I use for my network QLBAM algorithm

https://keio.pure.elsevier.com/en/publications/new-results-of-quick-learning-for-bidirectional-associative-memor

Thank you very much Mr. Lyakhovetskii for your prompt response but can you please confirm that QLBAM is applicable for training the neural networks or not?

Hm. QLBAM algorithm iteratively optimizes the weighting matrix for improvement of network capacity. Is this "training" or not?

In fact i dont understand your question? Do you want answer about all existing algorithms?

For example here you have some information about the most faster, but in fact it depends on given network, data, and of course problem you want to solve.

http://page.mi.fu-berlin.de/rojas/neural/chapter/K8.pdf

Actually I need the list of available training algorithms plus the various types of backpropagation algorithm

SO you have serious research task:) Nobody will give you ready answer for such question:)

Here are some articles about "training" of BAM networks

Oh H., Kothari S.C. A Pseudo-relaxation learning algorithm for bidirectional associative memory. International J. of Computing, Communications and Networking. 1992. V. 2. P. 208–213.

Xu Z.-B., Leung Y., He X.-W. Asymmetric bidirectional associative memories. IEEE Trans. Systems, Man and Cybernetics. 1994. V. 24. P. 1559–1564.

Wang Y.F., Cruz J.B. Jr., Mulligan J.H. Jr. Two coding strategies for bidirectional associative memory. IEEE Transactions on Neural Networks. 1990. V. 1. № 1. P. 81–92.

Any algorithm can be utilized for optimization can be used to train ANN. GA, swarm, Ant, bees, DA, gray wolf...etc. do not forget that training of ANN is an optimization problem of the cost function

You can use any optimisation algorithm depending upon your data. Levenberg-Marqaurdt algorithm (trainlm in matlab) or Levenberg-Marqaurdt algorithm with Bayesian regularization (trainbr) are widely used for optimisation problems. You can use trainrp ( resilient backpropagation algorithm) or some newton or quasi-Newton methods like trainbfg,  or trainscg (scaled conjugate gradient method) and many more standard training algorithms. Apart from backpropagation you can also use Generalised Regression method.

Are you using any  software package?

Thank you very much to all of you for your guidance and valuable suggestions. I am actually using the MATLAB tool box for training the ANN using different back-propagation based functions. like trainlm and trainbr but I am searching some more back-propagation based functions to compare them.

see here:

http://page.mi.fu-berlin.de/rojas/neural/chapter/K8.pdf