In this article, a new projection neural network (PNN) for solving L1 -minimization problem is proposed, which is based on classic PNN and sliding mode control technique. Furthermore, the proposed network can be used to make sparse signal reconstruction and image reconstruction. First, a sign function is introduced into the PNN model to design fixed-time PNN (FPNN). Then, under the condition that the projection matrix satisfies the restricted isometry property (RIP), the stability and fixed-time convergence of the proposed FPNN are proved by the Lyapunov method. Finally, based on the experimental results of signal simulation and image reconstruction, the proposed FPNN shows the effectiveness and superiority compared with that of the existing PNNs.