Recent research published in Scientific Reports demonstrates that the differential equation model of four-wave mixing in a nonlinear Schrödinger optical fiber system can be extracted via data-driven discovery with sparse regression.
Four-wave mixing (FWM) is a crucial phenomenon in nonlinear fiber optics. However, studying it in isolation has been difficult because wave mixing processes cascade with distance to produce several higher-order frequency components and sidebands.
However, developing new experimental techniques has enabled fiber FWM to be stimulated under near-ideal circumstances, opening up new avenues for studying ideal wave mixing dynamics.
Data-Driven Discovery and Nonlinear Systems
Machine learning (ML) tools and approaches are revolutionizing the research of complex dynamics, particularly in laser physics and ultrafast photonics.
The application of data-driven discovery in the study of nonlinear systems, where finding an underlying governing physical model is challenging, is an area of considerable potential.
While neural network approaches can accurately describe complicated systems, they do not give an analytic framework for understanding the underlying physics. This motivates the use of inverse problem-like methods.
Various new algorithms have recently been created to detect the system's mathematical structure using data generated by the system. These findings have sparked a great deal of multidisciplinary interest and have found application in a wide range of physical challenges.
Sparse Identification of Nonlinear Dynamics Technique
Sparse identification of nonlinear dynamics (SINDy) is a unique approach to data-driven discovery that selects the smallest terms from a library of functions to describe a given data set using a system of coupled differential equations. It has been employed in various disciplines, including mechanics, plasma physics, hydrodynamics, and chaotic systems.
The approach is based on the empirical finding that interactions between a small number of different physical processes often control the behavior of complex systems. This observation subsequently enables sparse regression to identify a parsimonious model with the smallest terms capable of repeating the observed behavior without the inclusion of excessive overfitting.
Although several types of research have been documented, finding driving factors in soliton dynamics and reducing impairments in communications networks, their applicability in nonlinear optics has been more restricted. However, there is excellent potential for extensive use of such approaches in nonlinear optics.
Using Sparse Regression to Study Optical Four-Wave Mixing
Ermolaev et al. demonstrated the application of sparse regression to study governing model for optical four-wave mixing in a nonlinear Schrödinger equation (NLSE) system.
The interaction between two frequency detuned sidebands and a single frequency pump, characterized by a simplified system of coupled differential equations, was studied using sparse identification of nonlinear dynamical systems (SINDy).
After examining each model's histogram distribution for the specific number of terms, the mean and uncertainty for each term's coefficient were computed.
The objective of the experiment was to evaluate the differential equation model of FWM by sparse regression (SINDy) in a nonlinear Schrodinger equation system.
Significant Findings of the Study
Ermolaev et al. verified that data-driven discovery utilizing sparse regression (SINDy) could effectively identify the governing model for nonlinear FWM.
A practical method for interpreting SINDy findings in the presence of noise is developed by assessing an ensemble of input data calculated in the dynamical phase space.
Compared to the recurrent sampling of just one set of beginning circumstances, sampling across numerous initial conditions enables thorough exploration of the dynamical space.
The findings imply that when a nonlinear system involves a separatrix border between qualitatively distinct dynamics, SINDy must sample initial conditions on both sides of the separatrix to return the underlying model reliably.
SINDy generated models with fewer than 6 terms when input data was examined solely for a small portion of initial conditions.
Any model(s) given by SINDy for an unidentified system should be accompanied by parallel analysis to direct the search for a suitable and physically sound theoretical description.
Similar systems of coupled equations in nonlinear optics, such as CW Raman scattering and parametric amplification, can be easily converted to operate with SINDy.
The ultimate goal of SINDy-like approaches is to develop physical models from experimental data. This work is crucial in demonstrating the practical viability of data-driven models in nonlinear optics.
Various systems such as plasma physics, cold atoms, and fluid dynamics depend on the nonlinear Schrödinger equation for their operation. Therefore, the findings of this study are easily transferable to studying optical four-wave mixing in different systems.
Ermolaev, A.V., Sheveleva, A., Genty, G., Finot, C., & Dudley, J.M. (2022). Data-Driven Model Discovery Of Ideal Four-Wave Mixing In Nonlinear Fiber Optics. Scientific Reports. 12, 12711 https://www.nature.com/articles/s41598-022-16586-5