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We assess a neural network (NN) method for reconstructing 3D cosmological density and velocity fields (target) from discrete and incomplete galaxy distributions (input). We employ second-order Lagrangian Perturbation Theory to generate a large ensemble of mock data to train an autoencoder (AE) architecture with a Mean Squared Error (MSE) loss function. The AE successfully captures nonlinear features arising from gravitational dynamics and the discreteness of the galaxy distribution. It preserves the positivity of the reconstructed density field and exhibits a weaker suppression of the power on small scales than the traditional linear Wiener filter (WF), which we use as a benchmark. In the density reconstruction, the reduction of the AE MSE relative to the WF is $\sim 15 \%$, whereas, for the velocity reconstruction, a relative reduction of up to a factor of two can be achieved. The AE is advantageous to the WF at recovering the distribution of the target fields, especially at the tails. In fact, trained with an MSE loss, any NN estimate approaches the unbiased mean of the underlying target given the input. This implies a slope of unity in the linear regression of the true on the NN-reconstructed field. Only for the special case of Gaussian fields, the NN and WF estimates are equivalent. Nonetheless, we also recover a linear regression slope of unity for the WF with non-Gaussian fields.