Soft Partition-based KAPI-ELM for Multi-Scale PDEs

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

Curriculum Learning-Driven PIELMs for Fluid Flow Simulations

This paper presents two novel, physics-informed extreme learning machine (PIELM)-based algorithms for solving steady and unsteady nonlinear partial differential equations (PDEs) related to fluid flow. Although single-hidden-layer PIELMs outperform deep physics-informed neural networks (PINNs) in speed and accuracy for linear and quasilinear PDEs, their extension to nonlinear problems remains challenging. To address this, we introduce a curriculum learning strategy that reformulates nonlinear PDEs as a sequence of increasingly complex quasilinear PDEs. Additionally, our approach enables a physically interpretable initialization of network parameters by leveraging Radial Basis Functions (RBFs). The performance of the proposed algorithms is validated on two benchmark incompressible flow problems: the viscous Burgers equation and lid-driven cavity flow. To the best of our knowledge, this is the first work to extend PIELM to solving Burgers' shock solution as well as lid-driven cavity flow up to a Reynolds number of 100. As a practical application, we employ PIELM to predict blood flow in a stenotic vessel. The results confirm that PIELM efficiently handles nonlinear PDEs, positioning it as a promising alternative to PINNs for both linear and nonlinear PDEs.