Machine learning meets Kepler: inverting Kepler’s equation for All vs All conjunction analysis

The number of satellites in orbit around Earth is increasing rapidly, with the risk of collision rising accordingly. Trends of the global population of satellites need to be analyzed to test the viability and impact of proposed rules and laws affecting the satellite population and collision avoidance strategies. This requires large scale simulations of satellites that are propagated on long timescales to compute the large amounts of actionable close encounters (called conjunctions), which could lead to collisions. Rigorously checking for conjunctions by computing future states of orbits is computationally expensive due to the large amount of objects involved and conjunction filters are thus used to remove non-conjuncting orbit pairs from the list of possible conjunctions. In this work, we explore the possibility of machine learning (ML) based conjunction filters using several algorithms such as eXtreme Gradient Boosting, TabNet and (physics-informed) neural networks and deep operator networks. To show the viability and the potential of ML based filters, these algorithms are trained to predict the future state of orbits. For the physics-informed approaches, multiple partial differential equations are set up using the Kepler equation as a basis. The empirical results demonstrate that physics-informed deep operator networks are capable of predicting the future state of orbits using these equations (RMSE: 0.136) and outperform eXtreme Gradient Boosting (RMSE: 0.568) and TabNet (RMSE: 0.459). We also propose a filter based on the trained deep operator network which is shown to outperforms the filter capability of the commonly used perigee-apogee test and the orbit path filter on a synthetic dataset, while being on average 3.2 times faster to compute than a rigorous conjunction check.