NOFE -- Neural Operator Function Embedding

2026-05-12

AI Summary

This work addresses a key limitation of traditional dimensionality reduction methods, which treat data as discrete point clouds and neglect the intrinsic continuous-domain structure of real-world processes. To overcome this, the authors propose a domain-aware continuous dimensionality reduction framework that learns function-to-function mappings via graph-theoretic operators, enabling mesh-free evaluation at arbitrary locations independent of input discretization. By extending neural networks to operate over continuous domains and constructing inter-layer operator mappings, the method preserves structural properties of the underlying function space. Evaluated on the ERA5 climate dataset, the model achieves a remarkably low local Stress of 0.111—significantly outperforming PCA, t-SNE, and UMAP—and reduces stitching error by up to 20-fold while effectively maintaining global structure.

Abstract

Most dimensionality reduction methods treat data as discrete point clouds, ignoring the continuous domain structure inherent to many real-world processes. To bridge this gap, we introduce Neural Operator Function Embedding (NOFE), a domain-aware framework for continuous dimensionality reduction. NOFE learns function-to-function mappings via a Graph Kernel Operator, enabling mesh-free evaluation at arbitrary query locations independent of input discretization. We establish NOFE as approximation of sheaf-to-sheaf mappings, generalizing Sheaf Neural Networks to continuous domains. We evaluate NOFE across different datasets, comparing it against PCA, t-SNE, and UMAP. Our results demonstrate that NOFE significantly outperforms baselines in local structure preservation, achieving a local Stress of 0.111 compared to 0.398 for PCA, 0.773 for t-SNE, and 0.791 for UMAP for the ERA5 climate reanalysis dataset. NOFE also exhibits robust sampling independence, reducing the Patch Stitching Error by up to $20.0\times$ relative to UMAP (59.0 vs. 267.6 under regional normalization) and ensuring consistency across disjoint domain patches. While maintaining competitive global structure preservation (Stress-1: 0.379 vs. PCA's 0.268), NOFE resolves fine-grained structures and produces smooth, consistent embeddings that generalize across varying sample densities, addressing key limitations of discrete reduction methods.

Problem

• dimensionality reduction
• continuous domain
• local structure preservation
• sampling independence
• discrete point clouds

Innovation

• Neural Operator
• Function Embedding
• Continuous Dimensionality Reduction
• Graph Kernel Operator
• Sheaf Neural Networks