Differential geometry and stochastic dynamics with deep learning numerics

TL;DR

This paper demonstrates how modern computational frameworks like Theano can efficiently implement deterministic and stochastic differential geometry on manifolds, enabling concise code and scalable high-dimensional problems.

Contribution

It introduces a method to implement differential geometric concepts and stochastic integrators using symbolic computation and automatic differentiation in Theano.

Findings

Efficient implementation of geometric concepts using Theano

Concise formulation of stochastic integrators and non-linear statistics

Scalable approach demonstrated on high-dimensional problems

Abstract

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.