Solving Variational Problems and Integral Equations with Neural Networks

Variational methods, a branch of functional analysis, seek a stationary function that maximizes or minimizes a functional. On the other hand, neural networks are artificial systems that perform complex information processing. The two are closely related, and neural networks may be used to solve variational problems. In the first part of this paper, we propose a new method for solving variational problems using neural networks. In the previous paper (2024), the Euler equation from variational method was applied to the problem of solving a differential equation. For this reason, a two-step approach is considered in which the Euler equation is derived and the solution is obtained by a neural network. The one-step approach proposed here finds the stationary function directly. Both methods yield the same solution, but finding the solution is not easy for complex Euler equations. Therefore, the applicable problem of two-step approach is limited. The second part deals with integral equations, which is another theme of functional analysis. Integral equations are closely related to differential equations, and depending on the problem, both have the same solution. For this reason, the neural method can be applied in the same way as for differential equations. We will show that accurate solutions can be obtained by applying this method to some problems.