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.

翻译：物理信息极限学习机（PIELMs）通常通过惩罚项施加边界和初始条件，这仅能实现近似满足，且对用户指定的权重敏感，可能导致误差传播至内部解。本文引入零空间投影PIELM（NP-PIELM），通过在系数空间进行代数投影实现精确约束实施。该方法利用可容许系数流形的几何结构，认识到其可通过边界算子的零空间进行分解。通过平移不变表示刻画该流形并投影至核分量，优化过程被限制在保持约束的方向上，从而将有约束问题转化为无约束最小二乘问题，其中边界条件在离散配置点上被精确满足。这消除了惩罚系数、对偶变量和针对特定问题的构造，同时保持了单次训练的效率。在包括复杂几何和混合边界条件的椭圆型及抛物型问题上的数值实验验证了该框架的有效性。