Solving differential equations by neural networks: goal functionals and verification of results

Machine learning methods for neural networks is 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 fine 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. 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.