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To overcome the intrinsic weaknesses of conventional back-propagation (BP) neural networks, a novel type of feed-forward neural network is constructed in this paper, which adopts a three-layer structure but with the hidden-layer neurons activated by a group of Euler polynomials. A weights-direct-determination (WDD) method is thus able to be derived for it, which obtains the optimal weights of the neural network directly (i.e., just in one step). Furthermore, a structure-automatic-determination (SAD) algorithm is presented to determine the optimal number of hidden-layer neurons of the Euler neural network (ENN). Computer-simulations substantiate the efficacy of such a Euler neural network with its WDD and SAD algorithms.