Scaling transformation of the multimode nonlinear Schr\"odinger equation for physics-informed neural networks

Single-mode optical fibers (SMFs) have become the backbone of modern communication systems. However, their throughput is expected to reach its theoretical limit in the nearest future. Utilization of multimode fibers (MMFs) is considered as one of the most promising solutions rectifying this capacity crunch. Nevertheless, differential equations describing light propagation in MMFs are a way more sophisticated than those for SMFs, which makes numerical modelling of MMF-based systems computationally demanding and impractical for the most part of realistic scenarios. Physics-informed neural networks (PINNs) are known to outperform conventional numerical approaches in various domains and have been successfully applied to the nonlinear Schr\"odinger equation (NLSE) describing light propagation in SMFs. A comprehensive study on application of PINN to the multimode NLSE (MMNLSE) is still lacking though. To the best of our knowledge, this paper is the first to deploy the paradigm of PINN for MMNLSE and to demonstrate that a straightforward implementation of PINNs by analogy with NLSE does not work out. We pinpoint all issues hindering PINN convergence and introduce a novel scaling transformation for the zero-order dispersion coefficient that makes PINN capture all relevant physical effects. Our simulations reveal good agreement with the split-step Fourier (SSF) method and extend numerically attainable propagation lengths up to several hundred meters. All major limitations are also highlighted.