论文标题
机器学习和计算数学
Machine Learning and Computational Mathematics
论文作者
论文摘要
基于神经网络的机器学习能够以前所未有的效率和准确性在很高的维度上近似功能。这不仅在人工智能的传统领域，而且在科学计算和计算科学领域中开辟了许多令人兴奋的新可能性。同时，机器学习还获得了一组“黑匣子”类型的技巧的声誉，而没有基本的原则。这是在机器学习方面取得进一步进展的真正障碍。在本文中，我们试图解决以下两个非常重要的问题：（1）机器学习如何影响，并将进一步影响计算数学，科学计算和计算科学？ （2）计算数学，尤其是数值分析，{can}如何影响机器学习？我们描述了在这些问题上取得的一些最重要的进展。我们的希望是将事情置于一个视角，将有助于将机器学习与计算数学集成在一起。
Neural network-based machine learning is capable of approximating functions in very high dimension with unprecedented efficiency and accuracy. This has opened up many exciting new possibilities, not just in traditional areas of artificial intelligence, but also in scientific computing and computational science. At the same time, machine learning has also acquired the reputation of being a set of "black box" type of tricks, without fundamental principles. This has been a real obstacle for making further progress in machine learning. In this article, we try to address the following two very important questions: (1) How machine learning has already impacted and will further impact computational mathematics, scientific computing and computational science? (2) How computational mathematics, particularly numerical analysis, {can} impact machine learning? We describe some of the most important progress that has been made on these issues. Our hope is to put things into a perspective that will help to integrate machine learning with computational mathematics.
