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The problem of blind separation of independent sources in non-linear mixtures is considered and the focus of this work is on a new type of non-linear mixture in which a linear mixing matrix is sandwiched between two mutually reverse non-linearities. The demixing system culminates to a novel Weierstrass network that is shown to successfully restore the original source signals under the non-linear mixing conditions. The corresponding parameter learning algorithm for the proposed network is presented through formal mathematical derivation. The authors show for the first time a new result based on the theory of forward series and series reversion that is integrated into a neural network to implement the proposed demixer. Simulations, including both synthetic and recorded signals, have been carried out to verify the efficacy of the proposed method. It is shown that the Weierstrass network outperforms other tested independent component analysis (ICA) methods (linear ICA, radial-basis function and multilayer perceptron network) in terms of speed and accuracy.