cs.LGOct 9, 2026

Learning Schwarzschild Geodesic Motion with Lagrangian Neural Networks

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

Organizations: Department of Physics, Bogazici University, Istanbul, Turkey · Department of Computer Engineering, Bogazici University, Istanbul, Turkey

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Abstract

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.

Figures & tables

Figure 1: Progression of loss graph for not normalized and normalized models. Y axis represents the ratio between the loss value in current epoch and the initial loss.
Figure 2: Effects of normalization and training dataset on the trajectory predictions. Analytical solution(left), legacy normalization training with balanced mixture dataset(middle left), baseline dataset training with normalization(middle right) and best mixture dataset training with decoupled normalization(right).Comparing the second and fourth panels isolates the effect of the normalization scheme on the mixed-velocity dataset. Comparing the third and fourth panels instead isolates the effect of dataset sampling while retaining the proposed normalization.
 r˙\dot{r} ( ×10−1\times 10^{-1} )ϕ˙\dot{\phi} ( ×10−3\times 10^{-3} )LLEEeerminr_{min} - rmaxr_{max}
1[[0.1,,0.2]][[2.2,,2.4]]8.46328.46320.99300.99300.19020.190257.5184.5357.5184.53
2[[0.1,,0.2]][[2.4,,2.6]]9.18589.18580.99480.99480.37690.376958.91130.1758.91130.17
3[[0.1,,0.2]][[2.7,,2.8]]10.012910.01290.99690.99690.62850.628559.40260.4259.40260.42
Table 1: Initial conditions and orbital metrics for close orbit scenarios.
Figure 3: Side-by-side comparison for e=e=0.3769$$ test orbit. Analytical to the left and model prediction to the right.
Figure 4: Overlay of analytic and predicted trajectories from scenario 3. The error accumulation in the precession is slight yet visible as the orbit completes new revolutions.
Figure 5: Energy conservation error (top) and angular momentum conservation error (bottom) for close region scenario 2. Analytical values for both quantities are computed from the Killing constants, which are conserved along the geodesic.
 r˙\dot{r} ( ×10−1\times 10^{-1} )ϕ˙(×10−3)\dot{\phi}(\times 10^{-3})LLEEeerminr_{min} - rmaxr_{max}
1[[0.1,,0.2]][[1.0,,1.2]]9.47229.47220.99440.99440.15330.153375.08102.2775.08102.27
2[[0.1,,0.2]][[1.25,,1.35]]11.639211.63920.99680.99680.39450.394594.83218.3994.83218.39
3[[0.0,,0.0]][[1.17,,1.19]]9.63389.63380.99450.99450.01010.010188.8190.6288.8190.62
Table 2: Initial conditions and orbital metrics for medium orbit scenarios.
Figure 6: Side-by-side comparison for e=e=0.0101$$ test orbit in medium region. Analytical to the left and model prediction to the right. This orbit is highly circular which the model accurately predicts.
Figure 7: Evolution of the radial distance over time for analytical and predicted trajectory. Accumulation of error is visible as the predicted orbit drifts and gains eccentricity.
 r˙\dot{r} ( ×10−1\times 10^{-1} )ϕ˙(×10−3)\dot{\phi}(\times 10^{-3})LLEEeerminr_{min} - rmaxr_{max}
1[[0.1,,0.2]][[0.35,,0.37]]14.72814.7280.99780.99780.23470.2347173.19279.39173.19279.39
2[[0.0,,0.01]][[0.353,,0.356]]14.23214.2320.99750.99750.00820.0082197.91201.16197.91201.16
Table 3: Initial conditions and orbital metrics for far orbit scenarios.
Figure 8: Side-by-side comparison for e=e=0.2347$$ test orbit. Slight precession is visible in both analytical and predicted trajectories.
Figure 9: Spatial coordinate evolutions for analytical vs. predicted orbit with their velocities.
Figure 10: Time velocities for elliptical orbits from each region with their coordinate-time and time-velocity errors. Shown here are: close region scenario 3 (left), medium region scenario 2 (middle) and far region scenario 1 (right).
ScenarioEERadial rangeMax t˙\dot{t}
Close0.99690.996959.4059.40 – 260.42260.421.03161.0316
Medium0.99680.996894.8394.83 – 218.39218.391.01831.0183
Far0.99780.9978173.19173.19 – 279.39279.391.00951.0095
Table 4: Conserved specific energy, radial range, and expected range of t˙=dt/dτ\dot{t}=dt/d\tau . The maximum values are obtained from t˙=E/(1−rs/r)\dot{t}=E/(1-r_{s}/r) .
Figure 11: Acceleration components comparisons. Plots for each component represents combinations of two coordinates on which the anaytical forms from Eq. 13 is dependent.
Figure 12: Lagrangian function comparisons in each coordinate. t and ϕ\phi plots are invariants of the system and the function should be constant under those coordinates. Therefore, relevant plots has been kept unnormalized to see the deviations better. The rest of the plots are normalized using the zscore for both analytical (blue) and learned functions (red), independently.
Figure 13: 0 velocity Lagrangian plots. Learned function contains r and phi dependent additional terms that is not present in the analytical form.
Figure 14: Acceleration magnitudes in r,ϕr,\phi space for r=[50,150]r=[50,150] with fixed r˙\dot{r} and ϕ˙\dot{\phi} .
Figure 15: Normalized Hessian components as function of r. Diagonal mismatch is most present in ϕ,ϕ\phi,\phi component.
Appendix figures & tables7 assets

Supplementary material from the paper’s appendix.

Appendix

ParameterValue
Architecture
Input dimension6 ( t,r,ϕ,t˙,r˙,ϕ˙t,r,\phi,\dot{t},\dot{r},\dot{\phi} )
Hidden layers4
Neurons per layer500
Activation functionGELU
OutputScalar LθL_{\theta} (1)
Appendix
Table AI: Model architecture and training hyperparameters. The values shown are those used to produce the main results.
Figure A1: Mean radial error for each scenario and model across the test time span 0≤τ/M≤5×1040\leq\tau/M\leq 5\times 10^{4} . The mixed-velocity sampler (orange) yields the best trajectory prediction accuracy across all eight test scenarios, while the baseline s ampler performs the worst. The energy-sampling method performs better than the baseline but worse than the velocity-based samplers at highly elliptical or far scenarios.
Datasetr˙\dot{r} range ( ×±1\times\pm 1 units/s)ϕ˙\dot{\phi} range ( ×±10−3\times\pm 10^{-3} rad)EmaxE_{\text{max}}Ueff2U_{\text{eff}}^{2} rangeBound Ratio
Baseline Sampling[[5.960,46\text{\times}{10}^{-8},,0.180,00]][[3.678,72\text{\times}{10}^{-5},,3.499,99]]–[[0.960,02,,2.081,32]]27 %27\text{\,}\mathrm{\%}
Energy Sampling[[3.569,60\text{\times}{10}^{-6},,0.178,73]][[3.921,56\text{\times}{10}^{-2},,3.930,16]]0.9990.999[[0.964,66,,0.998,00]]100 %100\text{\,}\mathrm{\%}
Constrained Velocity Sampling[[2.000,67\text{\times}{10}^{-3},,0.179,99]][[8.002,04\text{\times}{10}^{-2},,3.998,58]]1.0001.000[[0.964,74,,0.999,99]]100 %100\text{\,}\mathrm{\%}
Velocity Sampling[[1.618,09\text{\times}{10}^{-6},,0.179,71]][[5.008,41\text{\times}{10}^{-2},,3.499,92]]1.0011.001[[0.964,66,,1.001,98]]88 %88\text{\,}\mathrm{\%}
Best Mixture[[3.648,01\text{\times}{10}^{-9},,0.178,20]][[5.001,90\text{\times}{10}^{-2},,3.499,94]]1.0011.001[[0.964,74,,1.002,00]]92 %92\text{\,}\mathrm{\%}
(The baseline sampler has no energy cutoff; its maximum observed energy is E=1.443E=1.443 .)
Appendix
Table AII: Velocity ranges, energy-filter thresholds, effective-potential ranges, and fractions of samples satisfying (E<1)(E<1) for the five dataset variants. All quantities are expressed in geometric units with (G=c=1)(G=c=1) .
Figure A3: Progression of loss graph for baseline models and models trained with datasets created by energy sampling and velocity sampling. Y axis represents the ratio between the loss value in current epoch and the initial loss.
Figure A4: Trajectory predictions for all eight orbital scenarios arranged in a 4×24\times 2 grid across the close (1–3), medium (4–6), and far (7–8) regions. Each scenario block presents a side-by-side comparison between the analytical geodesic (left) and the neural network prediction (right).
Figure A5: Hexbin density analysis of the best-mixture (velocity sampling) dataset in the (r˙,ϕ˙)(\dot{r},\dot{\phi}) velocity phase-space. The balanced mixture combines two radial velocity windows to achieve the most uniform coverage of bound-orbit phase-space.
Figure A6: Hexbin density analysis of the baseline sampling dataset in the (r˙,ϕ˙)(\dot{r},\dot{\phi}) velocity phase-space. The baseline dataset exhibits strong density bias toward high-velocity regions and contains only 27.3 %27.3\text{\,}\mathrm{\%} bound orbits, compared to 92.3 %92.3\text{\,}\mathrm{\%} for the best-mixture dataset.
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