AI 中文总结
本文提出一个统一框架，将物理信息神经网络与有限元方法通过函数型Gauss--Newton问题的离散化联系起来，并利用Petrov--Galerkin离散化，在椭圆问题上验证了弱残差公式与混合有限元--神经逼近的有效性。
AI 中文摘要
物理信息神经网络和有限元方法为偏微分方程的数值逼近提供了两种不同的范式：前者通常通过最小化逐点强残差来训练，而后者则天然地基于弱变分公式以及离散化后得到的有限维系统。在本工作中，我们引入了一个基于通过有限族线性测量对函数型Gauss--Newton问题进行离散化的统一框架。我们证明，通过适当的对偶配对，线性测量可以用测试函数来表示。由此产生的Gauss--Newton系统恰好是线性化函数问题的Petrov--Galerkin离散化。这一视角将逐点配点和自然梯度构造作为特例加以恢复，同时使测试函数的选择成为显式的算法设计选择。我们将该框架专门应用于椭圆问题，在此类问题中，它自然地引出了弱残差公式以及一种作用于互补逼近空间的混合有限元--神经构造。数值实验支持所提出的框架，并展示了弱Gauss--Newton公式和混合有限元--神经逼近的有效性。
英文摘要
Physics-informed neural networks and finite element methods provide two different paradigms for the numerical approximation of partial differential equations: the former are commonly trained by minimizing pointwise strong residuals, whereas the latter are naturally built from weak variational formulations and the finite-dimensional systems obtained after discretization. In this work, we introduce a common framework based on the discretization of functional Gauss--Newton problems by finite families of linear measurements. We show that, through an appropriate duality pairing, the linear measurements can be represented by test functions. The resulting Gauss--Newton system is then precisely a Petrov--Galerkin discretization of the linearized functional problem. This perspective recovers pointwise collocation and natural-gradient constructions as particular cases, while making the choice of test functions an explicit algorithmic design choice. We specialize this framework to elliptic problems, where it naturally leads to weak residual formulations and to a hybrid finite element--neural construction acting on complementary approximation spaces. Numerical experiments support the proposed framework and demonstrate the effectiveness of weak Gauss--Newton formulations and hybrid finite element--neural approximations.
