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This work explores the combination of Sparse and Symbolic Regression, here called Deep Symbolic Regression, for the autonomous reconstruction of orbital anomalies. Orbital anomalies are detectable deviations from the state of an object that can be predicted from the propagation of some observable initial conditions. We contemplate anomalies that can derive from unmodelled natural phenomena or from intentional and unintentional orbital manoeuvres: an accurate modelling of atmospheric density fluctuations, for example, allows informing the space weather. Leveraging the powerful modelling capacity of symbolic regression and the sparse representability of dynamical systems in orbital mechanics , the proposed approach allows one to generate a symbolic representation of orbital anomalies from state observations only. In other words, we use sparse measurements of position and velocity, in general associated with uncertainty, to derive a symbolic representation of the unmodelled part of the dynamics that can explain the deviations of the propagated states. The advantage of such an approach, compared to more traditional filtering techniques, is twofold: it provides an explicit analytical representation of the phenomenon causing the anomaly and it provides a better long term prediction of the dynamics of the object under consideration. The use of Deep Symbolic Regression outdoes more traditional Genetic Programming-based approaches in that it is less prone to overfitting and far less computationally expensive. The explicit dependence, with respect to time, of the symbolic representation, allows one to indirectly model the evolution of unobservable states, whose behaviour can be later inferred from the analysis of the estimated equation itself. The proposed approach yields solutions that are robust against measurement noise: its estimation can be integrated into the derivation of the missing part of the dynamics. The performance of the Deep Symbolic Regression will be assessed against a number of case studies, with a focus on the interpretability of the obtained solutions, demonstrating the performance of this new tracking data-based algorithm.

The paper explores the use of symbolic regression to discover missing parts of the dynamics of space objects from tracking data. The starting assumption is that the differential equations governing the motion of an observable object are incomplete and do not allow a correct prediction of the future state of that object. Symbolic regression, making use of Genetic Programming (GP), coupled with a sensitivity analysis-based parameter estimation, is proposed to reconstruct the missing parts of the dynamic equations from sparse measurements of position and velocity. Furthermore, the paper explores the effect of uncertainty in tracking measurements on the ability of GP to recover the correct structure of the dynamic equations. The paper presents a simple, yet representative, example of incomplete orbital dynamics to test the use of symbolic regression.