Exact Constraint Enforcement in Physics-Informed Extreme Learning Machines using Null-Space Projection Framework
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arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866912833262845952 |
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| author | Mishra, Rishi Smriti Srinivasan, Balaji Natarajan, Sundararajan Krishnamurthi, Ganapathy |
| author_facet | Mishra, Rishi Smriti Srinivasan, Balaji Natarajan, Sundararajan Krishnamurthi, Ganapathy |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_10999 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Exact Constraint Enforcement in Physics-Informed Extreme Learning Machines using Null-Space Projection Framework Mishra, Rishi Smriti Srinivasan, Balaji Natarajan, Sundararajan Krishnamurthi, Ganapathy Numerical Analysis Machine Learning 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. |
| title | Exact Constraint Enforcement in Physics-Informed Extreme Learning Machines using Null-Space Projection Framework |
| topic | Numerical Analysis Machine Learning |
| url | https://arxiv.org/abs/2601.10999 |