Modeling complex systems with an unsupervised Lagrangian neural network

Abstract

Complex systems involve n-interacting bodies in the presence of simultaneous constraints. Understanding the dynamics of such systems is usually done by approximation based on some local symmetry and on the assumption that only nearby objects are responsible for the behavior of a particular particle. However, most observed patterns are very much dependent on global interaction and therefore the mentioned assumption is very restricted.
In this paper we introduce Lagrangian dynamics into unsupervised backpropagation neural networks (NN), and show that an adaptive algorithm can be created to determine the cost function from a knowledge of the constraints and boundary conditions without sacrificing the global nature of the interaction. The undetermined cost-function yields the dynamics of the system interaction, and usually exhibits highly non-linear behavior. This optimization procedure takes advantage of the inherent versatility of the Lagrangian approach to produce a functional relationship between the parameters of the complex behavior and the hyperdimensional-fitting ability of NN to provide converging solutions.
Our method is tested for a cylindrical nuclear reactor obeying the neutron diffusion theory (NDT). Knowing only the diffusion constant and the constraint of the reactor parameters (radius R and height H), we determine the function being optimized (which is the surface area in this case) with excellent accuracy.