Scholarlink

Abstract <p>Machine learning methods for neural networks are a new approach used in mathematical modeling. In recent years, they have been used to solve various problems based on differential equations. In this paper, we discuss two closely related questions: reliable verification of the accuracy and proper selection of the goal functional used in the process of supervised machine learning. For these purposes, we adapt the theory of a posteriori estimates of the functional type and show that they can provide reliable control of the accuracy of a network solution that works well for sufficiently accurate approximations as well as for coarse ones. Moreover, the proposed method allows you to identify situations, where the neural network is unable to construct an acceptable solution. Analysis of several negative examples leads to the conclusion that in some cases machine learning methods with collocation type goal functionals may lead to locking-type effects. The new goal functionals introduced in the paper are free from disadvantages of this kind. Network solutions generated by different goal functionals have been compared in numerous numerical tests. The results are discussed in the last part of the&#xa0;paper.</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

Solving Differential Equations by Neural Networks: Goal Functionals and Verification of Results

  • 期刊论文

  • 发表日期：

    2025年4月25日

查看全文
  • A. V. Muzalevskii,
  • S. I. Repin

摘要

Abstract

Machine learning methods for neural networks are a new approach used in mathematical modeling. In recent years, they have been used to solve various problems based on differential equations. In this paper, we discuss two closely related questions: reliable verification of the accuracy and proper selection of the goal functional used in the process of supervised machine learning. For these purposes, we adapt the theory of a posteriori estimates of the functional type and show that they can provide reliable control of the accuracy of a network solution that works well for sufficiently accurate approximations as well as for coarse ones. Moreover, the proposed method allows you to identify situations, where the neural network is unable to construct an acceptable solution. Analysis of several negative examples leads to the conclusion that in some cases machine learning methods with collocation type goal functionals may lead to locking-type effects. The new goal functionals introduced in the paper are free from disadvantages of this kind. Network solutions generated by different goal functionals have been compared in numerous numerical tests. The results are discussed in the last part of the paper.