In principle this is actually quite a reasonable idea and is a common pattern of many types of physical measurement. Often we have a detailed and accurate physical model of the forward dynamics of a system, given some system parameters, but we can't measure these parameters directly. Instead, we measure some data in a "sensor domain" and we'd like to map it back to the physical parameters.
This setup is known as an "inverse problem" and is often ill conditioned / singular or very complex, therefore requiring some regularization in the form of prior knowledge. Treating the inverse problem as a regression problem (given these observations in sensor domain, predict the state of the system) with a neural network as the regressor is one way of attacking these problems and is becoming very successful in some areas, for example MRI reconstructions, eg https://www.biorxiv.org/content/10.1101/278036v1. In this case you are adding the regularization / priors by constructing the training data with a physical model.
I think this kind of approach is interesting because it scales to input and output spaces with high dimensionality. However, it's not exactly clear to me what kind of estimate such a regressor provides (is it kind of like doing maximum likelihood?)
From a more standard statistical point of view, you'd like to estimate the full probability distribution over system parameters. In this case, the orbital elements and mass of the unknown bodies. Because this inference problem has relatively low dimensionality (I think?) you might do better to treat it as a problem of Bayesian inference and sample it using MCMC. Then you'd have a rigorous way to understand the uncertainty of the estimates and also to attack the problem of "unknown number of bodies" in a systematic way.