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Two novel neural data fusion algorithms based on a linearly constrained least square (LCLS) method are proposed. While the LCLS method is used to minimize the energy of the linearly fused information, two neural-network algorithms are developed to overcome the ill-conditioned and singular problems of the sample covariance matrix arisen in the LCLS method. The proposed neural fusion algorithms are samples for implementation using both software and hardware. Compared with the existing fusion methods, the proposed neural data fusion method has an unbiased statistical property and does not require any a priori knowledge about the noise covariance. It is shown that the proposed neural fusion algorithms converge globally to the optimal fusion solution when the sample covariance matrix is singular, and converge globally with exponential rate when the sample covariance matrix is nonsingular. We apply the proposed neural fusion method to image and signal fusion, and it is shown that the quality of the solution can be greatly enhanced by the proposed technique