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

2026-02-21

AI Summary

This work addresses the instability of neural operators in long-term autoregressive forecasting, which arises from error accumulation and high-frequency feedback. To mitigate this, the authors propose the Spectral Generative Neural Operator (SGNO), which performs stable exponential time-differencing updates in Fourier space by integrating a learnable diagonal generator constrained to have non-positive real parts, gated nonlinear forcing terms, spectral truncation, and smooth masking. Theoretical analysis provides bounds on single-step amplification and finite-time prediction errors. Evaluated on one-, two-, and three-dimensional partial differential equations from APEBench, SGNO substantially reduces long-term prediction errors and extends the duration of stable rollouts. Ablation studies confirm the contribution of each component to the overall performance.

Technology Category

• Machine Learning: Deep Generative Models & Autoencoders
• Search and Optimization: Learning to Search
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs

Abstract

Neural operators provide fast PDE surrogates and often generalize across parameters and resolutions. However, in the short train long test setting, autoregressive rollouts can become unstable. This typically happens for two reasons: one step errors accumulate over time, and high frequency components feed back and grow. We introduce the Spectral Generator Neural Operator (SGNO), a residual time stepper that targets both effects. For the linear part, SGNO uses an exponential time differencing update in Fourier space with a learned diagonal generator. We constrain the real part of this generator to be nonpositive, so iterating the step does not amplify the linear dynamics. For nonlinear dynamics, SGNO adds a gated forcing term with channel mixing within each Fourier mode, which keeps the nonlinear update controlled. To further limit high frequency feedback, SGNO applies spectral truncation and an optional smooth mask on the forcing pathway. We derive a one step amplification bound and a finite horizon rollout error bound. The bound separates generator approximation error from nonlinear mismatch and gives sufficient conditions under which the latent $L^2$ norm does not grow across rollout steps. On APEBench spanning 1D, 2D, and 3D PDE families, SGNO achieves lower long horizon error and longer stable rollout lengths than strong neural operator baselines. Ablations confirm the roles of the generator constraint, gating, and filtering.The code is available at https://github.com/lijy32123-cloud/SGNO.

Problem

• neural operators
• PDE rollout stability
• autoregressive instability
• error accumulation
• high-frequency feedback

Innovation

• Spectral Generator Neural Operator
• Exponential Time Differencing
• Fourier Space
• Stable Rollout
• Neural Operators