Learning Schwarzschild Geodesic Motion with Lagrangian Neural Networks

Technical Summary: Learning Schwarzschild Geodesic Motion with Lagrangian Neural Networks

Problem Statement
The paper addresses the challenge of learning the dynamics of Schwarzschild geodesics directly from trajectory data using Lagrangian Neural Networks (LNNs). While LNNs have shown success in Anti-de Sitter (AdS) spacetime, extending them to Schwarzschild geometry presents specific difficulties. The primary obstacles include:

1. Regime Diversity: The system encompasses near-circular bound orbits, high-eccentricity trajectories, and orbits near stability boundaries, all requiring simultaneous optimization.
2. Numerical Instability: The disparate scales of input coordinates (e.g., radial distance $r \sim 10^2$ vs. angular velocity $\dot{\phi} \sim 10^{-3}$) lead to poorly conditioned optimization and loss domination by specific channels.
3. Data Sampling Bias: Uniform sampling in phase space tends to overproduce unbound (plunge or escape) states while under-sampling the delicate bound-orbit regions essential for learning long-term orbital dynamics.

Methodology
The authors build upon a stabilized LNN baseline (Hamzaogullari and Ozakin [3]) that utilizes a Lorentzian structural prior and Sylvester regularization to enforce the correct indefinite signature on the learned mass matrix. To overcome the specific challenges of Schwarzschild dynamics, they introduce two coordinated improvements:

1. Decoupled Position–Velocity Normalization:
   Instead of applying a single scaling factor to spatial and velocity coordinates, the authors implement a scheme that independently rescales each coordinate $q_i$ and its time derivative $\dot{q}_i$.

   • Inputs are normalized using scale vectors $S$ for positions and $S/T$ for velocities, where $T$ represents characteristic time scales.
   • Target accelerations are reweighted to balance contributions across all channels. The loss function is modified to normalize the Mean Absolute Error (MAE) by the mean magnitude of each acceleration channel, ensuring that dimensions with smaller absolute values are not ignored during training.
   • The authors note that this transformation means the learned scalar $L_\theta$ is a latent generating function for normalized accelerations rather than a direct rescaling of the physical Schwarzschild Lagrangian.

2. Energy-Constrained Dataset Generation:
   To address sampling bias, the authors propose a rejection-sampling strategy informed by the Innermost Stable Circular Orbit (ISCO) structure.

   • Initial positions ($r, \phi$) and velocities ($\dot{r}, \dot{\phi}$) are sampled from predefined boundaries.
   • The time-velocity component $\dot{t}$ is computed to satisfy the timelike normalization constraint ($u^\mu u_\mu = -1$).
   • Samples are accepted only if they satisfy specific physical criteria: the specific angular momentum $|L|$ must exceed the ISCO threshold ($\sqrt{3}r_s$), and the specific energy $E$ must be below a maximum threshold ($E_{max} \approx 1$) to ensure the dataset is biased toward bound orbits. This strategy increases the fraction of bound-compatible samples from 27.3% to 92.3%.

Key Results
The model was evaluated across three orbital regimes (close: $r \in [50, 90]$, medium: $r \in [90, 200]$, far: $r \in [200, 300]$) and eight distinct initial configurations.

• Trajectory Prediction: The model successfully reproduces orbital paths, including relativistic perihelion precession, across all tested regimes. The average radial deviation was approximately $4.77 \pm 4.41$ in units of $r_s$.
• Conservation Laws: Energy and angular momentum were conserved to within a few percent over integration times of $5 \times 10^4$ proper time units.
• Time Dilation: The model accurately captures the evolution of the time-velocity $\dot{t} = dt/d\tau$, correctly modeling the coupling between temporal and spatial coordinates, particularly in the strong-field regime.
• Loss vs. Fidelity: The authors observed a dissociation between training loss minimization and trajectory quality. Baseline datasets with unbound samples showed faster loss decay but failed to learn stable bound dynamics, whereas the energy-constrained dataset forced the model to fit the harder, physically relevant bound regimes.
• Structural Analysis: Analysis of the learned Lagrangian and its Hessian revealed that while the model predicts accelerations correctly, the learned Lagrangian does not strictly converge to the purely kinetic form ($L = \frac{1}{2}g_{\mu\nu}\dot{x}^\mu\dot{x}^\nu$). The learned Hessian exhibits non-zero off-diagonal terms, and the Lagrangian retains residual position-dependent terms, indicating the model learns a mathematically equivalent but structurally distinct latent scalar.

Significance and Claims
The paper claims to present the first successful training of LNNs on Schwarzschild geodesics. The significance of this work lies in demonstrating that LNNs, when augmented with physically motivated normalization and sampling strategies, can reliably learn geodesic dynamics in curved spacetime.

The authors emphasize that their approach offers a promising direction for data-driven modeling of complex relativistic systems. However, they remain modest regarding the physical interpretability of the learned Lagrangian; while the model predicts dynamics accurately, extracting the exact spacetime metric directly from the learned Lagrangian is not guaranteed due to the non-uniqueness of the inverse problem and the effects of the decoupled normalization. The work establishes a foundation for future efforts to apply LNN-based methods to real astronomical observations spanning both strong and weak gravitational fields.