Q:

Can neural networks modify their own weights without back-propagation and gradient descent?

Can neural networks modify their own weights without back-propagation and gradient descent?

A:

Artificial neural networks (ANNs) do not modify their own weights! Humans create algorithms that modify the weights or architecture of ANNs.
Having said that, you can change the weights of ANNs using other methods other than gradient descent combined with back-propagation, such as Hebbian learning, evolutionary algorithms or reinforcement learning.

