Hamiltonian Graph Networks with ODE Integrators (1909.12790v1)

Abstract: We introduce an approach for imposing physically informed inductive biases in learned simulation models. We combine graph networks with a differentiable ordinary differential equation integrator as a mechanism for predicting future states, and a Hamiltonian as an internal representation. We find that our approach outperforms baselines without these biases in terms of predictive accuracy, energy accuracy, and zero-shot generalization to time-step sizes and integrator orders not experienced during training. This advances the state-of-the-art of learned simulation, and in principle is applicable beyond physical domains.

• The paper introduces a novel graph network framework that integrates Hamiltonian mechanics with ODE integrators to enhance predictive accuracy and energy conservation.
• It employs differentiable Runge-Kutta methods across architectures such as DeltaGN, OGN, and HOGN to accurately simulate multi-particle interactions.
• Experimental results demonstrate robust zero-shot generalization and improved performance on untrained time-steps, highlighting its potential for dynamic systems modeling.

An Expert Analysis of "Hamiltonian Graph Networks with ODE Integrators"

The paper "Hamiltonian Graph Networks with ODE Integrators" introduces a novel framework that incorporates physically informed inductive biases into learned simulation models. It combines graph networks with differentiable ordinary differential equation (ODE) integrators and employs a Hamiltonian representation for internal system dynamics. This approach is poised to improve predictive accuracy, energy conservation, and zero-shot generalization in simulation models.

Framework and Methodology

The work extends previous developments in interaction-based graph network models by integrating Hamiltonian mechanics and ODE structures. The graph networks represent particle systems through nodes and edges, ensuring detailed modeling of particle interactions. For dynamical system predictions, the authors leverage the Hamiltonian, $\mathcal{H}(\mathbf{q}, \mathbf{p})$, which is expressed as a function of position and momentum, maintaining the essential physical properties of the system.

Three model architectures were explored:

1. Delta Graph Network (DeltaGN) - Acts as a baseline, predicting state changes directly through node computations in a graph network.
2. ODE Graph Network (OGN) - Introduces an ODE integrator as an inductive bias by assuming dynamics follow a first-order ODE, hence learning the time derivatives $(\mathbf{\dot q}, \mathbf{\dot p})$.
3. Hamiltonian ODE Graph Network (HOGN) - Further constrains the model by employing the Hamiltonian function, differentiating it to produce system dynamics as per Hamilton's equations.

Both OGN and HOGN models use numerical integrators to solve the ODEs defining the system's trajectory, with an emphasis on differentiable Runge-Kutta (RK) integrators for accuracy.

Experimental Results

Empirical assessments were conducted on synthetic datasets representing multi-particle systems governed by Hooke's law-based interactions. The findings indicate that both OGN and HOGN models outperform the DeltaGN baseline in terms of prediction accuracy and energy consistency.

• Accuracy and Energy Conservation: The OGN and HOGN exhibit superior performance compared to DeltaGN, particularly when integrated with higher-order RK methods. The HOGN model, in particular, aligns closely with the true Hamiltonian model when trained with RK4, indicating its robustness and internal consistency.
• Generalization Across Time-Steps and Integrators: Both OGN and HOGN demonstrate remarkable generalization capabilities, extending their applicability to untrained time-step sizes and integrator orders. The HOGN maintains predictive fidelity even when applied with high-order integrators not seen during training, underlining the model's adaptability.
• Symplectic Integrators: Symplectic integration was explored, revealing that while lower order symplectic integrators can sometimes outperform higher order RK integrations, they potentially allow the model to diverge from learning a true Hamiltonian structure.

Implications and Future Directions

The marriage of graph networks with ODE integrators and Hamiltonian-based representations offers significant implications for the future of learned simulation models, particularly in domains involving complex, multi-entity systems. The approach promises essential progress in predictive modeling of physical systems — extending beyond immediate physical applications to other domains characterized by dynamic entities.

Future research directions may explore extending this framework with other physically-informed biases, such as Lagrangian mechanics or entropic constraints. Addressing challenges such as scalability to larger systems and incorporating stochastic dynamics may further enhance the flexibility and generalizability of these models.

Overall, the integration of Hamiltonian mechanics with learned simulation models via graph networks and ODE integrators marks a substantial step forward in capturing the essence of dynamic systems with enhanced accuracy and consistency.