I’m currently looking into the different training methods of NN. I’ve implemented a simple gradient descent method using backpropagation in my NN (just a very simple NN with 1 hidden layer). I’m now struggling with implementing a more sophisticated training (weights and biases) algorithm, i.e. the Levenberg-Marquardt. Is there also such an elegant method which uses backpropagation to calculate the jacobian?
Edit: I’ve found  which is a really interesting article and is almost what I’m looking for. Nevertheless, I don’t understand how I propagate through the different layers because I only have the error $e$ for the last layer (output layer). However, I want to adjust two weigth matrices (from input to hidden layer and from hidden layer to output).
 M. T. Hagan and M. B. Menhaj, "Training feedforward networks with the Marquardt algorithm," in IEEE Transactions on Neural Networks, vol. 5, no. 6, pp. 989-993, Nov. 1994, doi: 10.1109/72.329697.