Physically Plausible Multi-System Trajectory Generation and Symmetry Discovery

TL;DR

SPS-GAN addresses the challenge of generating physically plausible trajectories for multiple dynamical systems with unseen parameters and without prior configuration-space knowledge. It combines a configuration-space map, a Hamiltonian neural network latent motion model, and a cyclic-coordinate loss within a conditional GAN to produce both Cartesian trajectories and videos, while discovering the effective configuration-space dimensionality. The approach rivals supervised baselines in trajectory prediction, uncovers symmetry-induced low-dimensional latent manifolds for classic systems (e.g., 1D for two-body, 2D for double- and three-body), and delivers robust video generation under parameter variation, including real-world pendulum data and bifurcation scenarios. These results demonstrate the practical potential of unsupervised symmetry discovery and multi-system generative modeling for physics-informed AI, with strong implications for energy-aware control and simulation. The framework also highlights limitations related to non-Hamiltonian dynamics and decoding constraints, pointing to future work on integrating broader Hamiltonian structures into discriminators and extending learned representations to downstream tasks.

Abstract

From metronomes to celestial bodies, mechanics underpins how the world evolves in time and space. With consideration of this, a number of recent neural network models leverage inductive biases from classical mechanics to encourage model interpretability and ensure forecasted states are physical. However, in general, these models are designed to capture the dynamics of a single system with fixed physical parameters, from state-space measurements of a known configuration space. In this paper we introduce Symplectic Phase Space GAN (SPS-GAN) which can capture the dynamics of multiple systems, and generalize to unseen physical parameters from. Moreover, SPS-GAN does not require prior knowledge of the system configuration space. In fact, SPS-GAN can discover the configuration space structure of the system from arbitrary measurement types (e.g., state-space measurements, video frames). To achieve physically plausible generation, we introduce a novel architecture which embeds a Hamiltonian neural network recurrent module in a conditional GAN backbone. To discover the structure of the configuration space, we optimize the conditional time-series GAN objective with an additional physically motivated term to encourages a sparse representation of the configuration space. We demonstrate the utility of SPS-GAN for trajectory prediction, video generation and symmetry discovery. Our approach captures multiple systems and achieves performance on par with supervised models designed for single systems.

Figures (12)

• Figure 1: SPS-GAN-traj for generating Cartesian trajectory. The random motion sample $\epsilon_{m}$ is mapped onto the manifold for the dynamical system to create the latent initial condition of the system. The initial conditional is propagated in time using an HNN block to generate a trajectory on the latent space. Cartesian trajectories are generated by passing the latent trajectories through the $G_{\mathrm{T}}$.
• Figure 2: SPS-GAN-vid for generating video. The random motion sample $\epsilon_{m}$ is mapped onto the manifold for the dynamical system to create the initial condition of the system. The initial conditional is propagated in time using a HNN block to generate the trajectory on the latent space, which is concatenated with content sample $\epsilon_{c}$. Video frames are generated by passing the latent trajectories through $G_{\mathrm{I}}$. Discriminators $D_{\mathrm{V}}$ and $D_{\mathrm{I}}$ ensures realistic dynamics and content respectively.
• Figure 3: Accuracy of predicted trajectory when modelling a single system. Qualitative comparison between ground-truth (dotted) and generated (solid) trajectories across ideal pendulum, double pendulum, planar two-body systems, and planar three-body systems is shown. When generating dynamics from one system at a time, the generated trajectory from SPS-GAN closely follows reference dynamics.
• Figure 4: Accuracy of predicted trajectory when modelling multiple systems. Qualitative comparison between ground-truth (dotted) and generated (solid) trajectories across ideal pendulum, double pendulum, planar two-body systems, and planar three-body systems is shown. When generating dynamics from five distinct systems SPS-GAN is able to disentangle different dynamics and the generated trajectory from SPS-GAN closely follows reference dynamics.
• Figure 5: T-SNE projection of learned latent space.(a-c) t-SNE projection of learned latent spaces for the double pendulum, planar two-body, and planar three-body; (d-e) FastICA hyvarinen2000independent projection of trajectories for the double pendulum, planar two-body, and planar three-body. The t-SNE projections indicate that the learned latent dimensions is 1 for two-body and 2 for double pendulum and three-body while FastICA on the trajectory shows no discernable structure. This matches physical intuition for the three systems and indicates that SPS-GAN is able to learned the correctly sized latent dimension without supervision and a priori knowledge.