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

TL;DR

This paper introduces NP-PIELM, a novel method that enforces boundary and initial conditions exactly in physics-informed neural networks by using null-space projection, improving accuracy and efficiency.

Contribution

It develops a null-space projection framework for PIELMs, enabling exact constraint enforcement without penalty terms or additional variables.

Findings

Exact boundary condition satisfaction demonstrated on elliptic and parabolic problems.

Improved accuracy over traditional penalty-based PIELMs.

Efficient single-shot training preserved with the new method.

Abstract

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.