# Kernel-Adaptive PI-ELMs for Forward and Inverse Problems in PDEs with Sharp Gradients

arXiv: https://arxiv.org/abs/2507.10241 · HTML: https://mlsanity.com/paper/2507.10241/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 31 references that are in the map.
- First posted on arXiv 14 Jul 2025.

## Builds on (most cited first, 17 of 17)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [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)
- [Local Extreme Learning Machines and Domain Decomposition for Solving Linear and Nonlinear Partial Differential Equations](https://mlsanity.com/paper/2012.02895/) (2020)
- [Physics Informed Extreme Learning Machine (PIELM) -- A rapid method for the numerical solution of partial differential equations](https://mlsanity.com/paper/1907.03507/) (2019)
- [On Computing the Hyperparameter of Extreme Learning Machines: Algorithm and Application to Computational PDEs, and Comparison with Classical and High-Order Finite Elements](https://mlsanity.com/paper/2110.14121/) (2021)
- [Finite Element Operator Network for Solving Elliptic-type parametric PDEs](https://mlsanity.com/paper/2308.04690/) (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)
- [Curriculum Learning-Driven PIELMs for Fluid Flow Simulations](https://mlsanity.com/paper/2503.06347/) (2025)
- [Component Fourier Neural Operator for Singularly Perturbed Differential Equations](https://mlsanity.com/paper/2409.04779/) (2024)
- [Semi-analytic PINN methods for singularly perturbed boundary value problems](https://mlsanity.com/paper/2208.09145/) (2022)
- [Operator Learning Enhanced Physics-informed Neural Networks for Solving Partial Differential Equations Characterized by Sharp Solutions](https://mlsanity.com/paper/2310.19590/) (2023)
- [PICS in Pics: Physics Informed Contour Selection for Rapid Image Segmentation](https://mlsanity.com/paper/2311.07002/) (2023)

## Built on by (most cited first, 7 of 7)

- [AB-PINNs: Adaptive-Basis Physics-Informed Neural Networks for Residual-Driven Domain Decomposition](https://mlsanity.com/paper/2510.08924/) (2025)
- [Gated X-TFC: Soft Domain Decomposition for Forward and Inverse Problems in Sharp-Gradient PDEs](https://mlsanity.com/paper/2510.01039/) (2025)
- [Deep vs. Shallow: Benchmarking Physics-Informed Neural Architectures on the Biharmonic Equation](https://mlsanity.com/paper/2510.04490/) (2025)
- [Soft Partition-based KAPI-ELM for Multi-Scale PDEs](https://mlsanity.com/paper/2601.08719/) (2026)
- [Exact Constraint Enforcement in Physics-Informed Extreme Learning Machines using Null-Space Projection Framework](https://mlsanity.com/paper/2601.10999/) (2026)
- [Learning Where the Physics Is: Probabilistic Adaptive Sampling for Stiff PDEs](https://mlsanity.com/paper/2603.06287/) (2026)
- [Meta-Learned Basis Adaptation for Parametric Linear PDEs](https://mlsanity.com/paper/2604.09289/) (2026)

## Abstract (arXiv)

Physics-informed machine learning frameworks such as Physics-Informed Neural Networks (PINNs) and Physics-Informed Extreme Learning Machines (PI-ELMs) have shown great promise for solving partial differential equations (PDEs) but struggle with localized sharp gradients and singularly perturbed regimes, PINNs due to spectral bias and PI-ELMs due to their single-shot, non-adaptive formulation. We propose the Kernel-Adaptive Physics-Informed Extreme Learning Machine (KAPI-ELM), which performs Bayesian optimization over a low-dimensional, physically interpretable hyperparameter space governing the distribution of Radial Basis Function (RBF) centers and widths. This converts high-dimensional weight optimization into a low-dimensional distributional search, enabling targeted kernel refinement in regions with sharp gradients while also improving baseline solutions in smooth-flow regimes by tuning RBF supports. KAPI-ELM is validated on benchmark forward and inverse problems (1D convection-diffusion and 2D Poisson) involving PDEs with sharp gradients. It accurately resolves steep layers, improves smooth-solution fidelity, and recovers physical parameters robustly, matching or surpassing advanced methods such as the extended Theory of Functional Connections (X-TFC) with nearly an order of magnitude fewer tunable parameters. An extension to nonlinear problems is demonstrated by a curriculum-based solution of the steady Navier-Stokes equations via successive linearizations, yielding stable solutions for benchmark lid-driven cavity flow up to Re=100. These results indicate that KAPI-ELM provides an efficient and unified approach for forward and inverse PDEs, particularly in challenging sharp-gradient regimes.