Finsler Geometry, Graph Neural Networks, and You

2026-06-17 · Source: stat.ML updates on arXiv.org · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Mathematics & Computational Sciences · Depth: Expert, extended

Summary

This research introduces Finslerian graph neural networks (FGNNs) to address the isotropic limitations of traditional GNNs that rely on the graph Laplacian. The authors define an empirical Finsler Laplacian for point clouds, proving its uniform pointwise convergence to the continuous Finsler Laplacian on a manifold as the number of samples n increases, with ε=O(log n/n)^{1/(3d+4)}. This operator is then formulated as a GNN layer, enabling the creation of FGNNs that inherently express Finsler geometry, allowing for anisotropic, asymmetric, and non-elliptical structures. Numerical experiments validate the convergence on a torus, showing increased similarity between true and graph Laplacians as n grows from 250 to 9500. Furthermore, FGNNs significantly outperform standard GNNs in recovering Finsler metrics from observed heat diffusion, achieving an order of magnitude lower mean-squared error and demonstrating superior generalization to unseen graphs.

Key takeaway

For Machine Learning Engineers developing GNNs for complex physical simulations or geometric data, you should consider Finslerian GNNs to model anisotropic phenomena. Your current isotropic GNNs may oversimplify underlying geometries, leading to poor generalization. By incorporating Finsler geometry, you can achieve more accurate and interpretable models, especially for nonlinear diffusion or systems with directional biases. Explore using ReLU activations within FGNNs for optimal performance.

Key insights

Finslerian GNNs model anisotropic manifold geometry, overcoming isotropic limitations of standard graph Laplacians.

Principles

• Graph Laplacians approximate Laplace–Beltrami operators, implying Riemannian geometry.
• Finsler geometry allows anisotropic, asymmetric, non-elliptical manifold structures.
• Nonlinearities in neural architectures can implicitly impose geometric structures.

Method

The empirical Finsler Laplacian is defined using local PCA for gradient estimation, then expressed as a cellular sheaf neural network layer. A learnable matrix W(t) replaces J in Ŵ(xi)=W⁺(t)σ(W(t)ξ).

In practice

• Use Finslerian GNNs for problems requiring anisotropic geometric modeling.
• Employ ReLU activation in FGNNs to capture non-Riemannian Finsler metrics.
• Apply FGNNs to inverse problems like recovering diffusion geometry.

Topics

• Finsler Geometry
• Graph Neural Networks
• Laplace-Beltrami Operator
• Anisotropic Diffusion
• Cellular Sheaves
• Point Cloud Learning

Code references

• tmrod/finsler-gnn

Best for: AI Scientist, Machine Learning Engineer, Research Scientist

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Editorial summary, takeaway, and curation by AIssential. Original article published by stat.ML updates on arXiv.org.