Checkable NTK Positivity and Finite-Width Gradient Descent for Scalar- and Vector-Valued PINNs with Strong-Form, Weak-Form, and Nonlocal Linear Constraints

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 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.