beta_{2}\) is the second-order dispersion coefficient associated with group velocity dispersion (GVD) effects on pulse propagation, while \(\beta_{3}\) is the third-order dispersion coefficient affecting pulse asymmetry. The parameter \(\Omega_{S}\) represents the gain bandwidth, equivalent to the width of the gain spectrum. The nonlinear coefficient \(\gamma\) accounts for nonlinear effects such as self-phase modulation (SPM) and cross-phase modulation (XPM). \(\omega_{0}\) is the carrier frequency, representing the central frequency of the optical pulse. The term \(T_{R}\) is the Raman response time, typically around 3 fs for silica fibers. The left-hand side of the equation describes changes in the pulse envelope along the propagation distance, including linear loss and gain effects, second-order dispersion, third-order dispersion, and gain bandwidth effects. The right-hand side captures the effects of SPM, self-steepening which is represented by the term with \(\frac{i}{\omega_{0}}\), and Raman scattering which is represented by the last term with \(T_{R}\). Consider a traveling wave solution of the form \(A(Z,T)=A(\xi)\), where \(\zeta=T-vZ\) (with \(v\) being the group velocity). This transforms the partial differential equation (PDE) into an ordinary differential equation (ODE) with respect to \(\zeta\). Using the variable transformation  
\[\zeta=T-vZ, \tag{2}\]the derivatives are transformed, as follows:  
\[\frac{\partial A}{\partial Z}=-v\frac{dA}{d\zeta},\quad\frac{\partial A}{ \partial T}=\frac{dA}{d\zeta},\quad\frac{\partial^{2}A}{\partial T^{2}}=\frac{ d^{2}A}{d\zeta^{2}},\quad\frac{\partial^{3}A}{\partial T^{3}}=\frac{d^{3}A}{d \zeta^{3}} \tag{3}\]  
Substituting these expressions into the original equation, we obtain  
\[ \begin{split}-v\frac{dA}{d\zeta}+\frac{\alpha-g}{2}A+\frac{i \beta_{2}}{2}\frac{d^{2}A}{d\zeta^{2}}-\frac{\beta_{3}}{6}\frac{d^{3}A}{d\zeta ^{3}}-\frac{g}{2\Omega_{8}^{2}}\frac{d^{2}A}{d\zeta^{2}}=\\ i\gamma\bigg{(}|A|^{2}A+\frac{i\omega_{0}}{d\zeta}\frac{d(|A|^{2} A)}{d\zeta}\bigg{)}+i\gamma T_{R}\frac{d|A|^{2}}{d\zeta}A\end{split}  \tag{4}\]  
Thus, the equation is transformed from a PDE to an ODE with respect to the variable \(\zeta\). Our objective is to learn an operator \(\mathcal{G}\) that maps an input function \(A_{in}\) to its output values \(A_{out}\). After transforming the NLSE to its ODE form, the target operator can be simplified as follows:  
\[