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

Flora D. Salim; Hira Saleem; Jiayi Li; Penghao Jiang; Piotr Koniusz; Zhaonan Wang

arxiv: 2602.18801 · v2 · pith:UEO26EQOnew · submitted 2026-02-21 · 💻 cs.LG

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

Jiayi Li , Penghao Jiang , Hira Saleem , Zhaonan Wang , Piotr Koniusz , Flora D. Salim This is my paper

Pith reviewed 2026-05-21 12:02 UTC · model grok-4.3

classification 💻 cs.LG

keywords neural operatorsPDE forecastingspectral methodsautoregressive modelslong-horizon predictionFourier multipliersstable rollouts

0 comments

The pith

SGNO structures each autoregressive PDE step as a spectral evolution update to limit error accumulation over long rollouts

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper presents the Spectral Generator Neural Operator to address the accumulation of small one-step errors that distort spectral amplitudes, phases, and nonlinear interactions during extended forecasts of time-dependent PDEs. It organizes the learned map around a real-valued nonpositive diagonal generator that supplies a gain-controlled Fourier backbone together with a learned complex-valued correction pathway for residual mixing. This construction is intended for periodic linear and semilinear evolution equations whose linear parts admit Fourier multiplier representations. Experiments across ten mechanism-matched tasks show consistent reductions in a long-horizon error metric, with the largest gains appearing in dispersive, transport, and nonlinear-coupling regimes.

Core claim

SGNO organizes each learned one-step map as a structured spectral evolution update consisting of a real-valued nonpositive diagonal generator that provides a gain-controlled spectral backbone and a learned correction pathway that performs complex-valued spectral mixing, yielding lower accumulated spectral energy error and improved phase fidelity over long rollouts of periodic linear and semilinear PDEs.

What carries the argument

The spectral generator: a real-valued nonpositive diagonal matrix that supplies the gain-controlled Fourier-domain backbone within each structured autoregressive update.

If this is right

Spectral energy error remains lower and phase fidelity higher across the full rollout horizon
Relative gains are largest for dispersive, transport-dominated, and nonlinear mode-coupling regimes
Ablations confirm that the constrained generator, the structured update form, and the learned correction each contribute measurable accuracy

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same generator-plus-correction pattern could be transferred to other orthogonal bases for problems on non-periodic domains
Hybrid models that enforce additional conservation laws inside the correction pathway would test whether physical invariants can be preserved without sacrificing flexibility
Data-driven identification of the generator entries from short trajectories might allow the method to adapt to regimes where the linear operator is only approximately known

Load-bearing premise

The target equations are periodic linear or semilinear evolution PDEs whose linear dynamics admit Fourier multiplier representations.

What would settle it

Run identical long-horizon rollout comparisons on a non-periodic PDE or a fully nonlinear equation lacking a clear Fourier multiplier form; if SGNO loses its reported accuracy advantage, the central design premise does not hold.

Figures

Figures reproduced from arXiv: 2602.18801 by Flora D. Salim, Hira Saleem, Jiayi Li, Penghao Jiang, Piotr Koniusz, Zhaonan Wang.

**Figure 1.** Figure 1: (a) Overall architecture of SGNO. The input state is lifted to a latent feature space, propagated through stacked time advance blocks, and projected back to obtain the next state. (b) Time advance block. Features are propagated by a stabilized spectral ETD operator in the Fourier domain, with a nonlinear residual injected through a ϕ1 weighted forcing term, followed by a pointwise correction. 4.2. Spectral… view at source ↗

**Figure 2.** Figure 2: Long-horizon stability on 1D KdV. Top: nRMSE trajectories (first 100 steps shown for readability); both baseline and SGNO show the per-step median across test trajectories with a p10–p90 band. Bottom: empirical CDF of per-trajectory stable steps computed on full Teval = 200 rollouts with τ = 0.2 [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Figure 3: Long-horizon error trajectories on 1D Dispersion (first 100 steps shown). Under τ = 0.2, both methods are largely stable; the main difference is error suppression, where SGNO maintains consistently lower nRMSE throughout the horizon. 6.4. Mechanism oriented ablations Ablation analysis [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

read the original abstract

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

SGNO adds a real nonpositive diagonal generator plus complex correction to autoregressive neural operators and shows clear rollout gains on ten tasks, though the generator's limits on phase dynamics put extra weight on the learned term.

read the letter

SGNO structures each autoregressive step with a real-valued nonpositive diagonal generator as the spectral backbone and a learned complex correction for the residual. This split aims to give the model an evolution-like form while still fitting a range of PDE behaviors. The paper reports consistent improvements over strong baselines on ten APEBench tasks, with a median 74.8% drop in GMean100 error and the largest lifts on dispersive and transport cases. Ablations isolate the generator, the structured update, and the correction pathway, and each contributes. Code is released, which is helpful for checking the details.

Referee Report

2 major / 2 minor

Summary. The paper introduces the Spectral Generator Neural Operator (SGNO), a structured autoregressive neural operator for long-horizon PDE forecasting. Each one-step map uses a real-valued nonpositive diagonal generator as a gain-controlled spectral backbone together with a learned complex-valued correction pathway for the residual. Designed for periodic linear and semilinear evolution PDEs with Fourier-multiplier linear dynamics, SGNO is evaluated on ten mechanism-matched APEBench tasks spanning dissipative, dispersive, transport-dominated, and nonlinear regimes, where it reduces GMean100 by a median of 74.8% relative to the strongest non-SGNO baseline (per-task reductions 13.6–92.9%). Ablations isolate the contributions of the constrained generator, structured update, and correction pathway; spectral diagnostics indicate lower energy error and better phase fidelity. Code is released.

Significance. If the empirical results hold under the stated regime, SGNO demonstrates that embedding an evolution-like spectral structure (real nonpositive generator plus complex correction) can materially reduce long-horizon drift in neural PDE surrogates. The availability of code, the systematic ablations that isolate each design element, and the focus on mechanism-matched tasks across multiple regimes strengthen the contribution and make the work reproducible and extensible.

major comments (2)

[Abstract / design description] Abstract and design description: the real-valued nonpositive diagonal generator supplies only amplitude damping (or neutrality) and cannot encode the purely imaginary Fourier multipliers required for phase evolution in dispersive or transport-dominated PDEs. The largest reported gains occur precisely in these regimes, yet the manuscript provides no explicit structural constraint on the complex correction pathway (e.g., skew-Hermitian or unitary enforcement) to guarantee that it faithfully carries the linear operator without introducing accumulating phase or amplitude errors over 100-step rollouts.
[Ablation study] Ablation study (presumably §5 or equivalent): while the ablations isolate the generator, structured update, and correction pathway, they do not test whether the correction alone can stably represent imaginary linear dynamics when the generator is forced to be real and nonpositive. A controlled experiment that replaces the correction with an unconstrained complex operator on the same dispersive tasks would directly quantify whether the claimed stability benefit is robust or merely an artifact of the particular learned correction.

minor comments (2)

Clarify the precise definition of GMean100, the implementation details of all baselines, and whether any post-hoc task selection occurred.
Add a short paragraph or appendix entry that explicitly states the assumed form of the linear Fourier multiplier for each task and how the SGNO generator-plus-correction decomposition maps onto it.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and insightful comments on our manuscript. We address each of the major comments in detail below and indicate the revisions we have made or will make to the manuscript.

read point-by-point responses

Referee: [Abstract / design description] Abstract and design description: the real-valued nonpositive diagonal generator supplies only amplitude damping (or neutrality) and cannot encode the purely imaginary Fourier multipliers required for phase evolution in dispersive or transport-dominated PDEs. The largest reported gains occur precisely in these regimes, yet the manuscript provides no explicit structural constraint on the complex correction pathway (e.g., skew-Hermitian or unitary enforcement) to guarantee that it faithfully carries the linear operator without introducing accumulating phase or amplitude errors over 100-step rollouts.

Authors: We appreciate the referee's observation regarding the division of responsibilities between the generator and the correction pathway. As described in the manuscript, the real-valued nonpositive diagonal generator is designed to provide a gain-controlled spectral backbone focused on amplitude control, which is particularly relevant for dissipative components. For the phase evolution required in dispersive and transport-dominated regimes, the learned complex-valued correction pathway is responsible. We did not impose explicit structural constraints such as skew-Hermitian or unitary enforcement on the correction to maintain the flexibility required for modeling nonlinear and semilinear effects in the target PDEs. Instead, the correction is learned end-to-end to minimize the long-horizon rollout error. To clarify this design rationale and address concerns about potential error accumulation, we have revised the abstract and the design description section to more explicitly articulate the roles of each component and reference the empirical evidence from spectral diagnostics showing improved phase fidelity. We believe the empirical results across multiple regimes support the effectiveness of this approach. revision: partial
Referee: [Ablation study] Ablation study (presumably §5 or equivalent): while the ablations isolate the generator, structured update, and correction pathway, they do not test whether the correction alone can stably represent imaginary linear dynamics when the generator is forced to be real and nonpositive. A controlled experiment that replaces the correction with an unconstrained complex operator on the same dispersive tasks would directly quantify whether the claimed stability benefit is robust or merely an artifact of the particular learned correction.

Authors: The referee correctly notes that our ablation studies, while isolating the contributions of the constrained generator, structured update, and correction pathway, do not include the specific controlled experiment proposed. We agree that directly testing the correction pathway's ability to handle imaginary linear dynamics in isolation on dispersive tasks would provide additional insight into the robustness of the design. We will incorporate this experiment into the revised manuscript by adding a new ablation variant on the relevant dispersive tasks from the APEBench suite. This will help demonstrate whether the stability benefits are indeed due to the structured combination of generator and correction. revision: yes

Circularity Check

0 steps flagged

No circularity: empirical claims rest on external benchmarks

full rationale

The paper presents SGNO as an architectural design choice (real nonpositive diagonal generator plus complex correction pathway) motivated by the Fourier structure of target PDEs. All reported results are direct empirical comparisons of rollout error (GMean100) against independent baselines on fixed APEBench tasks. No derivation, prediction, or performance metric reduces by the paper's equations to a quantity defined solely in terms of internally fitted parameters or self-referential definitions. The central claims therefore remain self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The method adds a structured spectral update on top of standard neural-operator training; the ledger therefore records the domain assumption that enables the structure and the learned parameters that realize the correction.

free parameters (1)

weights of the learned correction pathway
Complex-valued parameters fitted during training to capture residual nonlinear and dispersive dynamics not captured by the diagonal generator.

axioms (1)

domain assumption Target PDEs are periodic and possess linear dynamics representable by Fourier multipliers.
This premise, stated in the abstract, justifies organizing the one-step map as a spectral evolution update with a diagonal generator.

pith-pipeline@v0.9.0 · 5859 in / 1369 out tokens · 87202 ms · 2026-05-21T12:02:35.141366+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel / Jcost unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We parameterize the generator so its real part is always nonpositive... αθ,c(k) = −softplus(ηθ,c(k)) ≤ 0... ETD update... exp(δt Λθ(k)) + δt ϕ1(δt Λθ(k)) F(k) Mθ(k) ĝ(k)
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We derive a one-step amplification bound and a finite-horizon rollout error bound... q(δt) = Lσ (e^{ω δt} + ...)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

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Jiang, Y ., Wang, Y ., Yang, H., and Wang, J

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Li, Z., Peng, W., Yuan, Z., and Wang, J

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doi: 10.1016/j.jcp.2022.111838. 8 Spectral Generator Neural Operator for Stable Long-Horizon PDE Rollouts Lippe, P., Veeling, B. S., Perdikaris, P., Turner, R. E., and Brandstetter, J. Pde-refiner: Achieving accurate long rollouts with neural pde solvers. InAdvances in Neural Information Processing Systems,

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Ye, Z., Zhang, C.-S., and Wang, W

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doi: 10.48550/arXiv.2505. 20721. You, H., Zhang, Q., Ross, C. J., Lee, C.-H., and Yu, Y . Learning deep implicit fourier neural operators (ifnos) with applications to heterogeneous material modeling. Computer Methods in Applied Mechanics and Engineer- ing, 398:115296,

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48550/arXiv.2512.11686. 9 Spectral Generator Neural Operator for Stable Long-Horizon PDE Rollouts A. Proofs A.1. Preliminaries Let F be the Fourier transform and PK be the projector onto retained modes K. We assume F is normalized so that Parseval’s identity holds onL2, henceP K is an orthogonal projector and is nonexpansive: ∥PKf∥ L2 ≤ ∥f∥ L2 .(28) IfF(k...

work page arXiv
[12]

We train with a one step MSE loss without multi step unrolling

We train with Adam using a warmup cosine learning rate schedule with warmup 2000 steps, base learning rate 10−3, minimum learning rate 0, weight decay 0, batch size 20, and 10000 optimizer updates. We train with a one step MSE loss without multi step unrolling. All experiments are run in float32 without AMP on the evaluation hardware described in the main...

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Guibas, J., Mardani, M., Li, Z., Tao, A., Anandkumar, A., and Catanzaro, B

doi: 10.1006/jcph.2002.6995. Guibas, J., Mardani, M., Li, Z., Tao, A., Anandkumar, A., and Catanzaro, B. Adaptive fourier neural operators: Efficient token mixers for transformers. InInternational Conference on Learning Representations,

work page doi:10.1006/jcph.2002.6995 2002

[2] [2]

Jiang, Y ., Wang, Y ., Yang, H., and Wang, J

doi: 10.1017/ S0962492910000048. Jiang, Y ., Wang, Y ., Yang, H., and Wang, J. Integrat- ing fourier neural operator with diffusion model for autoregressive predictions of three-dimensional turbu- lence.arXiv preprint arXiv:2512.12628,

work page arXiv

[3] [3]

Jiang, Y ., Wang, Y ., Yang, H., and Wang, J

doi: 10.48550/arXiv.2512.12628. Jiang, Y ., Li, Z., Wang, Y ., Yang, H., and Wang, J. An implicit adaptive fourier neural operator for long- term predictions of three-dimensional turbulence.Acta Mechanica Sinica, 42:325478,

work page doi:10.48550/arxiv.2512.12628

[4] [4]

Li, Z., Peng, W., Yuan, Z., and Wang, J

doi: 10.5555/3600270.3601490. Li, Z., Peng, W., Yuan, Z., and Wang, J. Long-term pre- dictions of turbulence by implicit u-net enhanced fourier neural operator.Physics of Fluids, 35(7):075145,

work page doi:10.5555/3600270.3601490

[5] [5]

Linot, A

doi: 10.1063/5.0158830. Linot, A. J., Burby, J. W., Tang, Q., Balaprakash, P., Gra- ham, M. D., and Maulik, R. Stabilized neural ordinary differential equations for long-time forecasting of dynam- ical systems.Journal of Computational Physics, 474: 111838,

work page doi:10.1063/5.0158830

[6] [6]

8 Spectral Generator Neural Operator for Stable Long-Horizon PDE Rollouts Lippe, P., Veeling, B

doi: 10.1016/j.jcp.2022.111838. 8 Spectral Generator Neural Operator for Stable Long-Horizon PDE Rollouts Lippe, P., Veeling, B. S., Perdikaris, P., Turner, R. E., and Brandstetter, J. Pde-refiner: Achieving accurate long rollouts with neural pde solvers. InAdvances in Neural Information Processing Systems,

work page doi:10.1016/j.jcp.2022.111838 2022

[7] [7]

Rahman, M

doi: 10.1098/rspa.2024.0819. Rahman, M. A., Ross, Z. E., and Azizzadenesheli, K. U-no: U-shaped neural operators.Transactions on Machine Learning Research,

work page doi:10.1098/rspa.2024.0819 2024

[8] [8]

Ye, Z., Zhang, C.-S., and Wang, W

URL https: //arxiv.org/abs/2402.04467. Ye, Z., Zhang, C.-S., and Wang, W. Recurrent neural opera- tors: Stable long-term pde prediction.arXiv preprint arXiv:2505.20721,

work page arXiv

[9] [9]

doi: 10.48550/arXiv.2505. 20721. You, H., Zhang, Q., Ross, C. J., Lee, C.-H., and Yu, Y . Learning deep implicit fourier neural operators (ifnos) with applications to heterogeneous material modeling. Computer Methods in Applied Mechanics and Engineer- ing, 398:115296,

work page doi:10.48550/arxiv.2505

[10] [10]

Zhang, R., Wan, H., Liu, Y ., and Sun, H

doi: 10.1016/j.cma.2022.115296. Zhang, R., Wan, H., Liu, Y ., and Sun, H. Stable spectral neural operator for learning stiff pde systems from limited data.arXiv preprint arXiv:2512.11686,

work page doi:10.1016/j.cma.2022.115296 2022

[11] [11]

9 Spectral Generator Neural Operator for Stable Long-Horizon PDE Rollouts A

48550/arXiv.2512.11686. 9 Spectral Generator Neural Operator for Stable Long-Horizon PDE Rollouts A. Proofs A.1. Preliminaries Let F be the Fourier transform and PK be the projector onto retained modes K. We assume F is normalized so that Parseval’s identity holds onL2, henceP K is an orthogonal projector and is nonexpansive: ∥PKf∥ L2 ≤ ∥f∥ L2 .(28) IfF(k...

work page arXiv

[12] [12]

We train with a one step MSE loss without multi step unrolling

We train with Adam using a warmup cosine learning rate schedule with warmup 2000 steps, base learning rate 10−3, minimum learning rate 0, weight decay 0, batch size 20, and 10000 optimizer updates. We train with a one step MSE loss without multi step unrolling. All experiments are run in float32 without AMP on the evaluation hardware described in the main...

work page 2000