Guided Inverse Gravity Modeling for Asteroids using Neural Networks

In this thesis, my goal was to include additional information about an asteroids internal structure into a gravity inversion process. I based my work on the GeodesyNet, a Neural Network representing the mass distribution of a body, and modified the authors approach to incorporate areas of predefined density into the process.

Description

The gravitational potential of a body, for example an asteroid, is produced by its internal mass distribution. Consequently, measurements of a gravitational field can be used to reconstruct the mass distribution inside the body that produced it. This process is called gravity inversion. In the past, constraints were imposed on the solution, and a solution was then derived using optimization methods. More recently, machine learning based methods where used successfully as well, which require fewer assumption about a body, and can yield more accurate results in some situations. I focused on these methods in my thesis.
While researching the topic, I found two different approaches to inverse gravity modeling with machine learning: Those based on convolutional neural networks and those inspired by Neural Radiance Fields, or NeRFs. The approach based on convolutional neural networks requires a dataset of mass distributions and their corresponding gravitational fields to train upon. In anticipation of using such an approach, I built a framework to generate such datasets at different resolutions, based on the 3D asteroid models in the 3D Asteroid Catalogue. I subdivide the bounding box of each asteroid, and distributed the total mass of the asteroid across the smaller boxes, that overlap with the asteroid, according to the mass concentration model. With this mass distribution I can then calculate the gravitational potential at a number of points around the asteroid, creating a dataset.
After evaluating both the convolutional and the NeRF inspired approaches with regard to incorporating areas of predefined densities, I decided to work with the NeRF inspired approach. With this approach a new neural network is trained for every asteroid, and it thus needs no dataset of the kind the convolutional approaches require. Therefore, I was unable to use my dataset framework, but it can still serve as a basis for future work, for approaches that require such a dataset.

The GeodesyNet is a neural network that learns a mapping between coordinates in three-dimensional space and a bodies density at these coordinates. This is how it represents a bodies mass distribution. The GeodesyNet learns this mass distribution by calculating the gravitational field its internal mass distribution produces and comparing it with the gravitational field it is supposed to match. The internal mass distribution is then continually optimized to generate a gravitational field that matches the reference as closely as possible. To incorporate regions of predefined density into this process, I added an additional term to the GeodesyNets Loss Function. This term measures how far the density represented by the GeodesyNet differs from the desired predefined density in a certain area. With this term, the GeodesyNet can match the predefined density, while still representing a mass distribution that matches the gravitational field well. I implemented a similar term for the masconCUBE, a comparative method the authors of the GeodesyNet used. The masconCUBE optimizes a fixed number of points masses directly.

Results

Both the GeodesyNet and the masconCUBE are able to match the predefined density in the desired area, when the guidance term is used. The GeodesyNet gets closer to the predefined density when large regions are defined, the masconCUBE works better for smaller regions. When no guidance term is used, both methods produce densities that differ strongly from the desired density in the target area. This suggests that a guidance term must be used, if adherence to a specific density in a specific region is desired.
Looking at adherence to the input gravitational field, the masconCUBE is generally the better performing method. In specific circumstances the GeodesyNet can perform similarly well, or better. My results suggest that the GeodesyNet performs comparatively well when looking at the gravitational field very close to the body. The GeodesyNet performed better than the masconCUBE with a large guidance region, when looking at the gravitational field close to the body. When a high resolution prediction of the density distribution is needed, the GeodesyNet might also be the better option. It produces a continuous function of the density, in comparison to the masconCUBE discrete point masses, resulting in a representation of higher resolution.

Files

Full version of the master's thesis

License

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