Hamiltonian Generative Networks (1909.13789v2)

Abstract: The Hamiltonian formalism plays a central role in classical and quantum physics. Hamiltonians are the main tool for modelling the continuous time evolution of systems with conserved quantities, and they come equipped with many useful properties, like time reversibility and smooth interpolation in time. These properties are important for many machine learning problems - from sequence prediction to reinforcement learning and density modelling - but are not typically provided out of the box by standard tools such as recurrent neural networks. In this paper, we introduce the Hamiltonian Generative Network (HGN), the first approach capable of consistently learning Hamiltonian dynamics from high-dimensional observations (such as images) without restrictive domain assumptions. Once trained, we can use HGN to sample new trajectories, perform rollouts both forward and backward in time and even speed up or slow down the learned dynamics. We demonstrate how a simple modification of the network architecture turns HGN into a powerful normalising flow model, called Neural Hamiltonian Flow (NHF), that uses Hamiltonian dynamics to model expressive densities. We hope that our work serves as a first practical demonstration of the value that the Hamiltonian formalism can bring to deep learning.

• The paper presents HGN, which learns a system's Hamiltonian dynamics directly from high-dimensional observations without restrictive assumptions.
• It integrates an inference network, a Hamiltonian network, and a decoder within a VAE framework using a leapfrog integrator to ensure energy conservation.
• Experiments on simulated systems demonstrate that HGN outperforms previous methods by enabling bidirectional rollouts and adjustable simulation speeds.

Hamiltonian Generative Networks: An Analysis

The paper "Hamiltonian Generative Networks" introduces a novel approach to learning Hamiltonian dynamics from high-dimensional observations such as images. The proposed model, Hamiltonian Generative Network (HGN), distinguishes itself by addressing two significant challenges in machine learning and physics: learning a system's Hamiltonian from data, and inferring a system's abstract phase space from high-dimensional observations.

The classical Hamiltonian formalism plays a pivotal role in modeling the continuous time evolution of physical systems. It is characterized by its smooth, reversible dynamics and conservation properties, making it suitable for capturing the dynamics of systems studied in physics and inspiring its application in machine learning. The authors extend this formalism by introducing HGN, a generative model capable of inferring abstract states from pixel data and utilizing these to learn Hamiltonian dynamics in phase space. Unlike the Hamiltonian Neural Network (HNN), HGN does not rely on restrictive domain assumptions, thus broadening its applicability to various physical systems.

Methodology

HGN consists of three core components: an inference network, a Hamiltonian network, and a decoder network. The inference network processes sequences of images and outputs a posterior over the initial state, mapping them into a low-dimensional space. The Hamiltonian network represents the Hamiltonian function, which governs the dynamics in this abstract space by inducing rollouts through time. Finally, the decoder network reconstructs images from the position components of these abstract states.

An essential aspect of HGN is its training method, which extends the variational autoencoder framework by incorporating Hamiltonian dynamics for rollout generation. The integration of the dynamics is performed using a leapfrog integrator, favored for its symplectic properties that ensure better energy conservation over time.

Results

The efficacy of HGN is demonstrated through experiments on four simulated physical systems: a pendulum, a mass-spring system, and two- and three-body systems. The model consistently outperforms previous approaches, including HNN, in learning dynamics and producing accurate image reconstructions. Notably, HGN allows for bidirectional rollouts in time and varying the speed of simulations by adjusting the integration time step.

The paper also proposes a modification to HGN's architecture to create a model known as Neural Hamiltonian Flow (NHF). NHF functions as a normalizing flow model that uses Hamiltonian dynamics to create expressive density models, capitalizing on the volume-preserving characteristics of Hamiltonian transformations. This approach offers computational advantages over traditional flow-based models by bypassing the need for computing Jacobian determinants.

Implications and Future Directions

The introduction of HGN and its extension to NHF has several implications for future research in machine learning and beyond. Practically, the model offers a robust framework for training generative models that respect physical laws, potentially enhancing applications in robotics, physics simulations, and video prediction tasks. Theoretically, it underscores the utility of Hamiltonian dynamics in representation learning, suggesting that further exploration into this formalism could yield new insights into efficient and interpretable machine learning models.

Looking forward, the authors acknowledge that this work serves as a foundational step in bridging physics-based modeling with deep learning techniques. Future research may explore more complex systems or incorporate additional physical constraints to further enhance the expressiveness and robustness of learned models. Additionally, integrating these methods with contemporary advances in neural networks could expand their application to real-world datasets and problems requiring nuanced dynamic modeling.

In conclusion, the "Hamiltonian Generative Networks" paper presents innovative contributions to the intersection of deep learning and physics, with potential to drive advances in machine learning systems through the application of structured, physically grounded models.