TF-SNO: Time-Frequency Gated Spectral Neural Operators for Learning Non-Stationary Partial Differential Equations

2026-06-19

AI Summary

Existing spectral neural operators struggle to capture the time-evolving spectral characteristics of non-stationary partial differential equations, leading to large and unstable errors in long-term predictions. This work proposes a state-adaptive spectral neural operator framework that introduces a learnable time–frequency gating mechanism to jointly extract statistical features from both the frequency and physical domains based on the current system state, thereby dynamically modulating the spectral response without requiring explicit temporal embeddings. This approach enables implicit time awareness and multi-scale adaptivity, significantly enhancing modeling capacity for non-stationary dynamics. Evaluated on six 1D and 2D non-stationary PDE benchmarks, the method demonstrates superior accuracy and robustness in long-term rollouts compared to strong baseline approaches.

Abstract

Non-stationary partial differential equations (PDEs) arise throughout scientific computing, where the dominant frequency content and energy distribution can drift over time. While efficient in PDE solving, many spectral neural operators apply a shared spectral response across rollout stages, leading to mismatch with time-varying spectra in non-stationary systems. To address this issue, we propose Time-Frequency Gated Spectral Neural Operator (TF-SNO), a state-adaptive framework with learnable time-frequency gating inside spectral blocks. TF-SNO extracts compact frequency-domain and physical-space statistics from the current state to generate modulation coefficients, enabling the spectral response to evolve with the dynamics. TF-SNO learns temporal variation implicitly from the evolving state without introducing an explicit time dimension or time embedding, keeping the modeling complexity low. We further embed the adaptive operator blocks to accurately capture the multi-scale features, thereby improving long-horizon stability. Experiments on six non-stationary PDE benchmarks in 1D and 2D demonstrate that TF-SNO significantly reduces prediction errors and improves robustness compared to strong baselines, with particularly clear gains in long rollout, suggesting the effectiveness of state-dependent spectral adaptation in modeling non-stationary physical systems.

Problem

• non-stationary PDEs
• spectral neural operators
• time-varying spectra
• long-horizon prediction
• state-dependent adaptation

Innovation

• Time-Frequency Gating
• Spectral Neural Operators
• Non-Stationary PDEs
• State-Adaptive Modeling
• Multi-Scale Feature Learning