Sparse POD Mode Selection and Manifold Dimensionality Reduction with Neural Networks

2026-05-26

AI Summary

This work addresses the limitations of traditional Proper Orthogonal Decomposition (POD) in convection-dominated and turbulent systems, where slow Kolmogorov n-width decay necessitates a large number of modes and often overlooks critical low-energy structures. The authors propose a novel framework that combines linear POD encoding with nonlinear neural network decoding, uniquely integrating LassoNet’s hierarchical sparsity mechanism into a joint optimization of mode selection and manifold learning. This approach simultaneously minimizes reconstruction error and automatically identifies the most informative modes, balancing representational capacity, accuracy, and physical interpretability. In benchmark tests involving convection-dominated and chaotic flows, the method matches or exceeds state-of-the-art performance; notably, for turbulent channel flow at $Re_\tau = 5200$, it reduces reconstruction error by 51%–78% compared to polynomial manifold-based methods.

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.

Problem

• model order reduction
• Proper Orthogonal Decomposition
• manifold learning
• mode selection
• high-dimensional systems

Innovation

• Sparse POD mode selection
• Neural network manifold
• LassoNet
• Model order reduction
• Nonlinear dimensionality reduction