Sparse POD Mode Selection and Manifold Dimensionality Reduction with Neural Networks

Linear dimensionality reduction methods such as proper orthogonal decomposition (POD) make high-dimensional data amenable to analysis by identifying the principal components, or modes, that capture the most variance, or energy, in the data and constructing a low-dimensional representation in the subspace they span.

Such linear methods struggle, however, for data with slowly decaying Kolmogorov $n$-widths, such as advection-dominated and turbulent flows, which require many modes for accurate reconstruction; moreover, energy-based truncation can discard 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, but fix the form of the nonlinear mapping a priori, limiting expressivity.

In contrast, neural network (NN) manifolds offer greater expressivity yet employ energy-based selection.

We present SparseModesNet, a dimensionality reduction framework that employs linear encoding and nonlinear NN decoding.

The decoder leverages LassoNet, a method enforcing hierarchical sparsity through a residual connection with a linear skip layer, to simultaneously select informative 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 $Re_\tau = 5200$, our method reduces reconstruction error by 51-78% compared to existing polynomial manifold methods while maintaining interpretability through physically meaningful mode selection.