Accurate modeling of spatiotemporal dynamics is crucial to understanding complex phenomena across science and engineering. However, this task faces a fundamental challenge when the governing equations are unknown and observational data are sparse. System stiffness, the coupling of multiple time-scales, further exacerbates this problem and hinders long-term prediction. Existing methods fall short: purely data-driven methods demand massive datasets, whereas physics-aware approaches are constrained by their reliance on known equations and fine-grained time steps. To overcome these limitations, we introduce an equation-free learning framework, namely, the Stable Spectral Neural Operator (SSNO), for modeling stiff partial differential equation (PDE) systems based on limited data. Instead of encoding specific equation terms, SSNO embeds spectrally inspired structures in its architecture, yielding strong inductive biases for learning the underlying physics. It automatically learns local and global spatial interactions in the frequency domain, while handling system stiffness with a robust integrating factor time-stepping scheme. Demonstrated across multiple 2D and 3D benchmarks in Cartesian and spherical geometries, SSNO achieves prediction errors one to two orders of magnitude lower than leading models. Crucially, it shows remarkable data efficiency, requiring only very few (2--5) training trajectories for robust generalization to out-of-distribution conditions. This work offers a robust and generalizable approach to learning stiff spatiotemporal dynamics from limited data without explicit \textit{a priori} knowledge of PDE terms.

翻译：准确建模时空动力学对于理解科学与工程中的复杂现象至关重要。然而，当控制方程未知且观测数据稀疏时，该任务面临根本性挑战。系统刚性——即多时间尺度的耦合——进一步加剧了这一问题，并阻碍了长期预测。现有方法存在不足：纯数据驱动方法需要海量数据集，而物理感知方法则受限于对已知方程和精细时间步长的依赖。为克服这些局限，我们引入了一种无方程学习框架，即稳定谱神经算子（SSNO），用于基于有限数据建模刚性偏微分方程（PDE）系统。SSNO并非编码特定方程项，而是在其架构中嵌入受谱启发的结构，从而为学习底层物理提供强归纳偏置。它能在频域中自动学习局部与全局空间相互作用，同时通过稳健的积分因子时间步进方案处理系统刚性。在笛卡尔与球面几何下的多个二维和三维基准测试中，SSNO实现了比主流模型低一至两个数量级的预测误差。关键的是，它展现出卓越的数据效率，仅需极少（2-5条）训练轨迹即可实现对外分布条件的稳健泛化。这项工作为在无需偏微分方程项显式先验知识的情况下，从有限数据中学习刚性时空动力学提供了稳健且可泛化的方法。