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

• Published in: Journal of scientific computing Vol. 92; no. 2; p. 49
• Main Authors: Chen, Qipin, Hao, Wenrui
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.08.2022; Springer Nature B.V
• Subjects: Algorithms; Artificial intelligence; Computational mathematics; Computational Mathematics and Numerical Analysis; Discretization; Finite element analysis; Machine learning; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Methods; Neural networks; Nonlinear differential equations; Nonlinear equations; Nonlinear systems; Optimization techniques; Partial differential equations; Theoretical; Neural networks; Random Newton’s method; Differential equations; Convergence analysis
• ISSN: 0885-7474; 1573-7691
• DOI: 10.1007/s10915-022-01905-9

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.