用SIREN提升FNO频谱学习能力，大幅减少参数量
SirenFNO: Efficient and Full Frequency Learning of Fourier Neural Operators
- 用SIREN实现全频谱参数化，无需频率截断
- 参数量减少4至15倍，保持离散化不变性
- 适合需要高效高精度解PDE的科研与工程场景
傅里叶神经算子（FNO）是高效近似偏微分方程（PDE）解并跨网格泛化的有效代理模型。然而，由于依赖频率截断以维持学习效率，实证研究表明FNO存在对低频信息的谱偏差，可能阻碍强高频振荡类PDE的学习能力。为此，我们提出SirenFNO，一种新框架：利用正弦表示网络（SIRENs）学习隐式神经表示，并实现逐模态核参数化。该SIREN参数化在固定参数量下学习全网格频谱，无需频率截断。我们进一步引入函数张量分解以增强参数与学习效率。实验表明，SirenFNO相比标准FNO在多个PDE基准上参数量减少约4至15倍，且保持离散化不变性；其函数分解变体更实现最高达73倍的参数缩减，同时性能提升。
原文摘要 · Abstract (English)
Fourier neural operators (FNOs) are effective and efficient surrogates for approximating solutions of PDEs and generalize across discretizations. However, owing to the reliance on frequency truncation to maintain learning efficiency of FNOs, empirical studies suggest that FNOs exhibit spectral bias toward low-frequency information, which may hinder the learning capability especially for certain PDEs with strong high-frequency oscillations. To address this limitation, we propose SirenFNO, a novel framework that leverages sinusoidal representation networks (SIRENs) to learn implicit neural representations and performs mode-wise kernel parameterization. Our SIREN parameterization learns a full-grid spectrum with a constant and discretization-independent parameter count, thereby eliminating the need for frequency truncation. We further extend SirenFNO with functional tensor decompositions to enhance parameter and learning efficiency. Empirical results show that our SirenFNO consistently outperforms FNO with approximately $4$ to $15$ times parameter reductions with preserved discretization invariance, and our functional decomposition variants obtain performance improvements with a maximum of $73$ times fewer parameters across multiple PDE benchmarks.
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