Exact Constraint Enforcement in Physics-Informed Extreme Learning Machines using Null-Space Projection Framework

Physics-informed extreme learning machines (PIELMs) typically impose boundary and initial conditions through penalty terms, yielding only approximate satisfaction that is sensitive to user-specified weights and can propagate errors into the interior solution. This work introduces Null-Space Projected PIELM (NP-PIELM), achieving exact constraint enforcement through algebraic projection in coefficient space. The method exploits the geometric structure of the admissible coefficient manifold, recognizing that it admits a decomposition through the null space of the boundary operator. By characterizing this manifold via a translation-invariant representation and projecting onto the kernel component, optimization is restricted to constraint-preserving directions, transforming the constrained problem into unconstrained least-squares where boundary conditions are satisfied exactly at discrete collocation points. This eliminates penalty coefficients, dual variables, and problem-specific constructions while preserving single-shot training efficiency. Numerical experiments on elliptic and parabolic problems including complex geometries and mixed boundary conditions validate the framework.