Fourier-Invertible Neural Encoder (FINE) for Homogeneous Flows

Invertible neural architectures have recently attracted attention for their compactness, interpretability, and information-preserving properties. In this work, we propose the Fourier-Invertible Neural Encoder (FINE), which combines invertible monotonic activation functions with reversible filter structures, and could be extended using Invertible ResNets. This architecture is examined in learning low-dimensional representations of one-dimensional nonlinear wave interactions and exact circular translation symmetry. Dimensionality is preserved across layers, except for a Fourier truncation step in the latent space, which enables dimensionality reduction while maintaining shift equivariance and interpretability. Our results demonstrate that FINE significantly outperforms classical linear methods such as Discrete Fourier Transformation (DFT) and Proper Orthogonal Decomposition (POD), and achieves reconstruction accuracy better than conventional deep autoencoders with convolutional layers (CNN) - while using substantially smaller models and offering superior physical interpretability. These findings suggest that invertible single-neuron networks, when combined with spectral truncation, offer a promising framework for learning compact and interpretable representations of physics datasets, and symmetry-aware representation learning in physics-informed machine learning.