Spectral-Inspired Neural Operator Learning with Limited Data and Unknown Physics

Learning PDE dynamics from limited data with unknown physics is challenging. Existing neural PDE solvers either require large datasets or rely on known physics (e.g., PDE residuals or handcrafted stencils), leading to limited applicability. To address these challenges, we propose Spectral-Inspired Neural Operator (SINO), which can model complex systems from just 5 trajectories without requiring explicit PDE terms. Specifically, SINO automatically learns Fourier multipliers as functions of frequency indices, capturing both high-order derivatives and global/local couplings, thus enabling a compact representation of the underlying differential operators in physics-agnostic regimes. To model nonlinear effects, it employs a ¶i-block that performs multiplicative interactions on derivative features, followed by a spectral low-pass filter for de-aliasing. Extensive experiments on both 2D and 3D PDE benchmarks demonstrate that SINO achieves state-of-the-art performance, with improvements of 1-2 orders of magnitude in accuracy. Particularly, with only 5 training trajectories, SINO outperforms data-driven methods trained on 200 trajectories and remains predictive on challenging out-of-distribution cases where other methods fail.