Exact Constraint Enforcement in Physics-Informed Extreme Learning Machines using Null-Space Projection Framework
Rishi Mishra, Smriti, Balaji Srinivasan, Sundararajan Natarajan, Ganapathy Krishnamurthi
TL;DR
This work introduces NP-PIELM, a method that enforces linear boundary and initial constraints exactly by projecting the PIELM coefficient vector onto the null space of the boundary operator, converting constrained optimization into an unconstrained least-squares problem on a constraint-preserving manifold. By parameterizing feasible coefficients as $\boldsymbol{\beta}=\boldsymbol{\beta}_{\star}+\mathbf{W}\boldsymbol{\eta}$, the approach guarantees $\mathbf{B}_{bc}\boldsymbol{\beta}=\boldsymbol{g}$ for all $\boldsymbol{\eta}$ and solves a reduced least-squares problem to obtain the optimal $\boldsymbol{\eta}$, with the final coefficients recovered as $\boldsymbol{\beta}^{\star}=\boldsymbol{\beta}_{\star}+\mathbf{W}\boldsymbol{\eta}^{\star}$. The framework avoids penalty terms, dual variables, and geometry-specific constraint constructions, while remaining scalable to arbitrary geometries and higher dimensions. Numerical experiments on 1D and 2D elliptic and parabolic problems—including complex domains and coupled flows—demonstrate machine-precision enforcement of constraints and superior accuracy. The methodology offers a dimensionally robust, computationally efficient alternative to penalty-based and variational constraint approaches, with broad applicability in physics-informed learning and scientific computing.
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.