Solving differential equations with unsupervised neural networks

A recent method for solving differential equations using feedforward neural networks was applied to a non-steady fixed bed non-catalytic solid–gas reactor. As neural networks have universal approximation capabilities, it is possible to postulate them as solutions for a given DE problem that defines an unsupervised error. The training was performed using genetic algorithms and the gradient descent method. The solution was found with uniform accuracy (MSE 10−9) and the trained neural network provides a compact expression for the analytical solution over the entire finite domain. The problem was also solved with a traditional numerical method. In this case, solution is known only over a discrete grid of points and its computational complexity grows rapidly with the size of the grid. Although solutions in both cases are identical, the neural networks approach to the DE problem is qualitatively better since, once the network is trained, it allows instantaneous evaluation of solution at any desired number of points spending negligible computing time and memory.