[Paper Review] Symplectic ODE-Net: Learning Hamiltonian Dynamics with Control

Yaofeng Desmond Zhong, Biswadip Dey+1 more|arXiv (Cornell University)|Sep 26, 2019

Model Reduction and Neural NetworksPhysics and Astronomy25 references91 citations

TL;DR

Symplectic ODE-Net (SymODEN) learns dynamics that obey Hamiltonian structure with external control, handling embedded angle data and enabling energy-based control insights with interpretable physical components.

ABSTRACT

In this paper, we introduce Symplectic ODE-Net (SymODEN), a deep learning framework which can infer the dynamics of a physical system, given by an ordinary differential equation (ODE), from observed state trajectories. To achieve better generalization with fewer training samples, SymODEN incorporates appropriate inductive bias by designing the associated computation graph in a physics-informed manner. In particular, we enforce Hamiltonian dynamics with control to learn the underlying dynamics in a transparent way, which can then be leveraged to draw insight about relevant physical aspects of the system, such as mass and potential energy. In addition, we propose a parametrization which can enforce this Hamiltonian formalism even when the generalized coordinate data is embedded in a high-dimensional space or we can only access velocity data instead of generalized momentum. This framework, by offering interpretable, physically-consistent models for physical systems, opens up new possibilities for synthesizing model-based control strategies.

Motivation & Objective

• Introduce Symplectic ODE-Net (SymODEN) that encodes generalized Hamiltonian dynamics with control.
• Achieve better generalization and data efficiency through physics-informed computation graphs.
• Enable learning from embedded angle data without requiring higher-order derivatives.
• Provide interpretable learnt components such as mass and potential energy for model-based control.

Proposed method

• Parametrize inverse mass matrix M^{-1}(q), potential V(q), and input map g(q) with neural nets to form Hamiltonian H and dynamics with control (Equations 12–13, 4).
• Use Neural ODE with augmented dynamics to include constant external forcing u (Equation 11).
• Extend to embedded angle data by deriving angle-aware dynamics with embedded representations (Equations 14–19).
• Handle hybrid state spaces R^n × T^m by using x1, x2, x3 representations and corresponding Hamiltonian dynamics (Equations 20–24).
• Enforce positive definiteness of M^{-1} via Cholesky-like factorization to stabilize training (Section 3.5).
• Synthesize energy-shaping and damping-based controllers using learned Hamiltonian components (Equations 7–10).

Experimental results

Research questions

• RQ1 Can a Hamiltonian dynamics framework with control be learned from time-series data using neural networks?
• RQ2 How can we learn dynamics when generalized coordinates are embedded (e.g., angles) rather than in Euclidean space?
• RQ3 Does incorporating physics priors improve generalization and enable interpretable physical quantities (mass, potential energy)?
• RQ4 Can the learned model support energy-based control strategies with guarantees and robustness?
• RQ5 How does SymODEN compare to unstructured baselines on classic robotic dynamics tasks?

Key findings

• SymODEN encodes dynamics that respect Hamiltonian structure with an additive control term, enabling energy-conserving trajectories.
• The learned components (mass, potential, and input mapping) align with physical interpretations, enabling insight into inertia and energy.
• SymODEN effectively handles embedded angle data and hybrid spaces without requiring higher-order derivatives.
• Energy shaping and damping injection can be derived from the learned Hamiltonian framework to synthesize nonlinear controllers.
• Compared to unstructured baselines, SymODEN achieves better generalization and can operate with smaller network sizes due to physics-based inductive bias.