仅用2-5条轨迹，无需物理方程即可学习复杂系统动态。
Spectral-inspired Operator Learning with Limited Data and Unknown Physics
- 基于频域索引自动捕捉局部与全局导数，实现无物理先验的微分算子建模。
- 在2D/3D PDE基准上，精度比现有方法提升1-2个数量级，5条轨迹胜过千条数据训练。
- 适合数据稀缺、物理未知场景，尤其擅长外分布泛化，对科研与工程应用价值高。
从少量数据中学习未知物理的偏微分方程（PDE）动力学极具挑战。现有神经PDE求解器要么需要大规模数据集，要么依赖已知物理信息（如PDE残差或手工设计的差分模板），适用性受限。为此，我们提出谱启发式神经算子（SINO），可在仅2-5条轨迹下建模复杂系统，无需显式给出PDE项。SINO通过频率索引自动提取局部与全局空间导数，实现无物理先验下的微分算子紧凑表示。为建模非线性效应，引入Pi-block对谱特征进行乘法操作，并辅以低通滤波抑制混叠。在2D和3D PDE基准上的大量实验表明，SINO达到当前最优性能，精度提升1-2个数量级。尤其在仅使用5条训练轨迹时，其表现超越在1000条轨迹上训练的数据驱动方法，且在其他方法失效的外分布情形仍具预测能力。
原文摘要 · Abstract (English)
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 2-5 trajectories, without requiring explicit PDE terms. Specifically, SINO automatically captures both local and global spatial derivatives from frequency indices, enabling a compact representation of the underlying differential operators in physics-agnostic regimes. To model nonlinear effects, it employs a Pi-block that performs multiplicative operations on spectral features, complemented by a low-pass filter to suppress 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 1000 trajectories and remains predictive on challenging out-of-distribution cases where other methods fail.
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