Sparse POD Mode Selection and Manifold Dimensionality Reduction with Neural Networks

• URL: http://arxiv.org/abs/2605.27756v1
• Date: Tue, 26 May 2026 23:10:05 GMT
• Title: Sparse POD Mode Selection and Manifold Dimensionality Reduction with Neural Networks
• Abstract summary: We present SparseModesNet, a dimensionality reduction framework that employs linear encoding via POD modes and nonlinear NN decoding.<n>On benchmark advectiondominated and chaotic flows, SparseModesNet matches or exceeds state-of-the-art performance.
• Score: 0.2609784101826761
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: High-performance computing enables simulation of high-dimensional physical systems, but downstream analyses such as inverse problems and control remain computationally expensive, 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 features. Recent nonlinear manifold methods using polynomial mappings with alternating or greedy 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 but employ energy-based selection. We present SparseModesNet, a dimensionality reduction framework that employs linear encoding via POD modes and nonlinear NN decoding. The decoder leverages LassoNet, a method enforcing 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 benchmark advection-dominated and chaotic flows, SparseModesNet matches or exceeds state-of-the-art performance. For turbulent channel flow at friction Reynolds number \(Re_τ=5200\), we reduce reconstruction error by 51--78\% compared to existing polynomial manifold methods while maintaining interpretability through physically meaningful mode selection.

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