In this paper, a new method for solving orbit determination problems is introduced named Physics Informed Least Squares (PILS). We use a particular kind of single layer feed-forward neural networks called Extreme Learning Machines to enable higher accuracy and flexibility than classical least squares. The least squares estimate is used as baseline for the loss function, to which a regularizing term based on the differential equations modelling the dynamics of the problem is then added. This ensure that the learned relationship between input and output is compliant with the physics of the problem. The results are comparable or better than the batch least squares solution, with the advantage of not requiring an initial guess and being able to solve for the complete trajectory without needing any integration.
The goal of this work is to build upon the foundation of ML applied to the OD problem using the recently developed Physics-Informed Extreme Learning Machines (PIELM) to regularize the learning algorithm and improve dynamical soundness of the estimated solution. Standard PIELM are particular neural networks, developed for solving forward and inverse problems governed by parametric Differential Equations (DEs). The training of these networks, as in any Physics Informed method, is regulated by the physics of the problem. In particular, the latent solution of the parametric DE is approximated via Neural Networks (NNs); the DE that models the physics of the problem, is then embedded, in its implicit form, into the NN loss function. Thus, the DE drives the training of the network acting as regulator that penalizes the network’s training when the physics of the problem is violated. Usually these NN are then trained via gradient-based methods. In case the method employs Extreme Learning Machines (ELM), as in this case, the training can be carried out in one single step using minimum norm least squares. With this work we harness the power and ease of implementation of neural network to solve a set of OD problems based on least squares. This leads to an algorithm capable of performance similar to batch least squares with some advantages. Specifically, the algorithm is capable of estimating the complete state evolution without any integration step and without the need to provide an initial guess.