A graph neural network emulator predicting planet formation through the giant impact stage

Terrestrial exoplanetary systems exhibit a wide diversity in masses and orbital architectures. Planetary population synthesis models are used to investigate the origin of this diversity. The final phase of terrestrial planet formation is the giant impact stage. During this stage, protoplanets undergo mutual collisions, altering their masses and orbits. This stage plays a key role in setting the final multiplicity, orbital excitation, and mass distribution of terrestrial planets. Planetary population synthesis models employ either semi-analytical models or N-body simulations for the giant-impact stage. However, both approaches suffer from important limitations: the latest semi-analytical models tend to overestimate the planet multiplicity over time and cannot treat orbital inclinations, while direct N-body simulations are computationally expensive and difficult to apply to large parameter surveys. Recently, machine-learning models have been proposed to accelerate these predictions; however, their applicability remains limited to close-in regions. 

To overcome these limitations, we develop a machine-learning emulator for the giant-impact stage that reproduces the time evolution of planetary systems and incorporates orbital inclinations. We adopt Graph Neural Networks (GNNs), as planetary systems can be naturally represented as graphs, with planets as nodes and mutual gravitational interactions as edges. To handle the decreasing number of planets caused by collisions, we employ a Transformer architecture that allows the model to process a variable number of planets. The emulator decomposes the planetary system evolution into two coupled tasks: (Task A) a GNN that predicts the next colliding planet pair and the time to collision, with predictive uncertainties quantified via Monte Carlo dropout; and (Task B) a probabilistic GNN that predicts changes in orbital elements due to collisions and secular perturbations occurring between collision events. By iteratively coupling Tasks A and B, the emulator generates sequences of collision events and emulates the full evolution of planetary systems through the giant impact stage. For training, we construct a dataset of 9000 samples derived from 800 N-body simulation runs by segmenting each simulation track into individual collisional events. 

The GNN emulator achieves a speedup of approximately four orders of magnitude compared to direct N-body simulations, while reproducing the key statistical properties of the final planetary systems.

As shown in Figure 1, the GNN emulator reproduces the N-body results across multiple statistical metrics, including the final number of surviving planets_, mean orbital spacing, mean eccentricity, and mass dispersion. 

Furthermore, Figure 2 shows that the emulator tracks the time evolution of the number of planets due to collisions across different regions. This capability represents a improvement over previous semi-analytical approaches, particularly in capturing the time evolution of planet multiplicity.

 

Comparisons with both semi-analytical models and previous machine-learning approaches show that our GNN emulator maintains robust performance over a wider radial range. These results demonstrate that our approach enables efficient and accurate exploration of the giant-impact stage in planetary population synthesis.