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

翻译：物理信息机器学习在求解微分方程方面前景广阔，然而现有方法因频谱偏差、昂贵的反向传播以及需手动调整的核函数或傅里叶频率等问题，在处理高度振荡、多尺度或奇异摄动偏微分方程时面临困难。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），这是一种确定性低维参数化方法，其中平滑分区长度联合控制配置点中心与高斯核宽度，无需傅里叶特征、随机采样或硬域界面即可实现从粗到细分辨率的连续调节。基于符号距离的加权策略进一步增强了不规则几何域上最小二乘学习的稳定性。在八个基准测试中——包括振荡常微分方程、高频泊松方程、不规则形状区域以及刚性奇异摄动对流扩散问题——所提方法在仅需单次线性求解的情况下，其精度达到或超越了当前最先进的物理信息神经网络（PINN）与函数连接理论（TFC）变体。尽管在稳态线性偏微分方程上验证，结果表明软分区核自适应技术为多尺度偏微分方程提供了一种快速、无需复杂架构的求解途径，在未来物理信息建模中具有广泛潜力。为保障可复现性，参考代码已发布于 https://github.com/vikas-dwivedi-2022/soft_kapi。