Neural networks for variational problems in engineering

Abstract

In this work a conceptual theory of neural networks (NNs) from the perspective of functional analysis and variational calculus is presented. Within this formulation, the learning problem for the multilayer perceptron lies in terms of finding a function, which is an extremal for some functional. Therefore, a variational formulation for NNs provides a direct method for the solution of variational problems.

This proposed method is then applied to distinct types of engineering problems. In particular a shape design, an optimal control and an inverse problem are considered. The selected examples can be solved analytically, which enables a fair comparison with the NN results