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

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.