arXiv:2512.11686v1 Announce Type: cross 
Abstract: 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.

يمثل إدخال المشغل العصبي الطيفي المستقر (SSNO) تقدمًا كبيرًا في نمذجة أنظمة المعادلات التفاضلية الجزئية (PDE) الصعبة، خاصة عندما تكون البيانات محدودة. يستخدم هذا الإطار التعليمي الخالي من المعادلات هياكل مستوحاة طيفيًا لتحسين عملية التعلم، مما يعالج التحديات التي تطرحها البيانات الملاحظة النادرة وصلابة النظام.

La introducción del Operador Neural Espectral Estable (SSNO) marca un avance significativo en la modelización de sistemas de ecuaciones en derivadas parciales (EDP) rígidas, especialmente cuando los datos son limitados. Este marco de aprendizaje libre de ecuaciones utiliza estructuras inspiradas espectralmente para mejorar el proceso de aprendizaje, abordando los desafíos planteados por datos de observación escasos y la rigidez del sistema.

L'introduction de l'Opérateur Neural Spectral Stable (SSNO) représente une avancée significative dans la modélisation des systèmes d'équations aux dérivées partielles (EDP) rigides, en particulier lorsque les données sont limitées. Ce cadre d'apprentissage sans équation utilise des structures inspirées du spectral pour améliorer le processus d'apprentissage, abordant les défis posés par des données d'observation rares et la rigidité du système.

The introduction of the Stable Spectral Neural Operator (SSNO) marks a significant advancement in modeling stiff partial differential equation (PDE) systems, particularly when data is limited. This equation-free learning framework utilizes spectrally inspired structures to enhance the learning process, addressing the challenges posed by sparse observational data and system stiffness.

Stable spectral neural operator for learning stiff PDE systems from limited data

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