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In this paper, we present two highly efficient second-order algorithms for the training of multilayer feedforward neural networks. The algorithms are based on iterations of the form employed in the Levenberg-Marquardt (LM) method for nonlinear least squares problems with the inclusion of an additional adaptive momentum term arising from the formulation of the training task as a constrained optimization problem. Their implementation requires minimal additional computations compared to a standard LM iteration which are compensated, however, from their excellent convergence properties. Simulations to large scale classical neural-network benchmarks are presented which reveal the power of the two methods to obtain solutions in difficult problems, whereas other standard second-order techniques (including LM) fail to converge. Index Terms--Algorithms, multilayer feedforward neural networks, nonlinear least-squares, optimization, supervised learning.