Differential geometry and stochastic dynamics with deep learning numerics
Differential geometry and stochastic dynamics with deep learning numerics
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微分几何和随机动力学与深度学习数值
DOI：
10.1016/j.amc.2019.03.044
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发表时间：
2017
期刊：
影响因子：
--
通讯作者：
S. Sommer
中科院分区：
文献类型：
--
作者：
Line Kühnel;Alexis Arnaudon;S. Sommer
随着深度学习方法的出现，新的计算框架被开发出来，将符号表达式与高效的数值计算相结合。在这项工作中，我们将演示如何在这些现代框架中实现流形上的确定性和随机动力学以及微分几何构造。特别是，我们使用了Python库Theano的符号表达和自动区分功能，该库最初是为深度学习中的高性能计算开发的。我们展示了微分几何和李群理论、联络、度量、曲率、左/右不变性、测地线和平行传输的各个方面如何用Theano通过自动计算任意阶导数来表示。我们还将展示如何用几行代码来表示和优化非线性统计中的符号随机积分器和概念。然后，我们将给出用于可视化的低维经典流形的显式例子，并演示该方法如何允许简明的实现和对高维问题的有效缩放。通过本文及其附带的代码，我们希望通过展示生成的代码如何允许在实现许多实验数学工作中的灵活性和简单性，来促进现代符号和数值计算框架在数学实验应用程序、应用数学计算和数据分析中的使用。
With the emergence of deep learning methods, new computational frameworks have been developed that mix symbolic expressions with efficient numerical computations. In this work, we will demonstrate how deterministic and stochastic dynamics on manifolds, as well as differential geometric constructions can be implemented in these modern frameworks. 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 orders. 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. With this paper and its accompanying code, we hope to stimulate the use of modern symbolic and numerical computation frameworks for experimental applications in mathematics, for computations in applied mathematics, and for data analysis by showing how the resulting code allows for flexibility and simplicity in implementing many experimental mathematics endeavors.
影响因子：
3
作者：
Arnaudon A
通讯作者：
Arnaudon A