Fourier-Invertible Neural Encoder (FINE) for Homogeneous Flows

TL;DR

This work introduces Fourier-Invertible Neural Encoder (FINE), a translation-equivariant, invertible autoencoder that uses a Fourier truncation bottleneck for interpretable dimension reduction of homogeneous flow fields. FINE combines invertible convolutions with strictly monotonic activations and spectral-parameterized filters to preserve information flow while retaining only select Fourier modes. Across 1D toy signals, the Kuramoto–Sivashinsky turbulence, a 2D separable field, and 2D isotropic turbulence benchmarks, FINE achieves substantially lower reconstruction errors with far fewer parameters than CNN-based autoencoders or attention models. The results demonstrate that symmetry-aware, invertible architectures yield high-fidelity reconstructions with compact latent representations, with potential impact on reduced-order modeling and physics-informed learning. Future work includes theoretical analysis, scaling to time-dependent 2D/3D flows, and integrating FINE with operator-learning frameworks such as Fourier Neural Operators or Koopman autoencoders.

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.

Figures (12)

• Figure 1: Schematic of the FINE architecture. The encoder consists of a cascade of invertible convolutions and monotonic nonlinearities, followed by a Fourier truncation bottleneck that performs interpretable, symmetry-preserving dimension reduction.
• Figure 2: The architecture of the CNN models used in comparison in the one-dimensional K-S dataset.
• Figure 3: Performance comparison for different latent space between convolutional neural network (CNN) and Fourier-Invertible neural encoder (FINE) models on Toy Problem.
• Figure 4: Reconstruction results on the Toy Problem 1, using a latent dimension of $N=2$.
• Figure 5: Sample evolution of the K--S equation solution.