# SGNO: Spectral Generator Neural Operators for Stable Long Horizon PDE Rollouts

arXiv: https://arxiv.org/abs/2602.18801 · HTML: https://mlsanity.com/paper/2602.18801/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 22 references that are in the map.
- First posted on arXiv 21 Feb 2026.

## Builds on (most cited first, 11 of 11)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [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)
- [Message Passing Neural PDE Solvers](https://mlsanity.com/paper/2202.03376/) (2022)
- [PDE-Refiner: Achieving Accurate Long Rollouts with Neural PDE Solvers](https://mlsanity.com/paper/2308.05732/) (2023)
- [Convolutional Neural Operators for robust and accurate learning of PDEs](https://mlsanity.com/paper/2302.01178/) (2023)
- [Accelerating Eulerian Fluid Simulation With Convolutional Networks](https://mlsanity.com/paper/1607.03597/) (2016)
- [Learned Coarse Models for Efficient Turbulence Simulation](https://mlsanity.com/paper/2112.15275/) (2021)
- [APEBench: A Benchmark for Autoregressive Neural Emulators of PDEs](https://mlsanity.com/paper/2411.00180/) (2024)
- [Towards Stability of Autoregressive Neural Operators](https://mlsanity.com/paper/2306.10619/) (2023)

## Abstract (arXiv)

Autoregressive neural PDE surrogates predict future states by repeatedly applying a learned one-step operator. This is a simple and widely used method, but small one-step errors can accumulate during long rollouts. The resulting drift often appears as spectral amplitude distortion, phase misalignment, and nonlinear mode-interaction error. These effects are especially important for time-dependent PDEs with clear Fourier structure. We introduce the Spectral Generator Neural Operator (SGNO), a structured autoregressive neural operator for long-horizon PDE forecasting. SGNO organizes each learned one-step map as a structured spectral evolution update. A real-valued nonpositive diagonal generator provides a gain-controlled spectral backbone, while a learned correction pathway with complex-valued spectral mixing completes the residual evolution. This design gives the autoregressive step an evolution-like structure while retaining the flexibility needed for dissipative, dispersive, transport-dominated, and nonlinear PDEs. SGNO is designed for periodic linear and semilinear evolution PDEs with Fourier multiplier linear dynamics. Across ten mechanism-matched APEBench tasks spanning this regime, SGNO consistently outperforms strong single-step autoregressive baselines in long-horizon rollout accuracy, reducing GMean100 by a median of 74.8% relative to the strongest available non-SGNO baseline, with per-task reductions ranging from 13.6% to 92.9%. The gains are strongest on dispersive and transport-dominated tasks, as well as tasks involving nonlinear closure and mode coupling. Spectral diagnostics show lower spectral energy error and improved rollout-level phase fidelity. Ablations show that the constrained generator, the structured update, and the learned correction pathway each contribute to performance. The code is available at https://github.com/cruiseresearchgroup/SGNO.