differential-geometry

The paper presents a unified geometric framework for understanding differentiable embeddings like t-SNE and UMAP, showing that existing diagnostics derive from a single object and proposing new curvature-based and integral-path trust measures.

Presents a continuous geometric framework modeling Transformer operations as integro-differential equations on a semantic fiber bundle, validated across multiple architectures.

A blog post explaining Hamiltonian Neural Networks through differential geometry, using a simple mass-spring system to demonstrate how imposing conservation laws via network architecture can lead to more efficient learning. The author builds up mathematical tools like symplectic manifolds and Poisson brackets from basic calculus.

This paper establishes a mathematically rigorous connection between shock-wave theory and symmetry-quotiented learning dynamics of stochastic gradient descent, showing that after symmetry reduction and coarse-graining, the dynamics satisfy viscous Hamilton-Jacobi and Burgers-type equations with shock formation times controlled by loss curvature.

A pictorial introduction to differential geometry, showing how Maxwell's equations emerge from geometric concepts, based on a 2017 arXiv paper.

A tweet emphasizes that differential geometry is a high leverage skill in the current AI landscape, suggesting that developments are still early.