Potential to Density via Poisson Equation: Application to Bespoke Learning of Gravitational Mass Density in Real Galaxy

In multiple real-world dynamical systems, structural properties can be deterministically linked to the evolution-driving function. For example, in self-gravitating systems, or in systems in which charge/current distributions dictate the dynamics, the evolution-driver or the system potential, is related in a known way to the density function that underlies system structure. In this chapter, the focus is on learning that density function, given observations that are possible of only some, instead of all of the phase space coordinates, in a dynamical system that we motivate to be in dynamic equilibrium. Then the embedding of the evolution-driver in the support of the probability density function (pdf) of the phase space variables—by exploiting the temporal evolution of the pdf—is equivalent to the embedding of the sought structural density function in the support of the pdf. Such learning demands generation of a training data, and this is illustrated via the bespoke learning of the value of the density of all gravitating mass in a real galaxy NGC 4649, at chosen locations inside the galaxy; values of the phase space pdf at chosen points in its support, are also bespoke learnt. For such learning, data on two types of galactic particles are implemented. Supervised learning of the gravitational mass density function and pdf are undertaken thereafter, along with predictions at test points. Such prediction suggests gravitational mass of about 10–100 billion solar masses inside the inner 0.001 kpc in this real galaxy.