[论文解读] Soft Partition-based KAPI-ELM for Multi-Scale PDEs

Vikas Dwivedi, Monica Sigovan+1 more|arXiv (Cornell University)|Jan 13, 2026

Model Reduction and Neural Networks被引用 0

一句话总结

本文介绍了一种基于软分区的 Kernel-Adaptive PI–ELM (KAPI–ELM)，在不使用傅里叶特征或反向传播的情况下通过单次线性最小二乘求解，自适应地解决多尺度、振荡及奇异摄动的偏微分方程。

ABSTRACT

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

研究动机与目标

• Motivate and address limitations of existing physics-informed learning approaches (PINNs, Fourier-based PINNs, and domain-decomposition methods) for multiscale and oscillatory PDEs.
• Propose a deterministic, low-dimensional soft partition framework that jointly governs collocation centers and Gaussian widths to enable coarse-to-fine resolution.
• Introduce geometry-aware, signed-distance–based residual weighting to stabilize learning on irregular domains.
• Demonstrate that the method achieves high accuracy across oscillatory, high-frequency, irregular-domain, and singularly perturbed problems with a single linear solve.

提出的方法

• Introduce a soft partition–based sampling strategy where partition lengths determine both collocation centers and Gaussian kernel widths.
• Define 1D and 2D sampling schemes with partition-length vectors that deterministically set center placement and kernel scales.
• Solve the PI–ELM linear system in closed form for given partition parameters and optimize partitions via Bayesian optimization on a validation objective.
• Apply signed-distance–based weighting to PDE residuals to stabilize learning near irregular boundaries.
• Explain how narrower kernels (via small partition lengths) increase high-frequency expressivity without Fourier features.
• Use a single linear least-squares solve and avoid backpropagation or neural architectures.

Figure 1: RBF spectral bandwidth induced by partition–adaptive sampling. Left: the proposed sampler generates a multimodal distribution of Gaussian widths $\sigma$ , including extremely small kernels. Right: corresponding Fourier magnitudes $\lvert\widehat{\phi}_{\sigma}(\omega)\rvert=\exp(-(\sigma\

实验结果

研究问题

• RQ1 Can a deterministic soft partition scheme provide adaptive, multiscale resolution without Fourier feature mappings or domain interface penalties?
• RQ2 How do partition lengths influence center density and kernel widths to capture high-frequency and boundary-layer structures?
• RQ3 Does SDF-based residual weighting improve stability and accuracy on irregular geometries and higher-order PDEs?
• RQ4 What is the comparative performance and speed of soft partition–based KAPI–ELM relative to PINN and FBPINN baselines on multiscale PDEs?

主要发现

Method	Training cost	Architectural complexity	Accuracy on high-frequency and multiscale tests
PINN	50,000 – 100,000 gradient steps; slow and unstable convergence	Single network; sensitive to depth, width, activations; often requires Fourier features	Fails for ω=15 ; large errors (10^{-2} – 10^{-3}); unstable on second-order ODEs
FBPINN	50,000 – 500,000 gradient steps depending on problem	20–30 subdomains; overlapping windows; multiple small networks per subdomain; handcrafted training schedules	Accurate but extremely expensive; sensitive to subdomain layout; best-case errors ~ 10^{-4}
Soft Partition KAPI–ELM	No backpropagation ; single least-squares solve ( ~ 0.1 s)	No neural architecture; few partition parameters; deterministic center and width placement	Near machine-precision accuracy (10^{-6} – 10^{-12}) on all tests; robust for oscillatory and stiff problems

• Matches or exceeds the accuracy of state-of-the-art PINN and TFC variants across eight benchmarks.
• Achieves near machine-precision accuracy (10^{-6} to 10^{-12}) in 1D oscillatory and multiscale tests.
• Solves all tested problems with a single linear least-squares solve, significantly faster than gradient-based methods (about 0.1 s in reported cases).
• Provides Fourier-free high-frequency accuracy by relying on a multimodal distribution of Gaussian widths induced by partition-based sampling.
• SDF-weighted residuals improve stability and reduce boundary leakage on irregular geometries and higher-order operators.
• Demonstrates strong performance on irregular-domain Poisson and biharmonic problems with modest additional computational cost.

Figure 2: KAPI–ELM approximation and exact solution for $u^{\prime}(x)=\cos(15x)$ on $[-2\pi,2\pi]$ .