用傅里叶正弦函数做测试函数，提升方程学习在噪声下的稳定性。
Fourier Weak SINDy: Spectral Test Function Selection for Robust Model Identification
- 采用正交正弦函数作为测试函数，构建弱形式稀疏回归框架。
- 通过多锥估计频谱，自动选出数据中的主导频率成分。
- 适合噪声环境下的动力系统建模，尤其适用于混沌系统。
我们提出 Fourier Weak SINDy，一种最小化噪声影响且可解释的无导数方程学习方法。该方法结合弱形式稀疏方程学习与谱密度估计，实现数据驱动的测试函数选择。受调制函数法系统辨识中正弦函数广泛使用的启发，采用正交正弦测试函数，使弱形式稀疏回归转化为对傅里叶系数的回归。通过多锥法估计数据频谱，筛选出主导频率。该框架将弱形式学习与谱估计统一于紧凑灵活的结构中。在多个混沌与超混沌常微分方程基准测试中验证了其有效性。
原文摘要 · Abstract (English)
We introduce Fourier Weak SINDy, a minimal noise-robust and interpretable derivative-free equation learning method that combines weak-form sparse equation learning with spectral density estimation for data-driven test function selection. By using orthogonal sinusoidal test functions inspired by their prevalence in Modulating Function-based system identification, the weak-form sparse regression problem reduces to a regression over Fourier coefficients. Dominant frequencies are then selected via multitaper estimation of the frequency spectrum of the data. This formulation unifies weak-form learning and spectral estimation within a compact and flexible framework. We illustrate the effectiveness of this approach in numerical experiments across multiple chaotic and hyperchaotic ODE benchmarks.
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