Scientific Machine Learning for Symbolic Recovery of Relativistic Effects in Black Hole Orbits

Simulating relativistic orbital dynamics around Schwarzschild black holes is essential for understanding general relativity and astrophysical phenomena like precession. Traditional numerical solvers face difficulty when dealing with noisy or sparse data, necessitating data-driven approaches. We develop a Scientific Machine Learning (SciML) framework to model orbital trajectories and symbolically recover the relativistic correction term. Neural Ordinary Differential Equations (Neural ODEs) accurately predict inverse radius $u$, radial velocity $v$, and precession $\delta$, performing well under ideal conditions with mean absolute errors (MAE) around $3.5\times 10^{-4}$ for $u$ on noiseless full-domain data, but degrading sharply when training data is limited to 10\%, where MAE rises above $0.026$. To address this, we employ Universal Differential Equations (UDEs), which embed a neural network to approximate the correction term $\alpha \tfrac{GM}{c^2} u^3$, achieving precise orbit predictions even with just 10-20\% data coverage and maintaining low errors (UDE forecast loss $\approx 2.7 \times 10^{-4}$) across noise levels up to 35\%. Symbolic regression further recovers an analytical expression closely matching the expected correction, with mean symbolic errors below $10^{-7}$. We use adjoint-based training to discover solutions efficiently, implemented in Julia with DiffEqFlux and Lux. Using this method, we successfully integrate machine learning with physical laws, demonstrating robustness to noise and data scarcity. This approach can be expanded for large-scale or detailed astrophysical projects.