无需傅里叶特征，用可学习分区解决多尺度微分方程求解难题
Soft Partition-based KAPI-ELM for Multi-Scale PDEs
- 基于软分区的自适应核函数，联合控制采样点与核宽
- 单次线性求解即达顶尖精度，8个基准测试全部超越或持平现有方法
- 适合多尺度、不规则区域的快速高精度物理建模，代码开源
物理信息机器学习在求解微分方程方面前景广阔，但现有方法在高振荡、多尺度或奇异摄动的偏微分方程（PDEs）上表现不佳，受限于谱偏差、代价高昂的反向传播以及人工调参的核函数或傅里叶频率。本文提出一种基于软分区的核自适应物理信息极限学习机（KAPI-ELM），通过平滑分区长度联合控制配点位置与高斯核宽度，实现无需傅里叶特征、随机采样或硬域界面的连续粗到细分辨率。基于符号距离的加权机制进一步提升了在不规则几何上的最小二乘学习稳定性。在八个基准测试中——包括振荡常微分方程、高频泊松方程、不规则区域及刚性奇异摄动对流-扩散问题——该方法在仅使用一次线性求解的情况下，达到或超过当前最优的物理信息神经网络（PINN）和函数连接理论（TFC）变体的精度。尽管演示限于稳态线性PDE，结果表明软分区核自适应为多尺度PDE提供了一种快速、无需架构设计的方法，具有广泛的应用潜力。为确保可复现性，参考代码已公开于 https://github.com/vikas-dwivedi-2022/soft_kapi。
原文摘要 · Abstract (English)
Physics-informed machine learning holds great promise for solving differential equations, yet existing methods struggle with highly oscillatory, multiscale, or singularly perturbed PDEs due to spectral bias, costly backpropagation, and manually tuned kernel or Fourier frequencies. This work introduces a soft partition--based Kernel-Adaptive Physics-Informed Extreme Learning Machine (KAPI-ELM), a deterministic low-dimensional parameterization in which smooth partition lengths jointly control collocation centers and Gaussian kernel widths, enabling continuous coarse-to-fine resolution without Fourier features, random sampling, or hard domain interfaces. A signed-distance-based weighting further stabilizes least-squares learning on irregular geometries. Across eight benchmarks--including oscillatory ODEs, high-frequency Poisson equations, irregular-shaped domains, and stiff singularly perturbed convection-diffusion problems-the proposed method matches or exceeds the accuracy of state-of-the-art Physics-Informed Neural Network (PINN) and Theory of Functional Connections (TFC) variants while using only a single linear solve. Although demonstrated on steady linear PDEs, the results show that soft-partition kernel adaptation provides a fast, architecture-free approach for multiscale PDEs with broad potential for future physics-informed modeling. For reproducibility, the reference codes are available at https://github.com/vikas-dwivedi-2022/soft_kapi
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