Self-attention-enhanced Deep Neural Networks for Simulating Post-Newtonian Dynamics of Three-body Systems

Abstract Simulating the long-term evolution of three-body systems remains a computational challenge due to the high cost of numerical integration, especially when incorporating post-Newtonian (PN) corrections. The first-order PN formulation introduces over 10 additional terms beyond the classical Newtonian equations, significantly increasing complexity. Although traditional neural networks have been used to approximate Newtonian dynamics, they struggle to capture PN effects, which are several orders of magnitude smaller. We propose a deep learning framework that combines a BERT-based encoder with an artificial neural network to model relativistic orbital dynamics. The motion is decomposed into two components—classical Newtonian and PN deviations—each learned independently and subsequently integrated to produce the final prediction. This architecture enables direct inference of positions and velocities at any target time from initial conditions, eliminating the need for stepwise acceleration-based integration. To ensure physical reliability, we introduce a relative energy error criterion derived from orbital energy conservation. During inference, predictions that exceed a predefined error threshold trigger local re-evaluation using smaller time steps. Applied to a hierarchical three-body system composed of a supermassive–intermediate-mass black hole binary and a star, the hybrid model achieves over 160× acceleration (100 s versus 5 hr) compared to full numerical integration when predicting the system’s state 150 yr into the future, while maintaining relative energy errors below 1%. These results demonstrate the framework’s potential for efficient and accurate simulation of relativistic multibody systems.