High-performance computing has enabled complex, high-dimensional physical simulations across many science and engineering applications. However, downstream analyses such as inverse problems and uncertainty quantification remain computationally expensive since they require many queries of the physical simulations, motivating model order reduction (MOR) to construct efficient low-dimensional surrogates. Proper Orthogonal Decomposition (POD), a widely adopted data-driven MOR method, projects dynamics onto linear subspaces spanned by the most energetic modes. However, POD struggles for problems with slowly decaying Kolmogorov \(n\)-widths, such as advection-dominated and turbulent flows, requiring many modes for accurate reconstruction. Moreover, energy-based selection can discard crucial low-energy modes needed to capture small-scale turbulent features. Recent nonlinear manifold methods using polynomial mappings with alternating~\cite{GW2023} or greedy~\cite{SP2024} mode selection achieve better reconstruction with fewer modes. However, these methods fix the nonlinear mapping form a priori, limiting expressivity. Conversely, neural network (NN) manifolds offer greater expressivity by learning the manifold from data but typically employ energy-based selection. We present SparseModesNet, a dimensionality reduction framework that enhances existing manifold reduction methods with machine learning by employing linear encoding via POD modes and nonlinear NN decoding. The decoder leverages LassoNet~\cite{LRAT2021}, a NN framework that enforces hierarchical sparsity through residual connections with linear skip layers, to simultaneously select informative POD modes and learn a nonlinear mapping that minimizes reconstruction error. On a turbulent channel flow problem at friction Reynolds number \(Re_\tau=5200\), SparseModesNet reduces reconstruction error compared to existing polynomial manifold methods while maintaining interpretability through physically meaningful mode selection.