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

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

cs.LGFeb 21, 2026

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

Authors: Jiayi Li, Penghao Jiang, Hira Saleem, Zhaonan Wang, Piotr Koniusz, Flora D. Salim

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.

Figures & tables

Figure 1: Overview of SGNO. (a) The current state and coordinates are lifted to a latent field, passed through stacked SGNO layers, and projected to a residual update. (b) Each SGNO layer combines a local path with a spectral generator branch. The spectral branch separates gain-controlled spectral carry from a learned correction term.

Regime	Task	Mechanism	Conv	Res	U-Net	Dil	FNO	CNO	SGNO	SGNO / best
Transport	adv1d	phase propagation	0.0800	0.2200	0.1040	0.0690	0.0290	10.8000	0.0130	0.4670
Dissipative	diff1d	damping	0.0120	0.0092	0.0570	0.0360	0.0180	0.0960	0.0024	0.2630
Dispersive	disp1d	phase propagation	0.0300	0.0340	0.0820	0.0590	0.0310	0.0110	0.0008	0.0710
Semilinear	kdv1d	nonlinear closure	0.1310	0.9060	0.0820	0.0630	0.2270	7.0400	0.0140	0.2230
Semilinear	ks1d	mode coupling	0.2990	0.2670	0.2070	0.1580	0.5980	0.3740	0.02940	0.1860
Dispersive	mixdisp2d	mixed phase	0.0650	0.0600	0.0680	0.0380	0.2840	0.0230	0.0146	0.6490

Table 1: Main long-horizon rollout results on ten evaluated APEBench tasks. Values are median

\mathrm{GMean100}

over independent network seeds, computed from the first 100 steps of a 200-step closed-loop rollout. Lower is better. Bold marks the best result; underlining marks the strongest available non-SGNO baseline. The

\mathrm{SGNO}/\mathrm{best}

column is computed against the strongest available non-SGNO baseline using unrounded recorded values. Dash entries denote unavailable baseline configurations and are excluded from this calculation. Values are not clipped, so entries larger than one indicate high closed-loop rollout error under the official metric.

Figure 2: Representative rollout error growth on dissipative, dispersive, and semilinear tasks. Each panel reports mean nRMSE over a 200-step closed-loop rollout and shows SGNO together with all available baselines. The strongest non-SGNO comparator from Table 1 is Res for diff1d , CNO for disp1d , and Dil for ks1d . Lower is better.

Task	Baseline	State ratio	Spectral ratio	Low-band ratio	Phase ratio
adv1d	FNO	0.499	0.396	0.334	0.455
diff1d	Res	0.247	0.178	0.224	–
disp1d	CNO	0.071	0.102	0.096	0.115
ks1d	Dil	0.139	0.135	0.125	–
kolm2d	FNO	0.789	0.815	0.713	–

Table 2: Spectral rollout diagnostics. Values are SGNO-to-baseline error ratios, and lower is better. For each row, the baseline is the strongest available comparator with saved rollout trajectories and Fourier diagnostics. Spectral ratio denotes spectral energy error, and low-band ratio denotes relative error on low-frequency energy. Phase ratios are reported where the phase diagnostic is defined; phase-reporting details are given in Appendix D .

Figure 3: Spectral diagnostic ratios relative to the listed diagnostic baseline. The dashed reference line at one denotes baseline parity. Values below one indicate lower SGNO error.

Task	Full SGNO	Unconstr. gen. / Full	Residual / Full	No correction / Full
adv1d	0.0130	1.360	75.880	non-finite
diff1d	0.0024	1.370	265.440	2.870
disp1d	0.0008	2.260	198.110	94.830
ks1d	0.0294	1.400	27.100	10.500
kolm2d	0.5560	1.022	1.564	1.925

Table 3: Controlled mechanism ablations. The Full SGNO column reports the matched full-model reference for each ablation suite, and the remaining columns report variant-to-full ratios. Larger values are worse. Ablation variants modify the corresponding SGNO component while keeping the dimension-level configuration fixed. Non-finite denotes invalid numerical values or rollout divergence under the same evaluation protocol.

Appendix figures & tables3 assets

Supplementary material from the paper’s appendix.

Appendix

Task	APEBench dynamic	Class	PDE and main mechanism
adv1d	adv	L-I	Linear advection: $\partial_{t}u=-c\,\partial_{x}u$ . Tests non-dissipative transport.
diff1d	diff	L-D	Diffusion: $\partial_{t}u=\nu\,\partial_{xx}u$ . Tests dissipative smoothing.
disp1d	disp	L-I	Linear dispersion: $\partial_{t}u=\xi\,\partial_{xxx}u$ . Tests phase propagation.
kdv1d	kdv	N-D	Korteweg–de Vries dynamics: $\partial_{t}u=-\frac{b}{2}\partial_{x}(u^{2})+\xi\,\partial_{xxx}u-\zeta\,\partial_{xxxx}u$ . Tests nonlinear convection with dispersion and hyper-diffusion.
ks1d	ks	N-I-C	Kuramoto–Sivashinsky dynamics: $\partial_{t}u=-\frac{b}{2}(\partial_{x}u)^{2}-\nu\,\partial_{xx}u-\zeta\,\partial_{xxxx}u$ . Tests chaotic nonlinear rollout.
mixdisp2d	mix_disp	L-I	Spatially mixed dispersion: $\partial_{t}u=\xi\,\mathbf{1}\cdot\nabla((\nabla\cdot\nabla)u)$ . Tests cross-dimensional phase mixing.

Appendix

Table 4: Descriptions of the APEBench PDE dynamics used in the main comparison.

Model	1D configuration	2D configuration	3D configuration
SGNO	width $=11$ , modes $=26$ , blocks $=7$ 30,573 params	width $=5$ , modes $=10$ , blocks $=9$ 56,402 params	width $=10$ , modes $=6$ , blocks $=2$ 192,827 params
Conv	width $=34$ , depth $=10$ , ReLU 31,757 params	width $=26$ , depth $=11$ , ReLU 61,595 params	width $=26$ , depth $=12$ , ReLU 202,489 params
Res	width $=26$ , depth $=8$ , ReLU 32,943 params	width $=26$ , depth $=5$ , ReLU 61,179 params	width $=25$ , depth $=6$ , ReLU 202,876 params
U-Net	width $=12$ , levels $=2$ , ReLU 27,193 params	width $=10$ , levels $=2$ , ReLU 55,661 params	width $=11$ , levels $=2$ , ReLU 200,322 params
Dil	depth $=2$ , width $=32$ , dilation blocks $=2$ , ReLU 31,777 params	depth $=2$ , width $=26$ , dilation blocks $=2$ , ReLU 61,699 params	depth $=2$ , width $=27$ , dilation blocks $=2$ , ReLU 192,722 params
FNO	width $=12$ , modes $=18$ , blocks $=4$ , GELU 32,527 params	width $=10$ , modes $=6$ , blocks $=4$ , GELU 57,787 params	width $=5$ , modes $=7$ , blocks $=4$ , GELU 196,246 params

Appendix

Table 5: Dimension-level model configurations and trainable parameter counts. Each entry reports the configuration used for all tasks of the corresponding spatial dimension, followed by the number of trainable parameters. SGNO is configured by dimension rather than tuned separately per task, and its parameter counts are comparable to the dimension-matched baselines. For CNO, CM denotes channel multiplier; all CNO runs use initial_step =1, in_dim =1, out_dim =1, kernel size 3, and lift/project latent dimension 64. The 3D CNO entry is unavailable because the local public-wrapper implementation does not support 3D CNO under our evaluation interface.

Train res.	Eval res.	GMean100	Ratio vs. $64^{2}$	Stable step $(\tau=0.2)$
$64^{2}$	$64^{2}$	0.5581	1.000	13.0
$64^{2}$	$128^{2}$	0.5550	0.995	14.5
$64^{2}$	$256^{2}$	0.5705	1.022	14.0

Appendix

Table 6: Resolution-extrapolation evaluation for kolm2d . SGNO is trained at resolution

64^{2}

and evaluated without retraining at

64^{2}

128^{2}

, and

256^{2}

. The transfer ratio is computed relative to the

64^{2}

evaluation. Lower GMean100 is better; ratios near one indicate stable resolution extrapolation under the GMean100 metric. Stable step uses threshold

\tau=0.2

, and larger values indicate later threshold crossing.

Explore similar work

May 11, 2026

Stable Long-Horizon PDE Forecasting via Latent Structured Spectral Propagators

Long-horizon forecasting of time-dependent partial differential equations (PDEs) is critical for characterizing the sustained evolution of physical systems. While neural operators have emerged as efficient surrogates, they typically learn implicit finite-time transitions from discrete observations. When deployed autoregressively, such propagators often suffer from rapid error accumulation and dynamic drift. To address this, we propose a neural forecasting framework that reformulates PDE rollout as learning a Structured Spectral Propagator (SSP) in a propagation-oriented latent space. Following an analysis-propagation-synthesis design, our framework: (i) maps physical states into a shared, time-consistent spatial representation; (ii) projects this space into a compact propagation state to isolate recurrent dynamics from fine-grained spatial details, thereby decoupling reconstruction fidelity from rollout regularity; and (iii) evolves retained spectral modes using a frequency-conditioned linear backbone complemented by a nonlinear spectral closure to account for truncated interactions. This explicit structuring endows the propagator with a strong inductive bias for coherent modal evolution. Extensive experiments demonstrate that SSP significantly outperforms state-of-the-art baselines, reducing relative

L_2

errors by up to 48.9% and exhibiting improved stability in temporal extrapolation beyond the supervised horizon.

Xiaoxiao Lu, Ye Yuan, Jiahao Shi

School of AIA, Huazhong University of Science and Technology

Latent Dynamics Modeling PDE Surrogate Modeling PDE Operator Learning Spatiotemporal Forecasting Spectral Methods

Ask PDF

Jul 31, 2026

Geometry-aware Incremental Neural Operator for Long-Horizon PDE prediction

Neural operators have shown strong potential for learning solution operators of partial differential equations (PDEs). However, long-horizon autoregressive prediction remains challenging: local errors accumulate as spectral inconsistency, phase misalignment, or mean drift. Existing methods mainly improve state representations and operator backbones, while leaving the repeatedly applied latent transition increment weakly structured, allowing spectral errors and unstable channel couplings to accumulate during rollout. To address these issues, we propose a geometry-aware incremental neural operator (GeoIncNO) for stable long-horizon PDE prediction. GeoIncNO predicts latent increments for residual advancement and uses lightweight low-rank projectors to regulate channel coupling within active frequency bands derived from the increment spectral energy distribution. To reduce physical-space reconstruction errors, GeoIncNO further introduces a mean--fluctuation decoupled reconstruction mechanism, where stable mean structures and dynamic fluctuations are fused separately, and phase correction is applied only to the zero-mean fluctuation component. Extensive experiments on six PDE benchmarks, covering 1D, 2D, and 3D dynamical systems, show that GeoIncNO achieves consistently strong prediction accuracy, improved rollout stability, and better spectral fidelity compared with competitive neural-operator baselines.

Jiaquan Zhang, Shuxu Chen, Haifan Meng +6

School of Information and Software Engineering, UESTC · Electronics and Information Convergence Engineering, KHU · School of Computer Science and Engineering, UESTC +4

PDE Surrogate Modeling PDE Operator Learning Spatiotemporal Forecasting Neural Operators

Ask PDF

May 25, 2026

Autoregression-Free Neural Operators for Time-Dependent PDEs

Neural operators learn mappings from function-dependent inputs to solutions, providing an effective framework for solving partial differential equations (PDEs). For time-dependent PDEs, existing methods typically perform long-horizon prediction through autoregressive rollout directly in high-dimensional physical field spaces, where each predicted state is recursively fed back as the input for the next step. Although effective for short-term prediction, this autoregressive rollout and the lack of continuous-time modeling lead to progressive error accumulation over long-horizon rollouts. In this work, we propose Autoregression-Free Neural Operators (AFNO), which map the time evolution of PDEs into a latent space and model continuous-time vector fields within it. AFNO uses flow matching to learn the latent vector field, thereby enabling continuous evolution over extended horizons, avoiding autoregressive rollout and capturing dynamics under varying parameter configurations through explicit conditioning on physical parameters. Theoretical analysis and extensive experiments on six PDEs demonstrate that AFNO improves long-horizon prediction stability and consistently reduces rollout errors compared with the baselines.

Jiaquan Zhang, Caiyan Qin, Haoyu Bian +7

School of Computer Science and Engineering, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, China · School of Robotics and Advanced Manufacture, Harbin Institute of Technology, Shenzhen, Guangdong 518055, China · School of Mathematical Sciences, Capital Normal University, Beijing 100048, China +3

Latent Dynamics Modeling PDE Operator Learning Spatiotemporal Forecasting

Ask PDF