Symplectic ODE-Net: Learning Hamiltonian Dynamics with Control (1909.12077v5)

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.

• The paper introduces SymODEN, a framework that extends Hamiltonian dynamics by incorporating external control inputs for improved prediction accuracy.
• The method enforces a symplectic structure to learn system behavior from limited samples, even in challenging non-Euclidean state spaces.
• Experimental results on pendulum, CartPole, and Acrobot tasks demonstrate superior performance and energy conservation compared to baseline models.

Symplectic ODE-Net: Learning Hamiltonian Dynamics with Control

The paper introduces the Symplectic ODE-Net (SymODEN), a framework for learning the dynamics of physical systems governed by ordinary differential equations (ODEs) from state trajectories. SymODEN enforces a symplectic structure on the system's dynamics, which aligns with Hamiltonian mechanics, allowing it to learn and generalize from fewer samples by incorporating physics-based inductive bias.

Overview and Contributions

SymODEN represents a significant advance in leveraging neural networks to understand physical system dynamics. It extends the Hamiltonian formalism, which transforms the state-space of generalized coordinates and momenta into a dynamical phase space, by incorporating control inputs. This integration enables learning from systems where control actions, like force or torque, influence the dynamics. Notably, the framework is capable of learning from data embedded in high-dimensional spaces or when direct observations of momentum are unavailable, further enhancing its utility in practical scenarios.

The two primary contributions are:

1. Framework for Hamiltonian Dynamics with Control: SymODEN adapts Hamiltonian mechanics to include external controls, learning dynamics that adhere to a symplectic structure. This is essential for synthesizing model-based controllers capable of executing complex tasks with performance guarantees.
2. Learning with Embedded Data: The framework effectively handles non-Euclidean spaces, addressing challenges in systems represented by angle data. This is achieved without requiring second-order derivatives, which are often inaccessible.

Relevance to Dynamical Systems and Control

SymODEN's architectural design provides interpretability and consistency with physical laws, making it especially adept at controlling systems like robotic manipulators and other mechanical setups. By basing the model on the underlying physics, it offers insight into system properties, such as inertia and energy. Moreover, the ability to incorporate external controls and still maintain a Hamiltonian structure sets the groundwork for developing energy-based control strategies like interconnection and damping assignment, which are crucial for altering the energy landscape to meet specific control objectives.

Experimental Validation

The paper illustrates the framework's effectiveness through four tasks involving a pendulum, a CartPole system, and an Acrobot. SymODEN consistently demonstrates lower prediction errors and better adheres to energy conservation principles compared to baseline models. This result is indicative of its superior generalization capacity achieved with fewer parameters and training samples. Importantly, it surpasses unstructured models in prediction accuracy, highlighting the value of imposing a structured Hamiltonian prior.

Future Directions

SymODEN opens various avenues for future research. For instance, exploring broader classes of physics-based priors, like the port-Hamiltonian system, could extend its applicability to more complex systems, including those in higher-dimensional spaces with embedded elements like 3D orientations. Additionally, integrating kinetic energy shaping could further enhance control synthesis for underactuated systems, expanding its utility in autonomous and robotic applications.

Conclusion

SymODEN exemplifies the confluence of deep learning and classical physics principles, offering a robust tool for inferring and controlling the dynamics of physical systems. By utilizing the structure of Hamiltonian dynamics and extending it with control capabilities, it cleverly bridges the gap between data-driven modeling and physical interpretability, paving the way for advanced, efficient control methodologies in dynamic environments.