Bidirectional Fourier-Enhanced Deep Operator Network for Spatio-Temporal Propagation in Multi-Mode Fibers

TL;DR

The paper tackles the computational bottleneck of simulating ultrashort-pulse propagation in graded-index multimode fibers by introducing a bidirectional Fourier-enhanced DeepONet that learns both forward and inverse spatio-temporal propagation operators. The model uses 2D and 1D spectral convolutions with Fourier feature embeddings to condition on physical parameters, yielding a fast, unified surrogate capable of predicting complex field evolution and recovering input fields from measurements. On a (3+1)D GNLSE-based MMF dataset, the approach achieves microsecond-scale forward/inverse predictions and substantial speedups (≈90×) over conventional solvers, with low forward prediction errors and robust inverse reconstructions. The work demonstrates that operator-learning can capture essential nonlinear physics in MMFs and outlines directions for extending to experimental data, noise robustness, and broader wave-propagation problems.

Abstract

Ultrashort-pulse propagation in graded-index multimode fibers is a highly nonlinear phenomenon driven by several physical processes. Although conventional numerical solvers can reproduce this behavior with high fidelity, their computational cost limits real-time prediction, rapid parameter exploration, experimental feedback, and especially inverse retrieval of input fields from measured outputs. In this work, we introduce an operator learning framework that learns both the forward and inverse propagation operators within a single unified architecture. By combining spectral filters for spatio-temporal representations with Fourier-embedded conditioning on physical parameters, the model functions as a fast surrogate capable of accurately predicting complex field evolution on previously unseen cases. To our knowledge, this represents one of the first demonstrations of a bidirectional operator-learning framework applied to ultrashort-pulse multimode fiber propagation. The resulting architecture enables orders-of-magnitude speedup over numerical solvers, paving the way for real-time beam diagnostics, data-driven design of complex input fields, and closed-loop spatio-temporal control. Moreover, the same framework can potentially be applied to a wide variety of wave systems exhibiting analogous nonlinear and dispersive effects in optics and beyond.

Figures (6)

• Figure 1: Schematic of the proposed bidirectional Fourier-enhanced DeepONet for joint forward and inverse modeling of nonlinear pulse propagation in graded-index multimode fibers. Two dedicated branch networks separately process the input transverse intensity distribution $I(x,y;0)$ (or $I(x,y;z)$) and temporal intensity trace $I(t;0)$ (or $I(t;z)$) using spectral convolutions in their respective domains. The resulting latent representations are concatenated (denoted by $\oplus$) and combined via element-wise multiplication (denoted by $\otimes$) with the output of a trunk network that encodes the physical parameters (peak power $P_0$, propagation distance $z$, and a binary direction flag $d \in \{0,1\}$ indicating forward or inverse operation) through Fourier feature embeddings. A shared decoder followed by two output projections generates the predicted transverse speckle pattern $I(x,y;z)$ and temporal trace $I(t;z)$ (forward mode) or recovers the initial state $I(x,y;0)$ and $I(t;0)$ (inverse mode). The depth, width, and number of retained Fourier modes in each branch were adjusted empirically to balance expressive power and computational efficiency for the present (3+1)D multimode propagation task.
• Figure 2: Left: Training and validation loss curves of the model, showing stable convergence and a small generalization gap at the selected checkpoint. Right: Illustration adapted from Goodfellow et al. Goodfellow-et-al-2016 showing the relationship between model capacity and generalization error, indicating the ideal stopping point prior to overfitting. The behavior observed in the left panel indicates that the model is stopped around this optimal zone.
• Figure 3: Forward propagation of transverse intensity $I(x,y;z)$ predicted by the trained network on a representative test sample (peak power $P_0 = 253.5$ MW) from the held-out test set. Leftmost plot: input transverse intensity $I(x,y;0)$. Top row plots: ground-truth intensity distributions $I(x,y;z)$ at three propagation distances ($z = 1.527$ mm, $2.869$ mm, and $5.392$ mm). Middle row plots: corresponding predictions from the trained network. Bottom row plots: signed difference between true and predicted speckle patterns.
• Figure 4: Forward propagation of the spatially integrated temporal intensity trace $I(t;z)$ for a representative test sample (peak power $P_0 = 906.0$ MW) from the held-out test set. First plot shows input trace $I(t;0)$. The three right plots show ground-truth traces (blue solid) at propagation distances $z = 1.585$ mm, $3.506$ mm, and $5.253$ mm, with network predictions (red dashed) overlaid.
• Figure 5: Inverse operation of the input transverse intensity $I(x,y;0)$ from a propagated observation in the held-out test set (peak power $P_0 = 375.8$ MW, observation distance $z = 2.881$ mm). First plot: propagated speckle pattern $I(x,y;z)$ used as input to the network. Second plot: ground-truth initial intensity $I(x,y;0)$. Third plot: network prediction conditioned only on the observed speckle, peak power $P_0$, and distance $z$ (inverse mode, direction flag $d=1$). Fourth plot: signed difference (prediction minus ground truth).