Functional Analysis using Neural Networks

Abstract

We propose a new method for solving mathematical problems by using neural networks. This method is a kind of analog computation and is quite different from conventional solving methods, so it is expected that this method can solve problems that are intractable for conventional methods.
The procedure for solution is as follows. We first prepare the circuit elements such as: function generators constructed by neural networks, differentiators, integrators, multipliers and adders. A problem equation is then implemented by using these circuit elements. A function signal is generated by the neural networks and the output of the problem circuit is observed, and the neural networks are trained so that the output will approache zero. If the training is completed, then we can obtain the solution to the problem. We simulated the method on differential equations and variation problems.