Discovering Phase Space Structure in Learned Hamiltonian Systems
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Introduction to the show: ident: AI Radio. Generated commentary on the latest Artificial Intelligence papers.
Tom: Today's paper: "Discovering Phase Space Structure in Learned Hamiltonian Systems".
Jane: The gist The Symplectic Phase Space GAN (SPS-GAN) is introduced as a novel framework capable of capturing the dynamics of multiple systems and discovering system symmetries without requiring prior knowledge…
Tom: First, who's behind it and why it matters.
Title and authors: Tom: Let's talk about the title and authors of "Discovering Phase Space Structure in Learned Hamiltonian Systems." The authors include Jiayin Liu, Yulong Yang, Vineet Bansal, and Christine Allen-Blanchette. They are from Princeton and Tsinghua University.
Jane: So we know these folks have done some heavy lifting in physics modeling before this paper started. The title itself points to the main goal: discovering phase space structure within systems that have been learned by neural networks.
Lu: The authors are clearly building on existing work in generative models, specifically using a conditional GAN backbone and then adding a Hamiltonian Neural Network to control the latent space dynamics according to Hamilton’s equations.
Meng: What I want to ask is how they handle the fact that these networks usually assume you already know things like the dimension of coordinates or how many constraints there are. That seems like a huge hurdle for any practical application of this research.
Lalam: It’s interesting because if they can discover the structure from arbitrary measurements, it means we don't need to pre-program every single physical detail into the model architecture anymore.
The paper's summary: Tom: To summarize what this paper is actually doing, it introduces SPS-GAN which takes a random motion sample and maps it onto an intermediate space that they interpret as the configuration space of the system. They then decode these latent trajectories back into real things like Cartesian paths or videos for checking.
Jane: That mapping process is key because it’s conditioned on system parameters, like whether you’re looking at a two-body system or something else. This lets it generalize beyond just one fixed set of physical rules.
Lu: They use a Hamiltonian Neural Network to govern the movement within this latent space, which ensures the latent trajectories are physically consistent in terms of energy conservation and dynamics.
Meng: So they are using the GAN structure for generation but controlling how those generated states evolve using mechanics, rather than just relying on statistical likelihood alone. That’s a sophisticated way to enforce physics during generation.
Lalam: It moves beyond just generating plausible *data* and aims to generate physically plausible *trajectories* for multiple systems at once, which is a step up in complexity.
The paper's improvements: Tom: The authors point out several improvements they made, particularly regarding how they find symmetries. They introduced a cyclic coordinate loss that encourages the configuration space learned by the model to have a minimal dimension by regularizing those momentum terms.
Jane: That cyclic coordinate loss is really smart because it directly tries to minimize the complexity of the phase space representation, which is exactly what we want when we're trying to discover inherent symmetries.
Lu: They also showed that this approach works well for identifying symmetries without needing any prior information about those symmetries, unlike some previous methods that relied on sweeping through constants of motion or Lie group structures.
Meng: The paper demonstrates this by successfully identifying the degrees of freedom for various systems, showing it can correctly identify the two-body system as one dimension and the double pendulum as two dimensions. That’s a concrete result.
Lalam: So they are showing that this framework is more general than existing tools because it doesn't rely on those specific prior knowledge assumptions about symmetry identification methods.
Conclusion: Tom: Wrapping things up, the main implication here is that SPS-GAN gives us a way to generate trajectory and video data for multiple systems under different settings while actively searching for system symmetries without needing a map of the configuration space first.
Jane: It achieves predictive performance on par with supervised models when it comes to identifying the degrees of freedom for both ordered and chaotic dynamical systems, which is quite impressive.
Lu: This work shows that we can discover the actual dimension of a system's phase space by minimizing the learned representation's complexity, which is a novel way to approach symmetry discovery.
Meng: From an engineering standpoint, it’s powerful because it moves us toward models that are less brittle when applied to physical systems we haven't perfectly characterized yet.
Lalam: Ultimately, this paper on "Discovering Phase Space Structure in Learned Hamiltonian Systems" provides a general framework for generating data and finding structure in learned dynamics across different physical setups.
Princeton University
cs.LG, cs.AI, cs.SY, eess.SY
Submitted: 2025-09-26
Updated: 2026-10-08
Comments: Accept to The 40th Annual Conference on Neural Information Processing Systems (NeurIPS 2026). Camera-ready to come
License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
Importance score: 82/100
The gist: The gist The Symplectic Phase Space GAN (SPS-GAN) is introduced as a novel framework capable of capturing the dynamics of multiple systems and discovering system symmetries without requiring prior
Key concepts
- Structured Phase Space GAN (SPS-GAN)
- A conditional generative model that learns structured dynamics from both Cartesian trajectories and video. It combines a port-Hamiltonian backbone with adversarial generation to represent conservative, dissipative, and forced systems in one architecture.
- Configuration Space Map $f(oldsymbol{ ho}_m, oldsymbol{ ho}_ heta)$
- This map takes a random motion sample ($oldsymbol{ ho}_m$) and system parameters ($oldsymbol{ ho}_ heta$) to generate an intermediate space. This intermediate space is interpreted as the configuration space of the underlying dynamical system being modeled.
- Cyclic Coordinate Loss ($L_{ ext{cyclic}}$)
- A loss function added to the GAN objective that penalizes momentum terms. It encourages the learned configuration space representation to have a minimal dimension by regularizing these terms, helping to identify system symmetries.
Terminology
Summary
The gist The Symplectic Phase Space GAN (SPS-GAN) is introduced as a novel framework capable of capturing the dynamics of multiple systems and discovering system symmetries without requiring prior knowledge of the configuration space structure.
SPS-GAN Architecture
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SPS-GAN is designed on a conditional GAN backbone with the latent space dynamics governed by a Hamiltonian Neural Network which allows us to generate physically consistent latent trajectories.
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The configuration space map f (ϵm, ξ): R → R 2×dlat, conditioned on ξ = [system label, physical parameters,...], maps a random motion sample ϵm ∼ N (0, 1) onto an intermediate space which we interpret as the configuration space of the underlying dynamical system.
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The latent trajectories are decoded into Cartesian trajectories or videos for discrimination.
Latent Space Dynamics and Constraints
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The motion on the latent space is governed by an HNN conditioned on the system label and physical parameters ξ.
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The initial condition of the system is given by z0 = f(ϵm, ξ) where ϵm is a motion sample and z0 = (q0, p0).
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A cyclic coordinate loss Lcyclic = λcyclicXp˙i is incorporated into the GAN objective to encourage identification of system symmetries. This loss encourages the learned configuration space to have a minimal dimension by regularizing the momentum terms.
Trajectory and Video Generation
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When generating Cartesian trajectories, SPS-GAN-traj uses an MLP GT: R 2×(dlat+dcont) → R 2×dout to decode each entry of the latent sequence.
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The final generated Cartesian trajectory is xˆ = [xˆ0,..., xˆT], where xˆt = x˜t · M(ξ).
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When generating video data, SPS-GAN-video leverages a CNN decoder GI: R 2×dlat+dcont → R dh×dw×3 to decode each entry of the latent sequence into RGB images.
Performance and Symmetry Discovery
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SPS-GAN is able to achieve predictive performance on par with supervised models; identify the number of degrees of freedom for a wide range of dynamical systems, both ordered and chaotic.
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SPS-GAN demonstrates its ability to identify symmetries by minimising the dimension learned phase space.
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In all experiments, the projection shows that SPS-GAN is able to correctly identify a the two-body system as 1-dimensional, and the double pendulum and three-body systems as 2-dimensional.
Ablation Studies
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The ablation study on cyclic coordinate loss shows that SPS-GAN is no longer able to learned the most compact representation with decreasing cyclic coordinate loss weight.
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Replacing the HNN (Greydanus et al., 2019) with a Gated Recurrent Unit (GRU) for the two-body system shows that SPS-GAN with HNN module outperforms the ablated SPS-GAN with GRU significantly.
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The study on non-Hamiltonian systems shows that when modelling non-Hamiltonian systems with SPS-GAN, a non-minimal latent space is learned.
Conclusion
SPS-GAN is a general framework to generate both Cartesian trajectory and video data of multiples systems under variable parameters and discover symmetry in dynamical systems. SPS-GAN demonstrates its ability to generate trajectory data with accuracy on par with supervised baselines and video data under different settings with consistency significantly higher than existing physics-informed baselines.
Improvements for AI systems
- Bold header: Discovery of system configuration space structure for unseen systems
SPS-GAN can discover the configuration space structure of the system from arbitrary measurement types (e.g., state-space measurements, video frames),
allowing it to generalize to unseen physical parameters.
- Bold header: Identification of minimal degrees of freedom
The model can identify the number of degrees of freedom for a wide range of dynamical systems, both ordered and chaotic,
as demonstrated by its ability to correctly identify that the minimal degrees of freedom required to represent the system is two
for a double pendulum.
- Bold header: Generation of physically consistent video data
SPS-GAN can generate physically plausible videos with more consistency than baseline physics informed methods,
specifically showing high consistency (FVD values) when modeling multiple systems under varying physical and visual characteristics.
Sources
- GeoHNNs: Geometric Hamiltonian Neural Networks
- LagNetViP: A Lagrangian Neural Network for Video Prediction
- Symplectic Recurrent Neural Networks
- On the Properties of Neural Machine Translation: Encoder-Decoder Approaches
- Lagrangian Neural Networks
- Adam: A Method for Stochastic Optimization
- Auto-Encoding Variational Bayes
- Latent Mixture of Symmetries for Sample-Efficient Dynamic Learning
- Deep Lagrangian Networks: Using Physics as Model Prior for Deep Learning
- Quantum dynamics of a one degree-of-freedom Hamiltonian saddle-node bifurcation
- Learning Interpretable Dynamics from Images of a Freely Rotating 3D Rigid Body
- FINDE: Neural Differential Equations for Finding and Preserving Invariant Quantities
- Conditional Generative Adversarial Nets
- Hamiltonian Generative Networks
- Towards Accurate Generative Models of Video: A New Metric & Challenges
- Behavior-Inspired Neural Networks for Relational Inference
- Learning Color Equivariant Representations
- GAGrasp: Geometric Algebra Diffusion for Dexterous Grasping
- Symplectic ODE-Net: Learning Hamiltonian Dynamics with Control
- Dissipative SymODEN: Encoding Hamiltonian Dynamics with Dissipation and Control into Deep Learning
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