可检验的NTK正定性与标量/向量值PINN在强形式、弱形式和非局部线性约束下的有限宽度梯度下降

中文摘要 AI 辅助

我们给出了极限神经正切核（NTK）的可检验正定性判据，以及具有线性约束的标量或向量值物理信息神经网络（PINN）的高概率有限宽度梯度下降保证。约束可以是任意固定有限阶的强形式微分行，包括耦合系统，并带有任意线性初始或边界条件；弱形式残差和边界泛函；或有限测度非局部观测，如积分、非局部扩散和狄拉克数据行，所有这些均适用于任意维度。对于每一类，极限NTK的正定性等价于固定设计的有限系数、泛函或矩矩阵上的秩条件：一个在训练前计算的证书，用于检测结构零模。模型假设为普通两层PINN的假设，即具有有界对称初始化的光滑非多项式激活函数，并且标准选择如带均匀初始化的$\ anh$满足这些假设。给定证书后，显式的宽度和步长条件确保以高概率经验NTK至少保留极限间隙的一半，并且全批量梯度下降在每次迭代中以几何速率降低训练损失。受控实验检验了证书和有限宽度机制。

英文摘要

We give checkable positive-definiteness criteria for the limiting neural tangent kernel (NTK) and high-probability finite-width gradient-descent guarantees for scalar- or vector-valued physics-informed neural networks (PINNs) with linear constraints. The constraints may be strong-form differential rows of any fixed finite order, including coupled systems, with any linear initial or boundary conditions; weak-form residual and boundary functionals; or finite-measure nonlocal observations such as integral, nonlocal-diffusion, and Dirac-data rows, all in any dimension. For each class, positive definiteness of the limiting NTK is equivalent to a rank condition on a finite coefficient, functional, or moment matrix of the fixed design: a certificate computed before training that detects structural zero modes. The model hypotheses are those of an ordinary two-layer PINN, a smooth nonpolynomial activation with bounded symmetric initialization, and are met by standard choices such as $\tanh$ with uniform initialization. Given a certificate, explicit width and step-size conditions ensure, with high probability, that the empirical NTK retains at least half the limiting gap and that full-batch gradient descent decreases the training loss geometrically at every iteration. Controlled experiments check the certificates and the finite-width mechanism.

发表机构

• ETH Zürich（苏黎世联邦理工学院）

机构由 AI 辅助整理，请以论文原文为准。