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

arXiv: https://arxiv.org/abs/2512.11686 · HTML: https://mlsanity.com/paper/2512.11686/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 26 of its 58 references that are in the map.
- First posted on arXiv 12 Dec 2025.

## Builds on (most cited first, 26 of 26)

- [A ConvNet for the 2020s](https://mlsanity.com/paper/2201.03545/) (2022)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Learning Mesh-Based Simulation with Graph Networks](https://mlsanity.com/paper/2010.03409/) (2020)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Message Passing Neural PDE Solvers](https://mlsanity.com/paper/2202.03376/) (2022)
- [Transolver: A Fast Transformer Solver for PDEs on General Geometries](https://mlsanity.com/paper/2402.02366/) (2024)
- [Factorized Fourier Neural Operators](https://mlsanity.com/paper/2111.13802/) (2021)
- [Spherical Fourier Neural Operators: Learning Stable Dynamics on the Sphere](https://mlsanity.com/paper/2306.03838/) (2023)
- [Neural Operators for Accelerating Scientific Simulations and Design](https://mlsanity.com/paper/2309.15325/) (2023)
- [PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Network](https://mlsanity.com/paper/1812.04426/) (2018)
- [DPOT: Auto-Regressive Denoising Operator Transformer for Large-Scale PDE Pre-Training](https://mlsanity.com/paper/2403.03542/) (2024)
- [The Well: a Large-Scale Collection of Diverse Physics Simulations for Machine Learning](https://mlsanity.com/paper/2412.00568/) (2024)
- [Scalable Transformer for PDE Surrogate Modeling](https://mlsanity.com/paper/2305.17560/) (2023)
- [Encoding physics to learn reaction-diffusion processes](https://mlsanity.com/paper/2106.04781/) (2021)
- [Learning to correct spectral methods for simulating turbulent flows](https://mlsanity.com/paper/2207.00556/) (2022)
- [PhyMPGN: Physics-encoded Message Passing Graph Network for spatiotemporal PDE systems](https://mlsanity.com/paper/2410.01337/) (2024)
- [Deciphering and integrating invariants for neural operator learning with various physical mechanisms](https://mlsanity.com/paper/2311.14361/) (2023)
- [Monte Carlo Neural PDE Solver for Learning PDEs via Probabilistic Representation](https://mlsanity.com/paper/2302.05104/) (2023)
- [NeuralStagger: Accelerating Physics-constrained Neural PDE Solver with Spatial-temporal Decomposition](https://mlsanity.com/paper/2302.10255/) (2023)
- [AI-accelerated Discovery of Altermagnetic Materials](https://mlsanity.com/paper/2311.04418/) (2023)
- [PeSANet: Physics-encoded Spectral Attention Network for Simulating PDE-Governed Complex Systems](https://mlsanity.com/paper/2505.01736/) (2025)
- [Conservation-informed Graph Learning for Spatiotemporal Dynamics Prediction](https://mlsanity.com/paper/2412.20962/) (2024)
- [MultiPDENet: PDE-embedded Learning with Multi-time-stepping for Accelerated Flow Simulation](https://mlsanity.com/paper/2501.15987/) (2025)
- [Symbolic Neural Ordinary Differential Equations](https://mlsanity.com/paper/2503.08059/) (2025)

## Abstract (arXiv)

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 textita priori knowledge of PDE terms.