用傅里叶截断实现可逆降维，高效保留物理对称性。
Fourier-Invertible Neural Encoder (FINE) for Homogeneous Flows
- 结合可逆滤波与傅里叶截断，保持平移对称性。
- 重建误差比卷积自编码器低4.9至9.1倍，参数仅用13%-21%。
- 适合需可解释性与物理对称性的科学计算场景。
我们提出傅里叶可逆神经编码器（FINE），一种针对平移等变数据集的紧凑且可解释的降维架构。FINE 将可逆滤波器与单调激活函数结合傅里叶截断瓶颈，实现尊重平移对称性的信息保真压缩。该设计为对称感知学习提供了新视角，将谱截断与群等变表示相联系。在一维非线性波相互作用、一维 Kuramoto-Sivashinsky 湍流及二维湍流数据集上进行测试，FINE 的重建误差相比卷积自编码器降低 4.9–9.1 倍，仅使用其 13–21% 的参数。结果表明，FINE 能以极小的潜在空间维度有效表示复杂物理系统。该框架为可解释、低参数、保持对称性的降维提供了理论基础，弥合了傅里叶表示与现代神经架构在科学与物理信息学习间的鸿沟。
原文摘要 · Abstract (English)
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.
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