# Multilayer Perceptrons for solving variational problems

Can we use a multilayer perceptron to solve variational problems?

By variational problem I mean something we might encounter in the calculus of variations, for example the geodesic problem: given two points A and B in the plane, find the curve of shortest length going from A to B. This is of course not the most interesting problem since the answer is just a line segment, but I'll use it anyways. The mathematical version of the constrained geodesic problem would be:

Find $$y(x)$$ defined on $$[0,1]$$ such that $$y(0)=1, y(1)=0$$ and for which the functional $$L[y(x)]=\int_{0}^{1}\sqrt{1+y'(x)^2}$$ is minimized.

For pattern recognition problems we might train an MLP with some training patterns and adjust the weights and biases so as to minimize and error function, which measures how accurately the net outputs come to the desired outputs for the training samples. We can do this for example by gradient descent. It is well known that multilayer perceptrons (MLP) can approximate any function we want, so if our algorithms work well we should be able to make the error function as small as possible.

Idea: Instead of using an error function based on training data, can we use a neural net to minimize a functional $$L[y(x)]$$ by gradient descent? The net would have one input $$x$$ and one output $$y(x)$$, and some hidden neurons and the goal would be to tweak the weights and biases until the functional is minimized.

Is there any merit to this?