Machine learning and computational mathematics

E. Weinan

Research output: Contribution to journalArticlepeer-review

34 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)1639-1670
Number of pages32
JournalCommunications in Computational Physics
Issue number5
StatePublished - Nov 2020

All Science Journal Classification (ASJC) codes

  • Physics and Astronomy (miscellaneous)


  • Machine learning-based algorithm
  • Neural network-based machine learning


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