Fourier-Invertible Neural Encoder (FINE) for Homogeneous Flows

TL;DR

FINE is a novel neural encoder that uses Fourier truncation and reversible filters to achieve efficient, interpretable, and symmetry-preserving dimension reduction in physical datasets, outperforming autoencoders in accuracy and parameter efficiency.

Contribution

Introduces FINE, a Fourier-based invertible neural architecture that preserves translational symmetry and enhances interpretability in dimension reduction tasks.

Findings

FINE achieves 4.9-9.1 times lower reconstruction error than convolutional autoencoders.

FINE uses only 13-21% of the parameters of traditional autoencoders.

FINE effectively models complex physical systems with minimal latent dimensions.

Abstract

We present the Fourier-Invertible Neural Encoder (FINE), a compact and interpretable architecture for dimension reduction in translation-equivariant datasets. FINE integrates reversible filters and monotonic activation functions with a Fourier truncation bottleneck, achieving information-preserving compression that respects translational symmetry. This design offers a new perspective on symmetry-aware learning, linking spectral truncation to group-equivariant representations. The proposed FINE architecture is tested on one-dimensional nonlinear wave interaction, one-dimensional Kuramoto-Sivashinsky turbulence dataset, and a two-dimensional turbulence dataset. FINE achieves an overall 4.9-9.1 times lower reconstruction error than convolutional autoencoders while using only 13-21% of their parameters. The results highlight FINE's effectiveness in representing complex physical systems with minimal dimension in the latent space. The proposed framework provides a principled framework for interpretable, low-parameter, and symmetry-preserving dimensional reduction, bridging the gap between Fourier representations and modern neural architectures for scientific and physics-informed learning.