The edge flow reconstruction task improves the integrity and accuracy of edge flow data by recovering corrupted or incomplete signals. This can be solved by a regularized optimization problem, and the corresponding regularizers are chosen based on prior knowledge. However, obtaining prior information is challenging in some fields. Thus, we consider exploiting the learning ability of neural networks to acquire prior knowledge. In this paper, we propose a new optimization problem for the simplicial edge flow reconstruction task, the simplicial ElasticNet, which is a regularized optimization problem that combines the advantages of the L1 and L2 norm. It is solved iteratively by the multi-block ADMM algorithm, and the convergence conditions are illustrated. By unrolling the simplicial ElasticNet's iterative steps, we propose a neural network with high interpretability and low requirement for the number of training data for the reconstruction task of simplicial edge flows. The unrolling network replaces the fixed parameters in the iterative algorithm with the learnable weights in the neural networks, thus exploiting the neural network's learning capability while preserving the iterative algorithm's interpretability. The core component of this unrolling network is simplicial convolutional filters with learnable weights to aggregate information from the edge flow neighbors, thus enhancing the learning and expressive ability of the network. We conduct extensive experiments on real-world and artificial datasets to validate the proposed approach. It is demonstrated that the simplicial unrolling network is significantly more advantageous than the traditional iterative algorithms and standard non-model-based neural networks in the case of limited training data.