1. A neural network with ReLU activations (the most common one) effectively has a piecewise linear optimisation surface, so the 2nd derivative is 0 almost everywhere. No curvature, no Hessian.
2. 2nd order methods make an assumption that the optimisation surface is (close to) convex, which is very much not the case with neural networks.
3. Even if the above were not true, to measure the Hessian you would have to iterate through the entire training set to measure it before you could make take an optimisation step.
4. Representing the entire Hessian for a network with N parameters needs O(N²) space, where N is on the order of 10⁹, so you couldn't use a method that needs the whole Hessian. There are so-called Quasi-Newton methods that need O(N) space though, like L-BFGS.
Neural networks need completely different optimisation methods, and there is no practically useful application of any of the Newton or Quasi-Newton methods for their optimisation.