Operator Learning Predicts Complex Fluid Dynamics with Geometry-Conditioned FNO

Researchers have developed a novel approach to model complex fluid dynamics, specifically the cubic nonlinear Schrödinger (NLS) equation, using a geometry-conditioned Fourier neural operator (FNO). This model is designed to operate on two-dimensional flat tori, which are mathematical constructs representing periodic domains, and can account for varying aspect ratios. The choice of aspect ratio significantly influences the Fourier resonance structure, leading to different high-frequency cascade behaviors in rational versus irrational geometries. The FNO takes as input the real and imaginary parts of the solution along with the aspect-ratio parameter. It is trained to approximate the one-step solution operator and has been validated against unseen trajectories generated from random-phase initial data. Numerical experiments show that the learned operator effectively captures the main solution dynamics on both types of tori. Crucially, the model reproduces the distinct Sobolev norm behavior observed in different geometries, with stronger H2-growth on rational tori and more constrained behavior on irrational tori, aligning with previous findings. Ablation studies confirmed that including the aspect-ratio parameter significantly enhances long-time predictive accuracy, particularly for rational geometries, highlighting the value of geometry-aware neural operators for understanding spectral-transfer phenomena in nonlinear dispersive partial differential equations.