# Differential geometry and stochastic dynamics with deep learning numerics

arXiv: https://arxiv.org/abs/1712.08364 · HTML: https://mlsanity.com/paper/1712.08364/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Dec 2017.

## Built on by (most cited first, 4 of 4)

- [A Survey of Geometric Optimization for Deep Learning: From Euclidean Space to Riemannian Manifold](https://mlsanity.com/paper/2302.08210/) (2023)
- [Legendre Deep Neural Network (LDNN) and its application for approximation of nonlinear Volterra Fredholm Hammerstein integral equations](https://mlsanity.com/paper/2106.14320/) (2021)
- [Learning landmark geodesics using Kalman ensembles](https://mlsanity.com/paper/2103.14076/) (2021)
- [Second-order differential operators, stochastic differential equations and Brownian motions on embedded manifolds](https://mlsanity.com/paper/2406.02879/) (2024)

## Abstract (arXiv)

In this paper, we demonstrate how deterministic and stochastic dynamics on manifolds, as well as differential geometric constructions can be implemented concisely and efficiently using modern computational frameworks that mix symbolic expressions with efficient numerical computations. In particular, we use the symbolic expression and automatic differentiation features of the python library Theano, originally developed for high-performance computations in deep learning. We show how various aspects of differential geometry and Lie group theory, connections, metrics, curvature, left/right invariance, geodesics and parallel transport can be formulated with Theano using the automatic computation of derivatives of any order. We will also show how symbolic stochastic integrators and concepts from non-linear statistics can be formulated and optimized with only a few lines of code. We will then give explicit examples on low-dimensional classical manifolds for visualization and demonstrate how this approach allows both a concise implementation and efficient scaling to high dimensional problems.