用投影法让神经网络精确满足物理边界条件，无需调参
Exact Constraint Enforcement in Physics-Informed Extreme Learning Machines using Null-Space Projection Framework
- 通过系数空间投影，直接在数学上保证边界条件成立
- 在椭圆和抛物型方程上实现零误差的边界满足，精度优于传统方法
- 适合需要高精度物理约束的科学计算场景，如流体模拟
物理信息极端学习机（PIELM）通常通过惩罚项施加边界和初始条件，仅能近似满足，且对用户设定的权重敏感，误差可能传播至内部解。本文提出零空间投影物理信息极端学习机（NP-PIELM），通过系数空间中的代数投影实现约束的精确满足。该方法利用可允许系数流形的几何结构，识别出其可通过边界算子零空间进行分解。通过构造平移不变表示并投影到核分量，优化被限制在保持约束的方向上，将有约束问题转化为无约束最小二乘问题，使边界条件在离散配点处精确满足。该方法无需惩罚系数、对偶变量或特定问题构造，同时保持单次训练效率。在椭圆与抛物型问题上的数值实验，包括复杂几何与混合边界条件，验证了该框架的有效性。
原文摘要 · Abstract (English)
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.
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