A Pontryagin Neural Network application to tracklets correlation of optical observations

As activity occurring in the region beyond the geostationary belt continues to grow, the number of satellites and space debris in this area is expected to rise significantly. As a direct consequence, the need of reliable methods to identify, correlate and catalog objects in this region will play a pivotal role to ensure the safety of space operations. This paper proposes a method based on Pontryagin Neural Networks (PoNNs), a specialized type of Physics-Informed Neural Networks (PINNs) designed to solve optimal control problems. By applying the Extreme Theory of Functional Connections, which combines the advantages of PINNs with the Theory of Functional Connections, the PoNN framework approximates the problem's unknowns through a single-layer feed-forward neural network. The correlation problem is addressed by formulating an energy-optimal control problem that links two tracklets, under the assumption that the objects are non-maneuvering. When a successful correlation is achieved, the resulting solution corresponds to a ballistic trajectory that minimizes control effort. The Mahalanobis distance is then used to evaluate the correlation, which includes residuals on the data, the computed fuel cost associated to the optimal trajectory and the residuals on the physics. As a secondary goal, Initial Orbit Determination capabilities of the method are also investigated. The developed algorithm is validated using both simulated and real angles-only observations of objects governed by a two-body dynamical model. The real optical measurements, consisting of right ascension and declination data, were supplied by the telescopes operated by the Space4 Center at The University of Arizona.