Stimulated Raman scattering (SRS) is a nonlinear optical phenomenon where energy is transferred between light waves due to interactions with molecular vibrations. In fiber-optic systems, SRS plays a dual role: it can be detrimental, causing interference and impairing performance in high-bandwidth systems, or beneficial, forming the basis for Raman amplifiers and lasers. Solving for the behavior of SRS in these systems involves complex mathematical models, often represented by coupled partial differential equations (PDEs). Traditional numerical methods, such as split-step algorithms and shooting methods, are often customized to specific scenarios (unidirectional vs. bidirectional transmission) and are computationally expensive, especially when dealing with combined forward and inverse problems. For instance, in Raman amplifiers, optimizing pump power to achieve a target gain profile requires iterative solutions of forward power evolution, increasing computation time and complexity. Similarly, accurate modeling in wideband multi-channel transmission systems necessitates simultaneous forward prediction and inverse parameter identification. The need for a more efficient and universal solution framework for these diverse scenarios, particularly combined problems, is a significant challenge in the field of nonlinear fiber optics. Physics-informed neural networks (PINNs) offer a promising alternative, leveraging the power of deep learning while integrating physical constraints, thereby reducing the need for large training datasets and improving the interpretability of the results.

Various numerical methods have been developed to tackle SRS problems, but these are often customized for specific scenarios and lack efficiency in handling combined forward and inverse problems. For forward problems, classical methods like split-step iterations are limited to unidirectional transmission, requiring more complex approaches like shooting algorithms for bidirectional cases, which increases computational cost and can lead to inaccuracies. Inverse problems, such as identifying fiber parameters or optimizing pump powers, are usually ill-posed, necessitating iterative search algorithms and increasing computational burden. Deep learning has emerged as a powerful tool for solving both forward and inverse problems, but data-driven neural networks (NNs) require vast datasets, and lack interpretability. They can also suffer from generalization problems when faced with scenarios outside the scope of the training data. While some studies have utilized data-driven NNs for SRS, these approaches often rely on extensive datasets or lack interpretability. To address these challenges, some researchers employed numerical integration of the SRS equation within autoencoder structures or used NN-based nonlinear interpolation of Raman gain coefficients, but these still have limitations. The emergence of PINNs offers a more promising approach by embedding physical laws into the loss function, guiding the training of neural networks to accurately solve PDEs. This approach has demonstrated effectiveness in solving various PDEs in diverse fields, paving the way for its application in the context of SRS.

SRS-Net is a physics-informed deep learning framework designed to solve SRS problems in fiber-optic systems. It leverages the automatic differentiation (AD) capabilities of neural networks to efficiently solve the coupled partial differential equations governing SRS. The core of SRS-Net lies in its loss function, which is composed of two parts: a data term (Le) and a physical-law regularization term (Lr). The data term enforces the constraints from known or measured data, such as input/output power profiles or initial/boundary conditions. The physical-law regularization term incorporates the SRS PDE, ensuring that the neural network's solution adheres to the underlying physics. This is achieved by using AD to calculate the derivatives of the neural network's output (the complex-valued signal field) with respect to the input variables (distance and time). This allows the direct incorporation of the differential terms from the SRS PDE into the loss function. The loss function is then minimized using an optimization algorithm (such as Adam), updating the neural network's parameters and, in the case of inverse problems, the unknown physical parameters or pump powers. The SRS PDE considered includes terms for attenuation, group velocity dispersion (GVD), Kerr nonlinearity, and SRS. The Raman gain coefficient, a crucial parameter in the SRS PDE, can be either treated as a known parameter or as an unknown parameter to be identified. SRS-Net can handle various SRS problems: forward problems (predicting power evolution given input conditions and parameters), inverse problems (identifying fiber parameters or optimizing pump powers given input and output data), and combined problems (simultaneously solving forward and inverse problems). For continuous-wave (CW) scenarios, the SRS PDE simplifies to ordinary differential equations (ODEs), and SRS-Net can efficiently solve these. For waveform propagation (non-CW), the full PDE is employed, allowing for accurate modeling of pulse shapes and evolution.

SRS-Net's performance was evaluated through extensive simulations and experiments. Simulations demonstrated high accuracy in various scenarios: * **Continuous-wave (CW) energy transfer:** SRS-Net accurately predicted power evolution between signal and pump waves with different power levels, showing excellent agreement with numerical split-step solutions (RMSE < 3 × 10⁻³). * **Time-domain waveform propagation:** SRS-Net successfully modeled the propagation and energy transfer of optical pulses, capturing the effects of group velocity dispersion and walk-off (RMSE < 6 × 10⁻⁴). The results accurately predicted the evolution of pulse shapes. * **Frequency-domain multi-channel power evolution:** In a 96-channel WDM system spanning the C+L-bands (approx. 10 THz), SRS-Net accurately predicted power evolution under full and partial loading conditions (RMSE < 2 × 10⁻²). * **Bidirectional power evolution prediction:** SRS-Net successfully addressed the challenge of bidirectional power evolution in Raman amplifiers under different pump configurations (co-propagating, counter-propagating, and bi-directional), where classical methods often fail to converge. An adaptive weight training strategy effectively handled the challenges associated with bidirectional transmission. * **Fiber parameter identification:** SRS-Net effectively identified frequency-dependent attenuation and Raman gain spectra from measured input and output power spectra, outperforming the differential evolution (DE) algorithm in terms of speed. * **Pump power optimization:** SRS-Net efficiently optimized pump powers to achieve a target flat output power spectrum, demonstrating a significant improvement in flatness compared to a genetic algorithm (GA) and improving ripple levels by over 4 dB. Experimental validation in a C+L-band WDM system with 96 channels further confirmed SRS-Net's capabilities. SRS-Net accurately predicted power evolution and GSNR (Generalized Signal-to-Noise Ratio) with maximum deviations of less than 0.3 dB and 0.8 dB, respectively. The speed of SRS-Net was significantly faster than conventional methods (two orders of magnitude faster for forward problems), highlighting its efficiency.

SRS-Net addresses the limitations of existing methods for solving SRS problems by offering a universal framework that can handle forward, inverse, and combined problems efficiently and accurately. The integration of physical laws through the physics-informed neural network approach improves generalization capabilities, reduces the dependence on large training datasets, and enhances the interpretability of the results. The significant speed advantage of SRS-Net over classical numerical methods, particularly evident in multi-channel wideband systems, has important implications for the design and optimization of fiber-optic systems. Its success in both simulations and experimental validation, especially in the complex C+L-band WDM system, highlights the potential of physics-informed deep learning for addressing challenging nonlinear problems in various domains.

SRS-Net offers a significant advancement in solving SRS problems in fiber-optic systems. Its universal framework, high accuracy, and computational efficiency demonstrate the potential of physics-informed deep learning for complex nonlinear problems. Future work could explore incorporating more sophisticated models of nonlinear effects, extending the framework to other types of optical fibers and system configurations, and investigating applications in other areas of physics and engineering governed by PDEs.

While SRS-Net demonstrates significant improvements over existing methods, there are some limitations to consider. The accuracy of SRS-Net relies on the accuracy of the underlying SRS PDE and the chosen parameters. In the experiments, efforts were made to minimize the impact of external factors such as laser power fluctuations, but these might still affect results in practical scenarios. Additionally, the training of the neural network requires careful selection of hyperparameters and can be computationally intensive.