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Learning Schwarzschild Geodesic Motion with Lagrangian Neural Networks

Sukru Caglar, Sena Kalabalik, Abdullah Umut Hamzaogullari, Arkadas Ozakin

arXiv:2610.13234v1cs.LGgr-qc

TL;DR

Schwarzschild geodesic learning faces difficulty sampling bound-orbit regimes, which naive phase-space sampling can underrepresent. The paper combines decoupled position–velocity normalization with an ISCO-informed, energy-filtered sampling strategy; these changes improve training performance.

  • Problem

    Naive phase-space sampling can overrepresent plunge or escape configurations while undersampling nearly circular and eccentric bound states needed for long-term orbital dynamics.

    Source: C. Relativistic Machine Learning and… · p. 4

  • Method

    The method independently normalizes position and velocity channels, normalizes acceleration errors by channel, and uses ISCO-informed energy-filtered velocity sampling.

    Source: §IV · p. 4, §1 · p. 5

  • Results

    Decoupled scaling and loss normalization greatly improved training performance and masked imbalances in coordinate magnitudes and their time derivatives.

    Source: §1 · p. 5

  • Takeaways & Limitations

    The energy-filtered sampling strategy concentrates training data on orbital regimes relevant to long-horizon bound motion.

    Source: §IV · p. 4

Abstract

from arXiv · show

Lagrangian Neural Networks (LNNs) learn dynamics from trajectory data by parameterizing a scalar Lagrangian and deriving accelerations through the Euler--Lagrange equation, providing a variational inductive bias that improves physical consistency over purely data-driven models. LNNs have recently been extended to general relativistic settings, leaving Schwarzschild geodesics as an open challenge. Training on complex relativistic systems is hindered by the diversity of the dynamics, as well as numerical instabilities and the difficulty of adequately sampling physically relevant, diverse orbital regimes. We introduce two coordinated improvements: a decoupled position--velocity normalization scheme that independently rescales each coordinate and its time derivatives, reweighting acceleration targets to balance contributions across all channels; and an energy-constrained log-uniformly sampled dataset generation strategy informed by the innermost stable circular orbit (ISCO) structure. With these improvements, our model accurately reproduces perihelion precession and relativistic time dilation, conserves energy and angular momentum to within a few percent even after completing multiple orbits around the central mass and predicts orbital trajectories across eight orbital scenarios in close, medium, and far-field regimes. These results establish that LNNs, augmented with physically motivated normalization and sampling, can reliably learn geodesic dynamics in curved spacetime. Our work also represents the first successful training of LNNs on Schwarzschild geodesics and offers a promising direction for data-driven modeling of complex relativistic systems.

I. INTRODUCTION

LNNs derive motion from a learned scalar Lagrangian, but Schwarzschild dynamics combine diverse orbital regimes, mixed numerical scales, and difficult sampling. The section motivates improvements to normalization and dataset generation while outlining the LNN acceleration and Lorentzian-structure framework.

I. INTRODUCTION

  • An LNN derives equations of motion by applying Euler–Lagrange structure to a learned scalar Lagrangian rather than fitting accelerations directly.The variational framing supplies a geometric inductive bias associated with physical fidelity and energy conservation.Source: §I · p. 1
  • Schwarzschild learning must handle diverse orbital regimes, mixed-scale coordinates, and sampling that underrepresents bound orbits; the proposed improvements target normalization and data generation.The baseline framework is extended with physics-aware normalization and an ISCO-informed, energy-constrained dataset strategy.Source: §I · p. 1

1. Acceleration targets from Blackbox Lagrangian:

  • The Euler–Lagrange formulation provides the acceleration targets by applying gradient operators to the black-box learned Lagrangian.The paper rearranges the vectorized equation and distributes time derivatives to obtain accelerations.Source: §1 · p. 2

2. Training objective and loss

  • Training minimizes mean absolute error between predicted accelerations computed from the learned Lagrangian and ground-truth accelerations from the system equations.The target accelerations are obtained by solving the system's Euler–Lagrange equations.Source: §2 · p. 2

3. Lagrangian Parameterization with Lorentzian Structural Prior and Sylvester Regularization:

  • The Lorentzian parameterization adds a fixed-signature kinetic term to a learnable perturbation and penalizes violations of the required Hessian signature.The Lorentzian base term biases the velocity Hessian toward its desired signature at initialization.Source: §3 · p. 2

B. Schwarzschild geodesics:

The section presents Schwarzschild geodesics as motion governed by a spacetime metric and motivates their study through perihelion precession. It reviews structured neural dynamics and stabilization methods relevant to learning this motion.

1. Geodesic motion in Schwarzschild spacetime

  • On the equatorial plane, timelike Schwarzschild geodesics can be represented by equations of motion in the six-dimensional state, with specific energy and angular momentum conserved.The work sets θ = π/2 and θ̇ = 0, and takes time derivatives with respect to affine proper time τ.Source: §1 · p. 3

2. Mercury’s Perihelion Precession

  • Mercury’s unexplained excess precession of roughly 43′′ per century motivated the Schwarzschild benchmark, while this work focuses on the strong-field regime r ∼50–300 in units of rs/2.In the weak-field limit, the relativistic correction to the effective potential accounts for the additional precession.Source: §2 · p. 3

III. RELATED WORK

  • Prior physics-informed learning includes PINNs for relativistic equations and automatic-differentiation solvers for recovering Schwarzschild solutions from Einstein’s equations.The cited applications include Teukolsky solutions, black-hole quasinormal modes, and Schwarzschild and charged Schwarzschild solutions.Source: §III · p. 3

A. Structure-Preserving Neural Dynamics

  • Neural ODEs lack explicit mechanical structure, HNNs require canonical coordinate–momentum pairs, and LNNs instead derive accelerations from a scalar Lagrangian.DeLaNs constrain the Lagrangian to a kinetic–potential decomposition; exact conservation during rollout is not guaranteed for approximate learned functions and numerical integrators.Source: A. Structure-Preserving Neural Dynamics · pp. 3–4

B. Stabilizing Lagrangian Neural Networks

  • Poorly conditioned velocity Hessians can amplify approximation errors and destabilize optimization; the stabilized LNN precursor uses Lorentzian signature regularization and is extended here to address Schwarzschild’s differing numerical scales.Alternative priors include positive-definite mass matrices or redundant Cartesian coordinates with known holonomic constraints.Source: B. Stabilizing Lagrangian Neural… · p. 4

C. Relativistic Machine Learning and Schwarzschild Phase-Space Sampling

Trajectory-based LNN learning differs from approaches that solve explicitly specified relativistic field equations, and bound Schwarzschild motion poses a data-selection challenge. The paper addresses these challenges with energy-filtered sampling and decoupled position–velocity normalization, while clarifying that the learned scalar is a latent representation rather than the physical Lagrangian.

C. Relativistic Machine Learning and Schwarzschild Phase-Space Sampling

  • Energy-filtered velocity sampling informed by the ISCO is designed to improve coverage of bound-orbit phase space.The accessible orbital regimes are structured by conserved specific energy, specific angular momentum, and the Schwarzschild effective potential.Source: C. Relativistic Machine Learning and… · p. 4

IV. METHODS

  • The method retains the baseline Lagrangian parameterization, Lorentzian initialization, and Sylvester regularization, adding decoupled channel scaling and energy-filtered sampling.These modifications target acceleration-objective conditioning and training-data concentration on regimes relevant to long-horizon bound motion.Source: §IV · pp. 4–5

1. Decoupled normalization of positions and velocities

  • Independent position and velocity scales balance input-channel ranges, while per-channel acceleration-loss weights make contributions comparable.The motivation includes the disparate scales of Schwarzschild coordinates, such as r ∼ 1 × 10^2 and ϕdot ∼ 1 × 10^-3.Source: §1 · p. 5
  • Decoupled scaling and acceleration-loss normalization improve training by reducing magnitude imbalances across dimensions and time derivatives.The learned scalar generally differs in functional form from the physical Lagrangian and serves as a latent representation for the normalized acceleration map.Source: §1 · p. 5, §2 · p. 6

2. Effect of decoupled normalization on the variational interpretation of the learned Lagrangian

  • Because independent velocity scales are not generally derivatives under one normalized time parameter, the learned scalar is not simply the physical Lagrangian with rescaled arguments.The paper describes it as a latent generating scalar for the normalized acceleration map.Source: §2 · pp. 5–6

B. Dataset Generation

The dataset strategy targets bound-compatible Schwarzschild orbits by sampling positions and velocities within observational boundaries, then filtering candidates using ISCO and energy constraints. In the reported loss comparison, decoupled normalization improves convergence over legacy normalization.

1. Bound Training Data:

  • The baseline coordinate-wise sampling produces plunging or escaping trajectories, motivating energy-constrained velocity sampling for bound orbits across eccentricities.Including plunge or escape motions increases the number of coordinate-range combinations needed and introduces normalization outliers.Source: §1 · p. 6

2. Sampling ˙r, ˙ϕ

  • Positions and velocities are sampled within predefined boundaries, with log-uniform r and independent uniform velocity distributions for this learning task.The timelike normalization constraint determines t-dot, after which the full state is used to compute conserved energy and angular momentum.Source: §2 · p. 6
  • Rejection sampling retains candidates meeting the ISCO and energy criteria; Emax ≈ 1 admits mostly, but not exclusively, bounded samples.The resulting dataset favors bound-compatible states while maintaining a controlled fraction of near-threshold configurations.Source: §2 · p. 6

V. EXPERIMENTS

  • The evaluation compares trajectories with analytical geodesics in close, medium, and far regimes, and examines time dilation and learned function surfaces.The function-surface analysis compares predicted accelerations, Lagrangian, and Hessian with analytical counterparts.Source: §V · p. 7

1. Loss Behavior:

  • Decoupled normalization significantly improves convergence, while legacy normalization fails to reduce the loss effectively.Because rescaling shifts raw MAE magnitudes, the comparison tracks each run’s fractional loss progression from its initial value.Source: §1 · p. 7

2. Predictions with and without Decoupled Normalization:

Decoupled normalization produces convergence and more informative trajectory predictions than legacy normalization, while dataset choice also affects prediction quality. The model performs across close, medium, and far orbital regimes, with time-velocity agreement strongest in close and medium regions.

2. Predictions with and without Decoupled Normalization:

  • Decoupled normalization converges where legacy normalization does not and enables trajectory predictions that capture system dynamics when coordinate signals are unbalanced.With decoupled normalization held constant, baseline and mixed-velocity datasets still produce discrepant predictions, suggesting random state sampling remains insufficient.Source: §2 · p. 7, Figure 1 · p. 7, Figure 2 · p. 8

B. Trajectory Accuracy

  • The trajectory evaluation tests eight initial configurations across the training region over τ/M = 5 × 10^4, assessing path agreement, conservation errors, coordinates, and velocities.Training uses non-plunging samples over r ∈ [50, 300].Source: B. Trajectory Accuracy · p. 8

1. Close Region:

  • Close-region prediction errors accumulate stepwise, producing slight visible precession error over successive revolutions.The close region spans r ∈ [50, 90], where time-dilation and precession effects are strongest.Source: §1 · p. 8, Figure 4 · p. 9

2. Medium Region

  • In the medium region, the highly circular test orbit is accurately predicted, though radial deviations can accumulate into drift and increased eccentricity.The authors report overall shape agreement and energy conservation across the medium-region predictions.Source: Figure 6 · p. 9, Figure 7 · p. 9, §2 · p. 9

3. Far Region:

  • Far-region energy conservation is consistent with close- and medium-region experiments, while spatial-coordinate and velocity predictions are used to assess weak relativistic effects.The far region spans r ∈ [200, 300] and is described as the weak-field limit.Source: §3 · pp. 9–10

C. Time Dilation:

  • Predicted time-velocity curves agree closely with analytical curves in close and medium regions, but far-region predictions show phase and amplitude deviations.The far-region discrepancy is attributed to limitations in global loss optimization.Source: C. Time Dilation · p. 10

D. Physical Consistency Check and investigation of the acceleration as a function of input state:

The authors probe acceleration functions, the learned Lagrangian, and its velocity Hessian against analytical forms. The learned functions show deviations from the expected purely kinetic form, although trajectory predictions can conceal some of them.

D. Physical Consistency Check and investigation of the acceleration as a function of input state:

E. Functional Analysis of Learned Lagrangian and Its Mass Matrix (Hessian):

  • Acceleration agreement does not establish Lagrangian identity because multiple Lagrangians can yield identical equations of motion, and a full functional characterization would require probing the entire six-dimensional input space.The learned Lagrangian is a black-box scalar function; the analysis compares its de-normalized form and velocity Hessian with analytical quantities.Source: E. Functional Analysis of Learned… · p. 11, §1 · p. 11, §2 · p. 13

1. Lagrangian:

  • The learned Lagrangian has slight ϕ dependence and more significant r dependence under zero velocities, despite these deviations having little impact on trajectory predictions.One-dimensional slices are consistent with an approximate form containing coordinate-dependent coefficients on quadratic velocity terms.Source: §1 · p. 11

2. Hessian:

  • The learned Hessian has nonzero cross terms involving ˙ϕ, and its r-form discrepancy shows that the learned Lagrangian does not inherit the purely kinetic form’s properties.The authors could not confirm the quadratic relation in ˙t because the test region for that coordinate was small.Source: §2 · p. 13, Figure 15 · p. 13

VI. SUMMARY OF FINDINGS AND CONCLUSIONS

The proposed normalization and energy-constrained sampling address numerical imbalance and poor coverage of bound-orbit regimes. Across close, medium, and far scenarios, the model reproduces key orbital behavior, while extrapolation to weak-field scales and recovery of a canonical Lagrangian remain unresolved.

A. Summary of Contributions:

  • The decoupled normalization addresses coordinate-scale imbalance, while energy-constrained sampling increases bounded-state coverage from 27.3% to 92.3%.The mixed-velocity strategy combines radial-velocity windows to provide more uniform velocity-phase-space coverage.Source: A. Summary of Contributions · p. 14

B. Trajectory Predictions:

  • Across three regimes and eight scenarios, the best model reproduces orbital paths with correct precession, averages 4.7662 ± 4.4051 in radial deviation, conserves energy and angular momentum to within a few percent over 5×10^4 s, and captures time dilation.The tested radial ranges are 50–90, 90–200, and 200–300.Source: B. Trajectory Predictions · p. 14
  • The baseline dataset shows faster apparent loss decay, but the mixed-velocity dataset yields better trajectory fidelity by emphasizing harder bound and low-velocity regimes.Those regimes are described as critical for stable integration.Source: B. Trajectory Predictions · p. 14

A. Physical Interpretability:

  • The model reproduces geodesics within r ∈ [50, 300], but its learned dynamics are not established across the much broader weak-field scales relevant to astronomical observations.The paper identifies generalization beyond the training range as an important direction for future work.Source: A. Physical Interpretability · p. 14
  • Time-coordinate velocity predictions degrade farther from the central body, which the authors attribute to weak-field acceleration gradients being overshadowed during global loss optimization.The passage reports ˙t starting near 1.005 farther out and approaching E as spacetime becomes asymptotically flat.Source: A. Physical Interpretability · p. 14

B. Inequivalent Lagrangians

  • The learned Lagrangian does not converge to a purely kinetic form; its velocity Hessian has off-diagonal components and it retains a position-dependent term at zero velocity.The passage therefore distinguishes successful geodesic prediction from convergence to the purely kinetic form.Source: B. Inequivalent Lagrangians · p. 15

Appendix A: Setup for the Model Training, Dataset Variances and performance comparisons

The appendix specifies the model and compares five dataset variants, finding that mixed-velocity sampling provides the strongest trajectory predictions across the tested scenarios. Training-loss reduction alone does not reliably rank long-horizon trajectory accuracy.

2. Dataset Variations and Generation parameters:

  • The five variants include baseline random-state sampling, direct energy sampling, velocity sampling with energy cutoffs, and a mixed-velocity sampler.The constrained and relaxed-energy velocity samplers use Emax values of 1 and 1.001, respectively.Source: §2 · pp. 15–16

1. Model Architecture:

  • The reported architecture uses a six-dimensional input, four hidden layers of 500 neurons, GELU activations, and a scalar Lθ output.The dataset variants contain 120,000 samples each, with an 80/20 training-validation split.Source: §2 · p. 16

2. Dataset Variations and Generation parameters:

  • The best mixture combines narrow and broad radial-velocity proposal windows, increasing representation of both circular and more eccentric states.The selected windows are {0, 1 × 10^-3} and {0, 1.8 × 10^-1}.Source: §2 · p. 16

3. Performance Comparison of the Datasets:

  • The mixed-velocity sampler yields the best trajectory prediction accuracy across all eight test scenarios, while the baseline performs worst.Energy sampling outperforms the baseline but trails velocity-based samplers in highly elliptical or far scenarios.Source: Figure A1 · p. 17
  • Baseline sampling has the largest fractional training-loss reduction, but that ranking does not match trajectory-level evaluation.The authors caution that relative single-step loss convergence alone is not a reliable indicator of long-horizon rollout accuracy.Source: §3 · p. 17, Figure A3 · p. 18
  • The mixed-velocity dataset increases representation of low-|˙r| and E < 1 configurations, which are harder to fit pointwise but important for stable integration.The authors describe the resulting tradeoff as worse average training loss but better trajectory fidelity.Source: §3 · p. 18

Appendix B: Alternative Dataset Sampling Methods and Energy Definitions

The appendix links Schwarzschild spacetime symmetries to conserved quantities and uses them to express a particle’s specific energy. The effective potential depends on distance from the black hole and angular momentum.

1. Killing Constants

  • Time-translation and azimuthal symmetries yield conserved specific energy E and specific angular momentum Lz along geodesics.The passage identifies the corresponding timelike and rotational Killing vectors.Source: §1 · p. 18
  • For a geodesic, contracting its four-velocity with a Killing vector gives a conserved quantity.The stated relation is Kµuµ = constant.Source: §1 · p. 18

a. Specific Energy Definition:

  • Using E and L from the Killing vectors, the appendix expresses ˙t and ˙ϕ and derives the specific-energy relation for Schwarzschild spacetime.The effective potential depends on the particle’s distance from the black hole and its angular momentum.Source: a. Specific Energy Definition · pp. 18–19

b. Innermost Stable Circular Orbit:

The appendix uses the ISCO structure to motivate bounds for stable-orbit sampling, then describes direct energy sampling over radius-energy space. It also reports that velocity sampling performs better in accuracy and generalization.

b. Innermost Stable Circular Orbit:

  • The stated stability condition is r > 3rs, which the authors use as a clue for setting a lower energy bound.The appendix connects the ISCO radius and angular momentum conditions to stable-orbit sampling.Source: b. Innermost Stable Circular Orbit · p. 19

c. Sampling E Based on LISCO:

  • The direct-energy sampler draws radius log-uniformly from 50–300 and energy between a radius-dependent Emin and 0.999 to cover bounded radius-energy space.E = 1 is described as an unbounded flyby orbit; energy above Emin is distributed to introduce eccentricity variation.Source: c. Sampling E Based on LISCO · p. 19
  • For small-precession-orbit prediction, the authors report better performance when 85% of α values are uniform and 15% are log-uniformly biased toward lower ˙r.They also report that velocity sampling outperforms direct energy sampling in accuracy and generalization.Source: c. Sampling E Based on LISCO · p. 19

Appendix C: Normalization, Euler–Lagrange Distortion, and Hessian De-normalization

The appendix describes per-coordinate normalization of positions, velocities, and accelerations, then explains how independent position and velocity scaling changes the form of the physical Euler–Lagrange equations. The network instead learns a latent scalar that generates normalized accelerations, rather than the physical Lagrangian in normalized coordinates.

1. Normalization map used in the methodology

  • The training pipeline applies per-coordinate scaling and trains or evaluates the model in normalized state coordinates.The passage identifies physical coordinates, velocities, and accelerations as the scaled quantities.Source: §1 · pp. 19–20

2. Euler–Lagrange equation and distortion under anisotropic scaling

  • Independent position and velocity scaling introduces T_j/T_i factors when the physical Euler–Lagrange equations are rewritten in normalized variables.When the time scales T_i differ, normalized velocity coordinates generally cannot be interpreted as derivatives of normalized positions under one common evolution parameter.Source: §2 · p. 20, a. Interpretation of the Tj/Ti factors · p. 20

a. Interpretation of the Tj/Ti factors

  • The model learns a latent scalar for normalized acceleration generation, not the transformed physical Lagrangian; the scalars need not share the same functional form.The observed learned-velocity-Hessian deviations are described as consistent with a distinct latent representation generating the normalized acceleration field.Source: a. Interpretation of the Tj/Ti factors · p. 20

3. Hessian de-normalization for model-vs-analytic comparability

The appendix describes comparing the learned velocity Hessian with its analytical counterpart in physical coordinates and adds trajectory plots for eight orbital scenarios. It also contrasts phase-space sampling, reporting a higher bound-orbit share in the best-mixture dataset than in the baseline.

3. Hessian de-normalization for model-vs-analytic comparability

  • The learned velocity Hessian is computed in normalized velocity coordinates and transformed back to physical coordinates for comparison with the analytical Hessian.The passage says the conversion applies the chain rule twice.Source: §3 · pp. 20–21

Appendix D: Additional Prediction Plots

  • The appendix plots all eight orbital scenarios in a 4 × 2 grid, comparing analytical geodesics with neural-network predictions across close, medium, and far regions.Each scenario block places the analytical geodesic on the left and the network prediction on the right.Source: Appendix D · p. 21, Figure A4 · p. 22
  • 92.3% of best-mixture samples are bound orbits, compared with 27.3% for baseline sampling.The best-mixture dataset combines two radial-velocity windows for more uniform coverage of bound-orbit phase space.Source: Figure A5 · p. 23, Figure A6 · p. 24
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