Deep Learning-Based Enhancement of Finite Volume Solutions for Non-Linear Schrödinger Equation

The accurate computation of complex wave behaviour, which includes soliton creation and high-frequency waves in nonlinear partial differential equations (PDEs), presents a major computational obstacle for both quantum mechanics and nonlinear optics. The high-resolution Finite Volume Methods (FVM) provide excellent conservation capabilities and stable numerical results, yet their operating expenses make them unsuitable for applications that require multiple queries or immediate processing. The research investigates how the Fourier Neural Operator (FNO) functions as a data-driven surrogate model, which speeds up the one-dimensional Nonlinear Schrödinger Equation (NLSE) solution process. The FNO system establishes a direct connection with the actual nonlinear operator mapping through its training process, which uses computationally efficient coarse-grid FVM simulations. The FNO serves as a statistical emulator that accurately replicates the environmental conditions from which it was trained. It reproduces the learned dynamics, including the numerical dissipation characteristics; however, it does not require correction of discretisation errors. The research defines the actual abilities and restrictions of operator-based learning systems while demonstrating how Fourier Neural Operators function as effective surrogate models for scientific computing processes that need fast and multiple computations.