  With the sustained rise in satellite deployment in Low Earth Orbits, the
collision risk from untracked space debris is also increasing. Often
small-sized space debris (below 10 cm) are hard to track using the existing
state-of-the-art methods. However, knowing such space debris' trajectory is
crucial to avoid future collisions. We present a Physics Informed Neural
Network (PINN) - based approach for estimation of the trajectory of space
debris after a collision event between active satellite and space debris. In
this work, we have simulated 8565 inelastic collision events between active
satellites and space debris. Using the velocities of the colliding objects
before the collision, we calculate the post-collision velocities and record the
observations. The state (position and velocity), coefficient of restitution,
and mass estimation of un-tracked space debris after an inelastic collision
event along with the tracked active satellite can be posed as an optimization
problem by observing the deviation of the active satellite from the trajectory.
We have applied the classical optimization method, the Lagrange multiplier
approach, for solving the above optimization problem and observed that its
state estimation is not satisfactory as the system is under-determined.
Subsequently, we have designed Deep Neural network-based methods and Physics
Informed Neural Network (PINN )based methods for solving the above optimization
problem. We have compared the performance of the models using root mean square
error (RMSE) and interquartile range of the predictions. It has been observed
that the PINN-based methods provide a better prediction for position, velocity,
mass and coefficient of restitution of the space debris compared to other
methods.
