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Physics-informed Neural Network for Nonlinear Dynamics in Fiber Optics

Xiaotian Jiang, Danshi Wang, Qirui Fan, Min Zhang, Chao Lu, Alan Pak Tao Lau

arXiv:2109.00526v1physics.opticscs.LG

TL;DR

Solving fiber-optic NLSE dynamics remains challenging because analytical solutions are unavailable in relevant cases and numerical methods can be costly. The paper applies a PINN with physical parameters as inputs across multiple effects and scenarios. It reports improved generalizability, higher accuracy than data-driven networks with much less data, and fewer multiplications than SSFM.

  • Problem

    The NLSE cannot be solved analytically when dispersion and nonlinearity coexist, while numerical approaches can become costly and PINN models have limited generalizability across changing physical constraints.

    Source: §1 · pp. 3–4

  • Method

    The PINN embeds pulse peak power and sub-pulse amplitudes as input controllers and incorporates governing equations and physical constraints into its loss.

    Source: §1 · p. 4, §2.1 · p. 6

  • Results

    PINN models multiple scenarios with robust generalizability, higher accuracy than DDNN using dozens of times the exact data, and usually two orders of magnitude fewer multiplications than SSFM.

    Source: §1 · p. 4, §5 · p. 26

  • Takeaways & Limitations

    The authors present PINN as a supplemental tool for advancing theoretical understanding of nonlinear dynamics in fiber optics.

    Source: §1 · p. 4

    For the soliton-like pulse under the self-steepening effect, error is relatively large near the peak, where approximating higher-order differential terms is difficult.

    Source: §3.1.3 · p. 13

Abstract

from arXiv · show

A physics-informed neural network (PINN) that combines deep learning with physics is studied to solve the nonlinear Schr\"odinger 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.

1 Introduction

Solving nonlinear dynamics in fiber optics requires methods for the NLSE, whose analytical solutions are unavailable in relevant cases and whose numerical solutions can be costly. This study applies a physics-informed neural network across physical effects and scenarios, using input physical parameters to improve generalizability.

1 Introduction

  • The NLSE governs optical-pulse propagation, but cannot be solved analytically when dispersion and nonlinearity coexist; SSFM can become costly at high nonlinearity or temporal resolution.For long-distance, high-nonlinearity scenarios, SSFM step lengths must be reduced to meet accuracy requirements.Source: §1 · pp. 2–3
  • Data-driven networks have modeled fiber-optic nonlinearities, but their performance depends heavily on data quantity and quality and they do not account for underlying physics.Conventional numerical methods can also struggle when initial or boundary conditions are missing and can be prohibitively expensive for inverse problems.Source: §1 · pp. 2–3
  • PINN incorporates governing equations and physical constraints into its loss, using automatic differentiation to calculate PDE derivatives without exact data at auxiliary coordinates.This converts PDE solving into loss-function optimization and reduces data dependence.Source: §1 · p. 3
  • PINN models multiple fiber-optic effects and scenarios, outperforming data-driven networks with much less data and usually requiring two orders of magnitude fewer multiplications than SSFM.The studied effects include chromatic dispersion, self-phase modulation, and higher-order nonlinear effects; the scenarios include soliton and multi-pulse propagation.Source: §1 · p. 4
  • Embedding pulse peak power and sub-pulse amplitudes as input controllers lets PINN learn differing physical constraints and model multiple scenarios without remodeling.The earlier PINN models described here had poor generalizability when physical constraints changed.Source: §1 · p. 4

2 Concept and Principles

PINN uses governing equations and initial and boundary conditions to train a neural approximation of PDE solutions. For fiber optics, the method applies this framework to the NLSE by splitting its complex form into real and imaginary components and enforcing their constraints during training.

2 Concept and Principles

  • PINN incorporates mathematical and physical prior knowledge while accounting for accuracy and computational complexity in modeling fiber-optic pulse distortions.The passage contrasts this approach with numerical methods that require more iterations and smaller steps, and data-driven methods that rely on large simulation or experimental datasets.Source: §2 · p. 4

2.1 Overview of Physics-Informed Neural Network

  • PINN maps temporal-spatial coordinates to a predicted solution and minimizes mean-square losses for governing-equation residuals and initial and boundary conditions.Automatic differentiation calculates derivatives and nonlinear terms used to evaluate the governing equation.Source: §2.1 · p. 6

2.2 PINN-based Modeling for Nonlinear Dynamics in Fiber Optics

  • The fiber-optics NLSE describes GVD, TOD, and self-phase modulation, with self-steepening and intrapulse Raman scattering included for ultrashort pulses.The passage specifies these higher-order effects for pulses with T0 < 1 ps.Source: §2.2 · pp. 6–7
  • The complex NLSE is separated into real and imaginary parts, with separate losses enforcing the initial pulse and governing-equation constraints.The initial-pulse losses constrain real and imaginary components; the other losses penalize violations of the governing equations.Source: §2.2 · pp. 7–8
  • Training samples known initial-pulse data and unknown auxiliary coordinates across the temporal-spatial region, using SSFM-generated evolution data as a reference.The PINN uses sampled coordinates to calculate the four loss terms during iterative training.Source: Figure 2 · p. 8, §2.2 · p. 9

3 Demonstrations and Results

The described implementation sets the computational region and sampling scheme, then specifies the network architecture and optimizer for learning.

3 Demonstrations and Results

  • The model uses 50,000 Latin-hypercube auxiliary coordinates and 256 uniformly sampled initial-condition coordinates over a region of [−24T0, 24T0] × [0, 4LD].The passage gives typical single-mode-fiber values of β2 approximately -20 ps/km, T0 approximately 10–100 ps, and LD approximately 5–500 km.Source: §3 · p. 9
  • The PINN has four hidden layers of 100 neurons with tanh activation, and L-BFGS is used to accelerate learning.The passage identifies L-BFGS as a quasi-Newton optimization method.Source: §3 · p. 9
  • The reported setup spans four dispersion lengths and samples the initial pulse separately from auxiliary coordinates in the two-dimensional region.The auxiliary coordinates are used to calculate governing-equation loss terms, while the initial coordinates govern initial-condition losses.Source: §3 · p. 9

3.1 Multiple Physical Effects in Fiber Modeled by PINN

PINN is evaluated across dispersion, coexisting dispersion and SPM, and higher-order nonlinear effects. It captures the reported pulse dynamics, though errors rise to the order of 10^-2 for the higher-order cases.

3.1.1 Dispersion

  • For GVD, the pulse broadens in the normal-dispersion regime, with maximum absolute error 4.6 × 10−3; for TOD, trailing-edge oscillations damp and error stays below 5.8 × 10−3.Dispersion changes spectral-component phase, so GVD- and TOD-affected pulses differ in time but retain the initial spectrum in the frequency domain.Source: §3.1.1 · pp. 10–11

3.1.2 Dispersion and SPM Coexisting

  • With GVD and SPM together, pulse broadening accelerates versus GVD alone, while PINN effectively models evolution despite larger, diverging relative L2 error.SPM generates red- and blue-shifted components at the pulse edges; the red-shifted component travels faster in the normal-dispersion regime, and amplitude is significantly lower than in the GVD-only case.Source: §3.1.2 · pp. 11–12
  • With TOD and SPM, trailing-edge oscillations multiply, their minima become non-zero, and PINN still characterizes evolution despite a maximum error of 1.56 × 10−2.The spectrum has an asymmetric double-peak structure; TOD introduces spectral asymmetry without removing SPM’s two-peak structure, unlike GVD’s hindrance of splitting in the normal-dispersion regime.Source: §3.1.2 · p. 12

3.1.3 Higher-order Nonlinear Effects

  • For self-steepening, the pulse peak moves more slowly than its sides and shifts toward the trailing side under anomalous dispersion; errors concentrate near the peak.The authors extended propagation distance to 10LD and set s to 0.2 for visibility; the pulse shape gradually stabilizes, especially for an initial soliton-like pulse.Source: §3.1.3 · p. 13
  • For intrapulse Raman scattering, even small τR induces pulse fission; the pulse also shifts in time and toward longer wavelengths, then decelerates after narrowing along a curved trajectory.The authors set τR = 0.1 and N = 1.4; fission is more apparent for a transmitted soliton, and the curved trajectory is attributed to slower group velocity at longer wavelength.Source: §3.1.3 · p. 14
  • For the same model configuration, PINN effectively models higher-order nonlinear effects with minimal computational-complexity increase from unchanged network structure; their relative L2 errors reach the order of 10^-2.The errors can be reduced through further optimization, unlike numerical methods such as SSFM, which the passage contrasts with PINN.Source: §3.1.3 · p. 14

3.2 PINN-Modeling Universality

PINN is tested on soliton and multi-pulse propagation, extending its evaluation from single-Gaussian pulses to more complex waveforms. It characterizes periodic soliton evolution and multi-pulse signals, though errors are larger around high-power peaks and inter-pulse interference.

3.2.1 Soliton Propagation

  • The fundamental soliton waveform remained unchanged during evolution, with a maximum error of 3.9 × 10^-3 for the stated setup.The passage notes that this error was lower than in the GVD-only case and that pulse-evolution complexity also affects accuracy.Source: §3.2.1 · pp. 15–16, Figure 9 · p. 16
  • The second-order soliton showed periodic evolution with period πLD/2 and was well characterized, with errors concentrated at its two high-power peaks.Pulse-center amplitudes were slightly lower in the time- and frequency-domain results.Source: §3.2.1 · p. 16, Figure 10 · p. 17

3.2.2 Multi-pulse Propagation

  • The multi-pulse tests covered regular equal-amplitude and random non-equal-amplitude five-pulse waveforms viewed as common pulse-amplitude modulation signals.These more complex waveforms were studied because practical applications use different pulse types.Source: §3.2 · p. 15, §3.2.2 · p. 16
  • 10^-2 was the error level for the equal-amplitude five-pulse waveform, where broadening caused crosstalk and newborn peaks between adjacent pulses.Errors were especially large at the central peak, where broadening from all pulses superimposed.Source: §3.2.2 · p. 17, Figure 11 · p. 18
  • 2.16 × 10^-2 was the maximum time-domain error for the non-equal-amplitude five-pulse waveform, yet PINN characterized its richer evolution with the same configuration.The error was relatively large near lower-amplitude leading-edge peaks; additional coordinate points and exact data could further improve performance.Source: §3.2.2 · p. 17

3.3 The Generalizability of PINN Modeling

To address scenario-specific PINN constraints, the model incorporates physical parameters as inputs and trains across multiple powers and random waveforms. It generalizes to held-out pulse powers and waveforms, with errors reported below 1.8 × 10^-2 for the power test and at the order of 10^-2 for unseen random signals.

3.3.1 Modeling Generalizability for Multi-power Scenario in Optical Fiber

  • The input vector (z_i, t_i, p_i) uses peak power as a physical controller, with the loss summing constraints for multiple scenarios.The real and imaginary parts of the complex envelope are predicted for each auxiliary coordinate and peak power.Source: §3.3 · p. 18, Figure 13 · p. 19, §3.3.1 · p. 20
  • The relative L2 error stayed below 1.8 × 10^-2 across the tested pulse powers, and the five verification powers between trained values showed generalization.Training constraints used six of 11 Gaussian pulses spanning p = 0.5 to 1.5; errors increased with peak power, and the p = 1.5 case had slightly greater error than p = 1.4.Source: §3.3.1 · p. 20, Figure 14 · p. 21

3.3.2 Modeling Generalizability for Random Waveforms in Optical Fiber

  • 10^-2 was the reported error order for two unlearned random three-sub-pulse waveforms, whose time- and frequency-domain results were nearly consistent with SSFM.The waveform closer to the learned scenarios had relatively small error, while the more different waveform had relatively large error; training sampled 10 of 34 possible waveforms.Source: §3.3.2 · pp. 20–21, Figure 15 · p. 22

4 Analysis and Discussions

The analysis examines how auxiliary coordinates, physical constraints, training data, and model structure relate to PINN accuracy and computational complexity. It reports a lower multiplication count than SSFM under the stated comparison and stronger accuracy than DDNN despite using less exact data.

4.1 PINN-modeling Accuracy Analysis

  • For GVD and SPM, 5000 auxiliary coordinates were sufficient for good accuracy, and using pulse constraints at three distances gave the best performance.With only the initial-pulse constraint, the model error was one order of magnitude higher than in the other two constraint cases; stronger constraints can include exact data or prior knowledge such as symmetry.Source: §4.1 · pp. 22–23, Figure 16 · p. 23
  • PINN outperformed DDNN even when DDNN used a large exact-data volume, while DDNN required exact data from at least five pulses to model evolution well.The comparison used the same four-hidden-layer, 100-neuron network configuration, and PINN used governing equations alongside data.Source: §4.1 · p. 23, Figure 17 · p. 24

4.2 PINN-modeling Complexity Analysis

  • At N0 = 512 and 10LD, PINN required 3.11 × 10^9 multiplications versus 2.14 × 10^10 for SSFM, or approximately 14.6% of SSFM's complexity.The comparison used equal data volumes; the paper notes that PINN may require less data in practice and that its multiplication count is mainly affected by network structure.Source: §4.2 · p. 25, Figure 18 · p. 25
  • PINN complexity grows linearly with selected coordinates, while SSFM complexity grows with FFT size and linearly with propagation steps.The SSFM nonlinear calculations include O(n^2) operations, making FFT size a stronger influence on SSFM complexity than on PINN complexity.Source: §4.2 · pp. 24–25

5 Conclusion

The paper concludes that PINN can model fiber-optic nonlinear dynamics across multiple physical effects and propagation cases. Parameter controllers support multi-scenario generalization, while comparisons report higher accuracy than DDNN with less exact data and fewer multiplications than SSFM.

5 Conclusion

  • The study verifies PINN for dispersion, SPM, and higher-order nonlinear effects, including soliton and multi-pulse propagation.It reports strong characterization of pulse evolution under multiple physical effects.Source: §5 · p. 25
  • Embedding initial pulse peak power and random-waveform sub-pulse amplitudes as physical controllers supports robust generalization across scenarios.The controllers let the network learn the physical constraints corresponding to each scenario.Source: §5 · p. 26
  • PINN showed higher accuracy than DDNN while using dozens of times less exact data, and generally required fewer multiplications than SSFM.The comparison cited for PINN accuracy used only initial- and received-pulse constraints.Source: §5 · p. 26
  • The authors present PINN as an effective PDE solver and a prospective technique for scientific computing and automatic modeling in fiber optics.Source: §5 · p. 26
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