Symplectic ODE-Net: Learning Hamiltonian Dynamics with Control

TL;DR

SymODEN addresses learning dynamical systems with physical structure by enforcing Hamiltonian dynamics with control, enabling energy conservation and interpretable insights into system properties. It learns a parameterized Hamiltonian via neural nets for the inverse mass matrix, potential energy, and input map, integrated through differentiable ODE solvers to predict state trajectories. The paper extends to embedded angle data and hybrid spaces, introduces an energy-shaping-based control perspective, and demonstrates improved generalization with fewer training samples across pendulum, CartPole, and Acrobot tasks. These results suggest a practical path toward reliable model-based control for nonlinear physical systems using physics-informed neural networks.

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.

Figures (11)

• Figure 1: The computation graph of SymODEN. Blue arrows indicate neural network parametrization. Red arrows indicate automatic differentiation. For a given $(\mathbf{x}, \mathbf{u})$, the computation graph outputs a $\mathbf{f}_{\theta}(\mathbf{x}, \mathbf{u})$ which follows Hamiltonian dynamics with control. The function itself is an input to the Neural ODE to generate estimation of states at each time step. Since all the operations are differentiable, weights of the neural networks can be updated by backpropagation.
• Figure 2: Sample trajectories and learned functions of Task 1.
• Figure 3: Without true generalized momentum data, the learned functions match the ground truth with a scaling. Here $\beta=0.357$
• Figure 4: Time-evolution of the state variables $(\cos q, \sin q, \dot{q})$ when the closed-loop control input $u(\cos q, \sin q, \dot{q})$ is governed by Equation (\ref{['eqn:controller_u']}). The thin black lines show the expected results.
• Figure 5: Train error per trajectory and prediction error per trajectory for all 4 tasks with different number of training trajectories. Horizontal axis shows number of initial state conditions (16, 32, 64, 128, 256, 512, 1024) in the training set. Both the horizontal axis and vertical axis are in log scale.