Geometry-Aware FNO Learns Nonlinear Wave Dynamics on Periodic Domains
New neural operator captures distinct energy cascades on rational vs irrational tori
A team led by Emmanuel Oguadimma has developed a geometry-conditioned Fourier neural operator (FNO) to solve the cubic defocusing nonlinear Schrödinger equation on two-dimensional flat tori. The key innovation is conditioning the operator on the torus aspect ratio ω², which controls the Fourier resonance structure. Rational aspect ratios lead to stronger high-frequency cascade and faster H² norm growth, while irrational ratios suppress such growth. The FNO takes as input both the real and imaginary parts of the solution along with ω², and is trained to approximate the one-step time evolution operator.
Numerical experiments on unseen random-phase initial data show the learned operator accurately reproduces these geometry-dependent dynamics. Ablation studies reveal that including ω² improves long-time predictive accuracy by 15-20% on rational tori, and that deeper Fourier layers and careful activation function choice further enhance performance. This work demonstrates that neural operators can capture subtle spectral-transfer phenomena in nonlinear dispersive PDEs, opening the door to geometry-aware surrogate models for wave propagation, plasma physics, and optical communications.
- Geometry-conditioned FNO uses aspect ratio ω² as input to capture resonance structure differences between rational and irrational tori
- Model accurately reproduces stronger H² norm growth on rational tori and constrained growth on irrational tori, matching known physics
- Including ω² improves long-term prediction accuracy by 15-20% for rational geometries; ablation studies identify optimal Fourier layer depth and activation functions
Why It Matters
Geometry-aware neural operators could accelerate simulations of nonlinear wave phenomena in optics, plasmas, and fluid dynamics.