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

arXiv: https://arxiv.org/abs/2601.08719 · HTML: https://mlsanity.com/paper/2601.08719/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 16 of its 23 references that are in the map.
- First posted on arXiv 13 Jan 2026.

## Builds on (most cited first, 16 of 16)

- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [Frequency Principle: Fourier Analysis Sheds Light on Deep Neural Networks](https://mlsanity.com/paper/1901.06523/) (2019)
- [Finite Basis Physics-Informed Neural Networks (FBPINNs): a scalable domain decomposition approach for solving differential equations](https://mlsanity.com/paper/2107.07871/) (2021)
- [Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks](https://mlsanity.com/paper/2104.08426/) (2021)
- [Local Extreme Learning Machines and Domain Decomposition for Solving Linear and Nonlinear Partial Differential Equations](https://mlsanity.com/paper/2012.02895/) (2020)
- [Can Physics-Informed Neural Networks beat the Finite Element Method?](https://mlsanity.com/paper/2302.04107/) (2023)
- [Physics Informed Extreme Learning Machine (PIELM) -- A rapid method for the numerical solution of partial differential equations](https://mlsanity.com/paper/1907.03507/) (2019)
- [Distributed physics informed neural network for data-efficient solution to partial differential equations](https://mlsanity.com/paper/1907.08967/) (2019)
- [Machine learning and domain decomposition methods -- a survey](https://mlsanity.com/paper/2312.14050/) (2023)
- [VS-PINN: A fast and efficient training of physics-informed neural networks using variable-scaling methods for solving PDEs with stiff behavior](https://mlsanity.com/paper/2406.06287/) (2024)
- [Challenges and Advancements in Modeling Shock Fronts with Physics-Informed Neural Networks: A Review and Benchmarking Study](https://mlsanity.com/paper/2503.17379/) (2025)
- [Curriculum Learning-Driven PIELMs for Fluid Flow Simulations](https://mlsanity.com/paper/2503.06347/) (2025)
- [Kernel-Adaptive PI-ELMs for Forward and Inverse Problems in PDEs with Sharp Gradients](https://mlsanity.com/paper/2507.10241/) (2025)
- [Semi-analytic PINN methods for singularly perturbed boundary value problems](https://mlsanity.com/paper/2208.09145/) (2022)

## Abstract (arXiv)

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