This short, self-contained article seeks to introduce and survey
continuous-time deep learning approaches that are based on neural ordinary
differential equations (neural ODEs). It primarily targets readers familiar
with ordinary and partial differential equations and their analysis who are
curious to see their role in machine learning. Using three examples from
machine learning and applied mathematics, we will see how neural ODEs can
provide new insights into deep learning and a foundation for more efficient
algorithms.

本短文自给自足地介绍和调查了基于神经常微分方程（神经 ODE）的连续时间深度学习方法，主要面向熟悉普通微分方程和偏微分方程及其分析的读者，意在揭示它们在机器学习中的作用。通过使用机器学习和应用数学领域的三个示例，我们将看到神经 ODE 如何提供深度学习的新见解，并为更高效的算法打下基础。

连续时间深度学习的微分方程

Differential Equations for Continuous-Time Deep Learning

In applied mathematics and related disciplines, the
modeling-simulation-optimization workflow is a prominent scheme, with
mathematical models and numerical algorithms playing a crucial role. For these
types of mathematical research data, the Mathematical Research Data Initiative
has developed, merged and implemented ontologies and knowledge graphs. This
contributes to making mathematical research data FAIR by introducing semantic
technology and documenting the mathematical foundations accordingly. Using the
concrete example of microfracture analysis of porous media, it is shown how the
knowledge of the underlying mathematical model and the corresponding numerical
algorithms for its solution can be represented by the ontologies.

使用本体论的方法，将应用数学中的建模 - 仿真 - 优化工作流的数学模型和数值算法表示出来，以构建 FAIR 数学研究数据。通过介绍多孔介质微裂纹分析的具体例子，展示了本体论如何表征基础数学模型及其相应的数值算法。