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| Format: | Preprint |
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2026
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| Online-Zugang: | https://arxiv.org/abs/2602.18801 |
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| _version_ | 1866916015134212096 |
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| author | Li, Jiayi Jiang, Penghao Saleem, Hira Wang, Zhaonan Koniusz, Piotr Salim, Flora D. |
| author_facet | Li, Jiayi Jiang, Penghao Saleem, Hira Wang, Zhaonan Koniusz, Piotr Salim, Flora D. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_18801 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | SGNO: Spectral Generator Neural Operators for Stable Long Horizon PDE Rollouts Li, Jiayi Jiang, Penghao Saleem, Hira Wang, Zhaonan Koniusz, Piotr Salim, Flora D. Machine Learning 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. |
| title | SGNO: Spectral Generator Neural Operators for Stable Long Horizon PDE Rollouts |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2602.18801 |