用傅里叶特征一步求解偏微分方程，速度快精度高。
FastLSQ: Solving PDEs in One Shot via Fourier Features with Exact Analytical Derivatives
- 基于三角傅里叶特征，直接解析计算任意阶导数。
- 线性问题0.07秒达10⁻⁷精度，非线性问题<9秒达10⁻⁸~10⁻⁹。
- 适合需要快速建模与反演的工程场景，如热源识别与方程发现。
我们提出FastLSQ，一种基于三角随机傅里叶特征求解偏微分方程（PDE）与逆问题的框架。该方法可对任意阶导数进行闭式解析计算，复杂度为$/mathcal{O}(1)$，无需自动微分即可实现无图运算符组装。线性PDE仅需一次最小二乘求解；非线性问题采用牛顿-拉夫逊法并复用解析组装。在17个不同维度（1–6维）的PDE上，线性问题0.07秒内达到10⁻⁷精度，非线性问题在<9秒内实现10⁻⁸至10⁻⁹精度，较迭代型PINNs快数个数量级且更精确。解析高阶导数支持可微数字孪生，我们在热源反演、线圈恢复等逆问题及PDE发现任务中进行了验证。代码开源：github.com/sulcantonin/FastLSQ，可通过pip install fastlsq获取。
原文摘要 · Abstract (English)
We present FastLSQ, a framework for PDE solving and inverse problems built on trigonometric random Fourier features with exact analytical derivatives. Trigonometric features admit closed-form derivatives of any order in $\mathcal{O}(1)$, enabling graph-free operator assembly without autodiff. Linear PDEs: one least-squares call; nonlinear: Newton--Raphson reusing analytical assembly. On 17 PDEs (1--6D), FastLSQ achieves $10^{-7}$ in 0.07s (linear) and $10^{-8}$--$10^{-9}$ in $<$9s (nonlinear), orders of magnitude faster and more accurate than iterative PINNs. Analytical higher-order derivatives yield a differentiable digital twin; we demonstrate inverse problems (heat-source, coil recovery) and PDE discovery. Code: github.com/sulcantonin/FastLSQ and \texttt{pip install fastlsq}.
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