TY - JOUR
T1 - Machine learning and computational mathematics
AU - Weinan, E.
N1 - Funding Information:
I am very grateful to my collaborators for their contribution to the work described here. In particular, I would like to express my sincere gratitude to Roberto Car, Jiequn Han, Arnulf Jentzen, Qianxiao Li, Chao Ma, Han Wang, Stephan Wojtowytsch, and Lei Wu for the many discussions that we have had on the issues discussed here. This work is supported in part by a gift to the Princeton University from iFlytek as well as the ONR grant N00014-13-1-0338.
Publisher Copyright:
© 2020 Global-Science Press
PY - 2020/11
Y1 - 2020/11
N2 - 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.
AB - 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.
KW - Machine learning-based algorithm
KW - Neural network-based machine learning
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U2 - 10.4208/CICP.OA-2020-0185
DO - 10.4208/CICP.OA-2020-0185
M3 - Article
AN - SCOPUS:85097479244
SN - 1815-2406
VL - 28
SP - 1639
EP - 1670
JO - Communications in Computational Physics
JF - Communications in Computational Physics
IS - 5
ER -