A Connectionist Network Approach to Find Numerical Solutions of Diophantine Equations
This addresses a fundamental problem in mathematics and computational theory for researchers in AI and number theory, but it appears incremental as it applies existing neural network techniques to a new domain.
The paper tackles the problem of finding numerical solutions for Diophantine equations, a key aspect of Hilbert's tenth problem, by proposing a connectionist network approach that uses a three-layer feed-forward neural network with backpropagation and momentum, and validates it on equations with varying numbers of variables and powers.
The paper introduces a connectionist network approach to find numerical solutions of Diophantine equations as an attempt to address the famous Hilbert's tenth problem. The proposed methodology uses a three layer feed forward neural network with back propagation as sequential learning procedure to find numerical solutions of a class of Diophantine equations. It uses a dynamically constructed network architecture where number of nodes in the input layer is chosen based on the number of variables in the equation. The powers of the given Diophantine equation are taken as input to the input layer. The training of the network starts with initial random integral weights. The weights are updated based on the back propagation of the error values at the output layer. The optimization of weights is augmented by adding a momentum factor into the network. The optimized weights of the connection between the input layer and the hidden layer are taken as numerical solution of the given Diophantine equation. The procedure is validated using different Diophantine Equations of different number of variables and different powers.