The article presents a method for learning the weights in one-layer feedforward neural networks minimizing either the sum of squared errors or the maximum absolute error, measured in the input scale. This leads to the existence of a global optimum that can be easily obtained solving linear systems of equations or linear programming problems, using much less computational power than the one associated with the standard methods. Another version of the method allows computing a large set of estimates for the weights, providing robust, mean or median, estimates for them, and the associated standard errors, which give a good measure for the quality of the fit. Later, the standard one-layer neural network algorithms are improved by learning the neural functions instead of assuming them known. A set of examples of applications is used to illustrate the methods. Finally, a comparison with other high-performance learning algorithms shows that the proposed methods are at least 10 times faster than the fastest standard algorithm used in the comparison.