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

TL;DR

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, provides an efficient and unified approach for forward and inverse PDEs, particularly in challenging sharp-gradient regimes.

Abstract

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.

Figures (14)

• Figure 1: Conceptual continuum of physics-informed learners. PINNs train all input and output weights via backpropagation, offering flexibility but at a high computational cost. PI-ELMs fix random input weights and determine only the output weights in a single shot least square solve. KAPI-ELM bridges these extremes through distributional optimization of input parameters, achieving adaptive kernel learning without backpropagation while retaining the analytic least-squares solution for output weights.
• Figure 2: Predicted and exact solutions (left) and corresponding RBF distributions (right) for the single boundary-layer problem across stiffness levels. As $\nu$ decreases, the adaptive component automatically concentrates RBF centers near the outflow boundary ($x=1$), capturing the thin layer without manual refinement.
• Figure 3: Predicted versus exact solutions (left) and corresponding RBF distributions (right) for the twin boundary-layer problem. The two adaptive Gaussian components automatically localize near $x=0$ and $x=1$, resolving both layers symmetrically as $\nu$ decreases.
• Figure 4: KAPI-ELM solution (left), finite-difference reference (center), and absolute error (right) for the 2D Poisson problem with $\nu=10^{-2}$. The adaptive component concentrates RBFs near $(0.5,0.5)$, sharply resolving the localized Gaussian source. Residuals remain below $10^{-4}$, and the mean-squared error is $\mathrm{MSE}=4.96\times10^{-8}$, demonstrating high accuracy and numerical stability.
• Figure 5: Evolution of the adaptive RBF kernel distribution during Bayesian optimization for the 2D Poisson problem ($\nu = 10^{-2}$). Each panel shows the RBF centers overlaid on the fixed baseline grid, colored by $\log_{10}(\sigma)$. As optimization proceeds (top-left to bottom-right), the adaptive RBFs progressively concentrate near the localized Gaussian source at $(x,y)\approx(0.5,0.5)$, while kernel widths shrink to resolve the steep gradients. The decreasing $\log_{10}(J)$ values indicate convergence toward the optimal configuration.