Hamiltonian Generative Networks

By Peter Toth, Danilo Jimenez Rezende, Andrew Jaegle, Sébastien Racanière, Aleksandar Botev, Irina Higgins. 2019

255 citations recorded by the source corpus.

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.

Subfields: Artificial Intelligence, Statistical and Nonlinear Physics, Generative Models

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Citations in the corpus

Stochastic Normalizing Flows (2020, methodology): “The molecule has 66 dimensions in x , and we augment it with 66 auxiliary dimensions in a second channel v , similar to “velocities” in a Hamiltonian ﬂow framework (Toth et al., 2019), resulting in 132 dimensions total.”
Equivariant Flows: exact likelihood generative learning for symmetric densities (2020, background): “Thus, if an isotropic normal distribution is used for ρ(p, q), having only access to ρ′(q) will require a variational approximation of ρ′(p|q) (Toth et al., 2019).”
Integrating Physics-Based Modeling with Machine Learning: A Survey (2020, background): “This is taken a step further in Toth et al. [249], where they show that not only can NNs learn the Hamiltonian, but also the abstract phase space (assumed to be known in Greydanus et al. [98]) to more effectively model expressive densities in similar physical systems and also extend more generally…”
Lagrangian Neural Networks (2020, background): “This was the core motivation behind Hamiltonian Neural Networks by Greydanus et al. (2019) and Hamiltonian Generative Networks by Toth et al. (2019).”
Integrating Scientific Knowledge with Machine Learning for Engineering and Environmental Systems (2020, background): “The Hamiltonian operator in physics is the primary tool for modeling the time evolution of systems with conserved quantities, but until recently the formalism had not been integrated with NNs. Greydanus et al. [112] design a NN architecture that naturally learns and respects energy conservation and ”
Discovering Symbolic Models from Deep Learning with Inductive Biases (2020, methodology): “This model is a combination of a Hamiltonian Neural Network [45, 46] and GN.”