Randomized Newton’s Method for Solving Differential Equations Based on the Neural Network Discretization

Qipin Chen, Wenrui Hao

Research output: Contribution to journalArticlepeer-review


We develop a randomized Newton’s method for solving differential equations, based on a fully connected neural network discretization. In particular, the randomized Newton’s method randomly chooses equations from the overdetermined nonlinear system resulting from the neural network discretization and solves the nonlinear system adaptively. We theoretically prove that the randomized Newton’s method has a quadratic convergence locally. We also apply this new method to various numerical examples, from one to high-dimensional differential equations, to verify its feasibility and efficiency. Moreover, the randomized Newton’s method can allow the neural network to “learn” multiple solutions for nonlinear systems of differential equations, such as pattern formation problems, and provides an alternative way to study the solution structure of nonlinear differential equations overall.

Original languageEnglish (US)
Article number49
JournalJournal of Scientific Computing
Issue number2
StatePublished - Aug 2022

All Science Journal Classification (ASJC) codes

  • Software
  • Engineering(all)
  • Computational Mathematics
  • Theoretical Computer Science
  • Applied Mathematics
  • Numerical Analysis
  • Computational Theory and Mathematics


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