[Paper Review] Physics-informed Neural Network for Nonlinear Dynamics in Fiber Optics

Xiaotian Jiang, Danshi Wang+4 more|arXiv (Cornell University)|Sep 1, 2021

Advanced Fiber Laser Technologies4 citations

TL;DR

This paper proposes a physics-informed neural network (PINN) to solve the nonlinear Schrödinger equation (NLSE) for modeling nonlinear dynamics in optical fibers, integrating physical laws like dispersion and self-phase modulation directly into the network. The PINN achieves high accuracy with minimal data and 14.6% of the computational complexity of the split-step Fourier method (SSFM), demonstrating strong generalization across soliton and multi-pulse scenarios by embedding physical parameters as input controllers.

ABSTRACT

A physics-informed neural network (PINN) that combines deep learning with physics is studied to solve the nonlinear Schrödinger equation for learning nonlinear dynamics in fiber optics. We carry out a systematic investigation and comprehensive verification on PINN for multiple physical effects in optical fibers, including dispersion, self-phase modulation, and higher-order nonlinear effects. Moreover, both special case (soliton propagation) and general case (multi-pulse propagation) are investigated and realized with PINN. In the previous studies, the PINN was mainly effective for single scenario. To overcome this problem, the physical parameters (pulse peak power and amplitudes of sub-pulses) are hereby embedded as additional input parameter controllers, which allow PINN to learn the physical constraints of different scenarios and perform good generalizability. Furthermore, PINN exhibits better performance than the data-driven neural network using much less data, and its computational complexity (in terms of number of multiplications) is much lower than that of the split-step Fourier method. The results report here show that the PINN is not only an effective partial differential equation solver, but also a prospective technique to advance the scientific computing and automatic modeling in fiber optics.

Motivation & Objective

• To develop a data-efficient method for solving the nonlinear Schrödinger equation (NLSE) in fiber optics, where analytical solutions are unattainable.
• To overcome the limitations of data-driven neural networks by embedding physical laws and constraints into the learning process.
• To enable generalization across diverse physical scenarios—such as soliton propagation and multi-pulse dynamics—using a single PINN architecture.
• To reduce computational complexity compared to conventional numerical methods like the split-step Fourier method (SSFM).
• To establish PINN as a viable alternative for scientific computing and automatic modeling in fiber optics.

Proposed method

• A physics-informed neural network (PINN) is trained to solve the NLSE by enforcing the governing partial differential equation as a loss function component.
• The network architecture uses fully connected layers with automatic differentiation to compute gradients of the NLSE residual at collocation points in space and time.
• Physical parameters such as initial pulse peak power and sub-pulse amplitudes are embedded as additional input controllers to enable scenario-specific learning and generalization.
• The loss function includes constraints on initial conditions, boundary conditions, and the NLSE residual, ensuring physical consistency.
• Computational complexity is analyzed by comparing the number of multiplications in PINN and SSFM, with PINN showing linear scaling with coordinate points and lower overall operations.
• The method is validated across multiple physical effects: dispersion, self-phase modulation (SPM), and higher-order nonlinearities.

Experimental results

Research questions

• RQ1 Can a PINN effectively model nonlinear pulse propagation in optical fibers with minimal training data?
• RQ2 How does the inclusion of physical parameter controllers (e.g., pulse peak power) affect the generalization of PINN across different physical scenarios?
• RQ3 What is the computational complexity of PINN relative to the split-step Fourier method (SSFM) for the same simulation accuracy?
• RQ4 How does PINN performance compare to data-driven neural networks (DDNN) when trained on significantly less data?
• RQ5 Can PINN accurately simulate complex dynamics such as soliton formation and multi-pulse interactions in fiber optics?

Key findings

• The PINN achieves higher accuracy than a data-driven neural network (DDNN) even when trained on only one-tenth the data, demonstrating superior data efficiency.
• For a 1000 km fiber span with a 512-point FFT, the PINN's computational complexity is approximately 14.6% of that of the SSFM, significantly reducing required multiplications.
• The PINN model generalizes robustly across different physical scenarios, including soliton propagation and multi-pulse dynamics, by using physical parameters as input controllers.
• The number of multiplications in PINN scales linearly with the number of collocation points, unlike SSFM, which scales quadratically with FFT size due to nonlinear operations.
• The PINN maintains high accuracy even with sparse data, outperforming DDNNs that require orders of magnitude more training data for comparable results.
• The method effectively captures higher-order nonlinear effects, dispersion, and self-phase modulation within a single unified framework.