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.
@article{arxiv.2505.15329,
title = {Fourier-Invertible Neural Encoder (FINE) for Homogeneous Flows},
author = {Anqiao Ouyang and Hongyi Ke and Qi Wang},
journal= {arXiv preprint arXiv:2505.15329},
year = {2025}
}