# Finsler Geometry, Graph Neural Networks, and You

arXiv: https://arxiv.org/abs/2606.17185 · HTML: https://mlsanity.com/paper/2606.17185/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 30 references that are in the map.
- First posted on arXiv 15 Jun 2026.

## Builds on (most cited first, 15 of 15)

- [On Neural Differential Equations](https://mlsanity.com/paper/2202.02435/) (2022)
- [GRAND: Graph Neural Diffusion](https://mlsanity.com/paper/2106.10934/) (2021)
- [Neural Sheaf Diffusion: A Topological Perspective on Heterophily and Oversmoothing in GNNs](https://mlsanity.com/paper/2202.04579/) (2022)
- [Equinox: neural networks in JAX via callable PyTrees and filtered transformations](https://mlsanity.com/paper/2111.00254/) (2021)
- [Sheaf Neural Networks](https://mlsanity.com/paper/2012.06333/) (2020)
- [Error estimates for spectral convergence of the graph Laplacian on random geometric graphs towards the Laplace--Beltrami operator](https://mlsanity.com/paper/1801.10108/) (2018)
- [Sheaf Neural Networks with Connection Laplacians](https://mlsanity.com/paper/2206.08702/) (2022)
- [Analysis of p-Laplacian Regularization in Semi-Supervised Learning](https://mlsanity.com/paper/1707.06213/) (2017)
- [A variational approach to the consistency of spectral clustering](https://mlsanity.com/paper/1508.01928/) (2015)
- [MSGNN: A Spectral Graph Neural Network Based on a Novel Magnetic Signed Laplacian](https://mlsanity.com/paper/2209.00546/) (2022)
- [Rates of Convergence for Laplacian Semi-Supervised Learning with Low Labeling Rates](https://mlsanity.com/paper/2006.02765/) (2020)
- [A Manifold Perspective on the Statistical Generalization of Graph Neural Networks](https://mlsanity.com/paper/2406.05225/) (2024)
- [Finsler-Laplace-Beltrami Operators with Application to Shape Analysis](https://mlsanity.com/paper/2404.03999/) (2024)
- [Finsler Multi-Dimensional Scaling: Manifold Learning for Asymmetric Dimensionality Reduction and Embedding](https://mlsanity.com/paper/2503.18010/) (2025)
- [Differentiable Randers-Finsler Eikonal Solvers](https://mlsanity.com/paper/2603.00035/) (2026)

## Built on by (most cited first, 1 of 1)

- [mathbbSL(n) Representation Learning: An Intrinsic Mixed-Curvature Space with Higher Curvature Capacities and Deeper Order-Aware Composition](https://mlsanity.com/paper/2609.15083/) (2026)

## Abstract (arXiv)

Graph neural network architectures based on the graph Laplacian approximate the Laplace-Beltrami operator, thus limiting their application to isotropic operators. As a nonlinear alternative to the Laplace-Beltrami operator, we consider estimates of the Finsler Laplacian on point clouds sampled from a manifold. We prove that these discrete estimates converge to the true operator on the manifold as the number of point samples grows. Moreover, we show that this operator can be expressed as a graph neural network layer, which we use to define a family of Finslerian graph neural networks constrained to express Finsler geometry. We show that Finslerian graph neural networks recover the geometry underlying nonlinear diffusion equations in practice.