cs.LGSep 26, 2025

Discovering Phase Space Structure in Learned Hamiltonian Systems

Authors: Yulong Yang, Jiayin Liu, Vineet Bansal, Christine Allen-Blanchette

Organizations: Princeton University, USA · Tsinghua University, China

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Abstract

Mechanics-inspired neural dynamical models improve interpretability and physical consistency, but they usually assume a known configuration space, fixed physical parameters, and deterministic prediction from a prescribed initial condition. We introduce Structured Phase Space GAN (SPS-GAN), a conditional generative model that learns structured dynamics from both Cartesian trajectories and video. SPS-GAN combines a port-Hamiltonian backbone with adversarial conditional generation, allowing conservative, dissipative, and forced systems to be represented in a single architecture. To infer latent mechanical structure from raw observations, we use a cyclic-coordinate loss that recovers both the system degrees of freedom and canonical coordinates without access to the true phase space. Experiments on simulated and real-world systems show that SPS-GAN improves trajectory accuracy over supervised dynamics baselines and video consistency over generative baselines, while consistently recovering the correct latent mechanical coordinates across observation modalities and dynamical regimes.

Figures & tables

Figure 1: SPS-GAN generates trajectories and videos across multiple systems through a shared phase-space backbone. Random motion samples ϵm\boldsymbol{\epsilon}_{\mathrm{m}} and system information ξ\xi are passed through the phase-space map ff to produce an initial phase-space state z0\mathbf{z}_{0} . This state is evolved forward in time by a Port-Hamiltonian Neural Network ( Desai et al., 2021 ) conditioned on ξ\xi , yielding the phase-space trajectory [z0:T]\left[\mathbf{z}_{0:T}\right] , which is decoded to (a) Cartesian trajectory [x^0:T][\hat{\mathbf{x}}_{0:T}] and (b) video frames [v^0:T][\hat{\mathbf{v}}_{0:T}] .
NetworkMass SpringSingle Pend.Double Pend.Two BodyThree Body
SPS-GAN6.16×10−46.16\times 10^{-4}4.29×10−4\boldsymbol{4.29\times 10^{-4}}3.25×10−2\boldsymbol{3.25\times 10^{-2}}3.07×10−3\boldsymbol{3.07\times 10^{-3}}1.26×10−3\boldsymbol{1.26\times 10^{-3}}
HNN3.53×10−43.53\times 10^{-4}6.27×10−46.27\times 10^{-4}1.25×10−11.25\times 10^{-1}3.61×10−3‾\underline{\smash{3.61\times 10^{-3}}}2.81×10−32.81\times 10^{-3}
SymODEN6.40×10−46.40\times 10^{-4}6.56×10−46.56\times 10^{-4}9.89×10−29.89\times 10^{-2}6.20×10−36.20\times 10^{-3}8.08×10−38.08\times 10^{-3}
TRS-ODEN6.72×10−46.72\times 10^{-4}9.57×10−49.57\times 10^{-4}6.12×10−26.12\times 10^{-2}6.36×10−36.36\times 10^{-3}6.22×10−36.22\times 10^{-3}
NeuralODE1.62×10−4‾\underline{\smash{1.62\times 10^{-4}}}1.59×10−31.59\times 10^{-3}5.99×10−25.99\times 10^{-2}5.72×10−35.72\times 10^{-3}3.67×10−33.67\times 10^{-3}
LatentODE1.13×10−4\boldsymbol{1.13\times 10^{-4}}9.27×10−49.27\times 10^{-4}4.66×10−24.66\times 10^{-2}1.98×10−21.98\times 10^{-2}1.37×10−3‾\underline{\smash{1.37\times 10^{-3}}}
Table 1: Trajectory generation accuracy on conservative systems. We report MSE between each generated or predicted trajectory and the simulator rollout initialized from the same state. SPS-GAN is trained per system, while cSPS-GAN is trained jointly across all five systems with system labels. Best results are bolded and second-best results are underlined.
NetworkIdeal Pend.Damped Pend.Forced Duffing
SPS-GAN4.77×10−4\boldsymbol{4.77\times 10^{-4}}4.57×10−4\boldsymbol{4.57\times 10^{-4}}5.22×10−4\boldsymbol{5.22\times 10^{-4}}
pHNN8.42×10−4‾\underline{\smash{8.42\times 10^{-4}}}4.76×10−4‾\underline{\smash{4.76\times 10^{-4}}}6.92×10−46.92\times 10^{-4}
NeuralODE1.59×10−31.59\times 10^{-3}6.85×10−46.85\times 10^{-4}7.35×10−47.35\times 10^{-4}
LatentODE9.27×10−49.27\times 10^{-4}6.69×10−46.69\times 10^{-4}1.18×10−31.18\times 10^{-3}
TRS-ODEN9.57×10−49.57\times 10^{-4}1.44×10−31.44\times 10^{-3}6.50×10−4‾\underline{\smash{6.50\times 10^{-4}}}
Table 2: Trajectory generation accuracy on non-conservative systems. We report MSE under the same initial-condition alignment protocol as Table 1 . SPS-GAN uses the port-Hamiltonian formulation for dissipative and forced systems; the ideal pendulum is included as a conservative control. Best results are bolded and second-best results are underlined.
 NetworkMass-SpringSingle Pend.Double Pend.Two-BodyThree-Body
FVDSPS-GAN25.6340.5724.1287.1289.08
HGAN45.6891.6473.21105.851981.10
HGN385.08688.12331.94830.91451.40
ΔE\Delta ESPS-GAN2.33%2.33\%2.89%2.89\%6.83%6.83\%5.96%5.96\%3.47%3.47\%
Simulator1.64%1.64\%2.62%2.62\%6.32%6.32\%6.11%6.11\%3.57%3.57\%
Table 3: Video generation quality and physical consistency. We report FVD against baselines and energy variance ( ΔE\Delta E ) of centroid-extracted trajectories. SPS-GAN achieves the lowest FVD on all rendered conservative systems, while its ΔE\Delta E closely matches the simulator reference floor.
Figure 2: Generated videos of real-world systems. On non-ideal system videos, SPS-GAN produces temporally consistent videos of (left) a single ( Garcia et al., 2025 ) and (right) a double pendulum ( Kaheman et al., 2023 ) . HGN collapses to a static frame and HGAN fails to capture correct dynamics. For the single pendulum, the energy drift of centroid-extracted SPS-GAN videos ( 2.9%2.9\% ) closely matches that of real recordings ( 2.7%2.7\% ).
 MassSin.Dou.TwoThreeDamp.Force.Sin.Dou.Sin.Dou.
 SpringPend.Pend.BodyBodyPend.Duff.Pend.Pend.Pend.Pend.
ModeTraj.Traj.Traj.Traj.Traj.Traj.Traj.Traj.Traj.VideoVideo
SourceSimSimSimSimSimSimSimRealRealRealReal
PVE1\mathrm{PVE}_{1}98.8597.1678.7095.8781.8795.7996.4796.2861.3998.6970.47
PVE2\mathrm{PVE}_{2}0.480.7918.391.3217.401.281.021.9135.410.6424.63
PVE3\mathrm{PVE}_{3}0.180.580.901.050.240.790.760.701.960.460.88
Table 4: Identifying active degrees of freedom. We report the percentage variance explained (PVE) by the leading principal components of the learned phase space. Components contributing to 95% cumulative PVE are highlighted in green . The resulting active dimension matches the ground-truth degrees of freedom across simulated and real trajectory/video observations.
Figure 3: Interpretable coordinates from active latent directions. Sweeping active latent directions and decoding to Cartesian space reveals coordinates aligned with known mechanical variables: (a) radial distance for two-body; (b) orbital angle and radial scale for three-body; (c) two angular positions for double pendulum.
 SPS-GAN+SympNet+GRUw/o Phase Space Mapw/o Cyclic Loss
MSE3.07×10−3\boldsymbol{3.07\times 10^{-3}}4.43×10−34.43\times 10^{-3}1.60×10−11.60\times 10^{-1}4.93×10−24.93\times 10^{-2}4.40×10−24.40\times 10^{-2}
DoF12373
Table 5: Ablation analysis on two-body system. We ablate the phase-space map, cyclic-coordinate loss, and pHNN module. Each ablation degrades both accuracy and recovered active dimension.
Appendix figures & tables11 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 4: Example Cartesian trajectory rollout. Comparison between ground-truth (dotted) and generated (solid) trajectories across the mass-spring, single pendulum, double pendulum, two-body, and three-body systems. Whether generating dynamics from (a) one system at a time or (b) multiple systems jointly, SPS-GAN trajectories closely follow the reference dynamics.
Figure 5: Example video rollout under fixed parameters. Real system (top) and generated video (bottom) for (a) mass-spring oscillator, (b) ideal pendulum, (c) double pendulum, (d) two-body, and (e) three-body system, all under fixed parameters.
Figure 6: Generated video under color and physics variation. SPS-cGAN generated video of mass-spring and two body systems with (i) physics and color varied (FVD 182.15), (ii) only physics varied (FVD 135.38), and (iii) only color varied (FVD 135.25). The default color setting uses yellow, red, and green.
Figure 7: Accuracy of generated trajectory for actuated pendulum. SPS-GAN generated trajectory under constant external actuation closely matches the reference dynamics from gymnasium [ 47 ] .
 MassSin.Dou.TwoThreeDamp.Force.Sin.Dou.Sin.Dou.
 SpringPend.Pend.BodyBodyPend.Duff.Pend.Pend.Pend.Pend.
ModeTraj.Traj.Traj.Traj.Traj.Traj.Traj.Traj.Traj.VideoVideo
SourceSimSimSimSimSimSimSimRealRealRealReal
PVE1\mathrm{PVE}_{1}98.8597.1678.7095.8781.8795.7996.4796.2861.3998.6970.47
PVE2\mathrm{PVE}_{2}0.480.7918.391.3217.401.281.021.9135.410.6424.63
PVE3\mathrm{PVE}_{3}0.180.580.901.050.240.790.760.701.960.460.88
Appendix
Table 6: Identifying system degrees of freedom. SPS-GAN correctly recovers the degrees of freedom for conservative and non-conservative systems, under observations of Cartesian trajectory and video recordings. PCA components contributing to the 95% are highlighted in green .
a1\mathbf{a}_{1}t=0t=0t=15t=15t=30t=30
t=0t=01.0000.99970.9988
t=15t=150.99971.0000.9997
t=30t=300.99880.99971.000
Appendix
Table 7: Alignment of phase space at different times. We report the absolute cosine similarity between the first two eigenvectors of the phase space for the three-body system at t={0,15,30}t=\left\{0,15,30\right\} , where the phase space at different times are geometrically equivalent
Figure 8: Recovering system canonical coordinates. We show the recovered canonical coordinates for (a) simulated mass-spring and single pendulum, and (b) real world single pendulum videos.
dlatd_{\mathrm{lat}}10204080
DoF2222
s/ep0.2360.2360.2350.233
MB389390391393
Appendix
Table 8: Robustness of phase space dimension. We train SPS-GAN with phase space with dimension dlat={10,20,40,80}d_{\mathrm{lat}}=\left\{10,20,40,80\right\} , where SPS-GAN is able to consistently recover a two degrees of freedom representation for the three-body system. Furthermore, increase phase space dimension yields marginal increase in computational cost, both in terms of memory and training time, suggesting that moderate overparameterization is practical.
λcyclic\lambda_{\mathrm{cyclic}}0.050.050.030.030.010.010.000.00
PVE1\mathrm{PVE}_{1}96.4896.1784.9955.59
PVE2\mathrm{PVE}_{2}1.301.4313.5832.48
PVE3\mathrm{PVE}_{3}0.810.960.439.65
PVE4\mathrm{PVE}_{4}0.450.740.210.88
PVE5\mathrm{PVE}_{5}0.240.520.120.56
PVE6-10\mathrm{PVE}_{6\text{-}10}0.720.180.670.84
Appendix
Table 9: Ablation study of the cyclic coordinate loss for the two-body system. SPS-GAN correctly identifies a one degree of freedom phase space for the two-body system at λcyclic=0.03\lambda_{\mathrm{cyclic}}=0.03 , while progressively lower weights results in a non-minimal phase space. PCA components contributing to the 95% are highlighted in green .
NetworkMass SpringSingle Pend.Double Pend.
SPS-GAN(6.16±0.34)×10−4\left(6.16\pm 0.34\right)\times 10^{-4}(4.29±0.64)×10−4\boldsymbol{\left(4.29\pm 0.64\right)\times 10^{-4}}(3.25±0.14)×10−2\boldsymbol{\left(3.25\pm 0.14\right)\times 10^{-2}}
HNN(3.53±0.15)×10−4\left(3.53\pm 0.15\right)\times 10^{-4}(6.27±0.32)×10−4\left(6.27\pm 0.32\right)\times 10^{-4}(1.25±0.89)×10−1\left(1.25\pm 0.89\right)\times 10^{-1}
SymODEN(6.40±0.36)×10−4\left(6.40\pm 0.36\right)\times 10^{-4}(6.56±0.45)×10−4\left(6.56\pm 0.45\right)\times 10^{-4}(9.89±0.51)×10−2\left(9.89\pm 0.51\right)\times 10^{-2}
TRS-ODEN(6.72±0.68)×10−4\left(6.72\pm 0.68\right)\times 10^{-4}(9.57±1.87)×10−4\left(9.57\pm 1.87\right)\times 10^{-4}(6.12±0.31)×10−2\left(6.12\pm 0.31\right)\times 10^{-2}
NeuralODE(1.62±0.60)×10−4\left(1.62\pm 0.60\right)\times 10^{-4}(1.59±1.22)×10−3\left(1.59\pm 1.22\right)\times 10^{-3}(5.99±1.32)×10−2\left(5.99\pm 1.32\right)\times 10^{-2}
LatentODE(1.13±0.03)×10−4\boldsymbol{\left(1.13\pm 0.03\right)\times 10^{-4}}(9.27±1.06)×10−4\left(9.27\pm 1.06\right)\times 10^{-4}(4.66±0.49)×10−2\left(4.66\pm 0.49\right)\times 10^{-2}
Appendix
Table 10: Trajectory generation accuracy on conservative systems. The MSE of SPS-GAN generated trajectory is compared to predicted trajectory from baselines. SPS-GAN models single systems, while cSPS-GAN jointly models all five conditionally. When generating single system dynamics, unsupervised SPS-GAN outperforms supervised baselines. When jointly modeling systems, supervised conditional cSPS-GAN performs on par with single system baselines.
NetworkIdeal Pend.Damped Pend.Forced Duffing
SPS-GAN(4.77±0.13)×10−4\boldsymbol{\left(4.77\pm 0.13\right)\times 10^{-4}}(4.57±0.06)×10−4\boldsymbol{\left(4.57\pm 0.06\right)\times 10^{-4}}(5.22±0.62)×10−4\boldsymbol{\left(5.22\pm 0.62\right)\times 10^{-4}}
pHNN(8.42±0.88)×10−4\left(8.42\pm 0.88\right)\times 10^{-4}(4.76±0.19)×10−4\left(4.76\pm 0.19\right)\times 10^{-4}(6.92±0.84)×10−4\left(6.92\pm 0.84\right)\times 10^{-4}
NeuralODE(1.59±1.22)×10−3\left(1.59\pm 1.22\right)\times 10^{-3}(6.85±0.57)×10−4\left(6.85\pm 0.57\right)\times 10^{-4}(7.35±0.59)×10−4\left(7.35\pm 0.59\right)\times 10^{-4}
LatentODE(9.27±1.06)×10−4\left(9.27\pm 1.06\right)\times 10^{-4}(6.69±0.27)×10−4\left(6.69\pm 0.27\right)\times 10^{-4}(1.18±0.31)×10−3\left(1.18\pm 0.31\right)\times 10^{-3}
TRS-ODEN(9.57±1.87)×10−4\left(9.57\pm 1.87\right)\times 10^{-4}(1.44±0.32)×10−3\left(1.44\pm 0.32\right)\times 10^{-3}(6.50±0.38)×10−4\left(6.50\pm 0.38\right)\times 10^{-4}
Appendix
Table 11: Trajectory generation accuracy on non-conservative systems. The MSE error of SPS-GAN generated trajectory is reported compared to predicted trajectory from baselines. Unsupervised SPS-GAN outperforms supervised single-system baselines.
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