Technology Category
Machine Learning: Kernel MethodsSearch and Optimization: Learning to SearchConstraint Satisfaction and Optimization: Constraint Learning and Acquisition
Kernel-Adaptive PI-ELMs for Forward and Inverse Problems in PDEs with Sharp Gradients
🤖 AI Summary
Traditional physics-informed extreme learning machines (PI-ELMs) struggle to resolve sharp gradients in partial differential equations (PDEs) for both forward and inverse problems, as their fixed, randomly initialized input weights lack adaptability to localized steep features. To address this, we propose a kernel-adaptive PI-ELM framework: the input-layer weights are modeled as learnable parameters drawn from a parametric probability distribution, and a lightweight Bayesian optimization scheme is integrated to adaptively estimate distribution hyperparameters—requiring only a single least-squares solve. Our method synergistically combines radial basis function kernels, hard physical constraints, the ELM architecture, and distributed hyperparameter optimization, reducing trainable parameters by ~90%. Evaluated on multiple stiff PDE benchmarks, it achieves state-of-the-art accuracy, matching or surpassing advanced methods such as XTFC.
Technology Category
Application Category
📝 Abstract
This paper introduces the Kernel Adaptive Physics-Informed Extreme Learning Machine (KAPI-ELM), an adaptive Radial Basis Function (RBF)-based extension of PI-ELM designed to solve both forward and inverse Partial Differential Equation (PDE) problems involving localized sharp gradients. While PI-ELMs outperform the traditional Physics-Informed Neural Networks (PINNs) in speed due to their single-shot, least square optimization, this advantage comes at a cost: their fixed, randomly initialized input layer limits their ability to capture sharp gradients. To overcome this limitation, we introduce a lightweight Bayesian Optimization (BO) framework that, instead of adjusting each input layer parameter individually as in traditional backpropagation, learns a small set of hyperparameters defining the statistical distribution from which the input weights are drawn. This novel distributional optimization strategy -- combining BO for input layer distributional parameters with least-squares optimization for output layer network parameters -- enables KAPI-ELM to preserve PI-ELM's speed while matching or exceeding the expressiveness of PINNs. We validate the proposed methodology on several challenging forward and inverse PDE benchmarks, including a 1D singularly perturbed convection-diffusion equation, a 2D Poisson equation with sharp localized sources, and a time-dependent advection equation. Notably, KAPI-ELM achieves state-of-the-art accuracy in both forward and inverse settings. In stiff PDE regimes, it matches or even outperforms advanced methods such as the Extended Theory of Functional Connections (XTFC), while requiring nearly an order of magnitude fewer tunable parameters. These results establish the potential of KAPI-ELM as a scalable, interpretable, and generalizable physics-informed learning framework, especially in stiff PDE regimes.
Problem
Research questions and friction points this paper is trying to address.
Solving PDEs with sharp gradients using adaptive RBF networks
Overcoming fixed input layer limitations in PI-ELMs via Bayesian Optimization
Achieving high accuracy in forward and inverse PDE problems efficiently
Innovation
Methods, ideas, or system contributions that make the work stand out.
Kernel-adaptive RBF extension for sharp gradient PDEs
Lightweight Bayesian Optimization for input layer distribution
Combines BO and least-squares for speed and accuracy
🔎 Similar Papers
No similar papers found.
💼 Related Jobs
No related jobs found.
