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NOFE – Neural Operator Function Embedding

NeurIPS 2026 · Probabilistic and scientific modeling › Molecules and proteins · approach: Sparse graphical and tensor models · 0 citations (OpenAlex)

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Proposes NOFE, a graph-kernel-operator neural operator that maps a continuous vector field to a lower-dimensional continuous field for domain-aware, mesh-free dimensionality reduction, framed as a sheaf morphism.

Problem

Standard dimensionality reduction (PCA, t-SNE, UMAP) treats data as discrete point clouds and ignores the continuous domain structure of underlying functions, yielding inconsistent embeddings across patches and no evaluation at arbitrary query locations.

Approach

Learn a function-to-function dimensionality reduction operator with a graph kernel operator whose graph is built from domain coordinates and whose node features are function values, trained with a t-SNE-style KL loss on domain-neighbor affinities; the operator is motivated as a sheaf morphism generalizing sheaf neural networks to continuous domains.

Key results

• On ERA5, NOFE achieves local Stress 0.111 vs 0.398 (PCA), 0.773 (t-SNE), 0.791 (UMAP).
• Patch-stitching error (region normalization) 13.0 for NOFE vs 22.2 PCA, 59.0 t-SNE, 267.6 UMAP; gluing MSE 0.080 vs 1.369 for PCA.
• Global Stress-1 is worse for NOFE (0.379) than PCA (0.268), which is best globally.

Surprising findings

• PCA is best on global Stress-1 and continuity (Lipschitz ratio) despite NOFE winning local metrics.
• Ablation shows lifting dimension W has the largest effect on loss while other hyperparameters have negligible effect; some stress metrics are near-identical across configurations.

Limitations

• Scaling with spatial points and neighborhood size yields high memory and training time on large grids.
• Performance is sensitive to hyperparameters (k, T), requiring dataset-specific tuning.
• k-NN neighborhoods break the strict locality claimed by the theory; reliance on a meaningful distance metric.

Experimental setup

Datasets: ERA5 reanalysis (550 hPa, Europe grid) Human Connectome Project Young Adult 7T diffusion MRI (16-subject subset) satellite images

Benchmarks: ERA5 climate reanalysis (Jan 2019 evaluation) Human Connectome Project diffusion MRI (frontal region)

Metrics: Stress-1 Stress-local Patch-Stitching Error (region and local normalization) Lipschitz ratio Gluing MSE Pairwise correlation super-resolution stress

Baselines: PCA t-SNE UMAP

Builds on: Graph Kernel Operator Sheaf Neural Networks t-SNE Vector Diffusion Maps

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