TF-SNO：用于学习非平稳偏微分方程的时频门控谱神经算子
TF-SNO: Time-Frequency Gated Spectral Neural Operators for Learning Non-Stationary Partial Differential Equations
AI总结：
针对非平稳PDE中频谱随时间漂移的问题，提出TF-SNO，通过可学习的时频门控机制使谱响应随状态演化，在不引入显式时间维度下隐式学习时间变化，提升长期预测稳定性。
摘要：
非平稳偏微分方程（PDE）在科学计算中广泛出现，其主导频率内容和能量分布会随时间漂移。尽管许多谱神经算子在PDE求解中高效，但它们在整个展开阶段应用共享的谱响应，导致与非平稳系统中时变频谱不匹配。为解决此问题，我们提出时频门控谱神经算子（TF-SNO），一种状态自适应框架，在谱块内部具有可学习的时频门控。TF-SNO从当前状态提取紧凑的频域和物理空间统计量以生成调制系数，使谱响应随动力学演化。TF-SNO从演化状态隐式学习时间变化，无需引入显式时间维度或时间嵌入，保持建模复杂度低。我们进一步嵌入自适应算子块以准确捕捉多尺度特征，从而提高长期稳定性。在1D和2D的六个非平稳PDE基准上的实验表明，与强基线相比，TF-SNO显著降低了预测误差并提高了鲁棒性，在长展开中尤其明显，表明状态依赖的谱自适应在建模非平稳物理系统中的有效性。
英文摘要：
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.
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