Geometric Contact Flows: Contactomorphisms for Dynamics and Control

Andrea Testa, Søren Hauberg, Tamim Asfour, Leonel Rozo

TL;DR: Leveraging Riemannian and Contact geometry as inductive biases to learn and control complex dynamical systems.

Abstract: Accurately modeling and predicting complex dynamical systems, particularly those involving force exchange and dissipation, is crucial for applications ranging from fluid dynamics to robotics, but presents significant challenges due to the intricate interplay of geometric constraints and energy transfer. This paper introduces Geometric Contact Flows (GFC), a novel framework leveraging Riemannian and Contact geometry as inductive biases to learn such systems. GCF constructs a latent contact Hamiltonian model encoding desirable properties like stability or energy conservation. An ensemble of contactomorphisms then adapts this model to the target dynamics while preserving these properties. This ensemble allows for uncertainty-aware geodesics that attract the system’s behavior toward the data support, enabling robust generalization and adaptation to unseen scenarios. Experiments on learning dynamics for physical systems and for controlling robots on interaction tasks demonstrate the effectiveness of our approach.

Lay Summary:

AI models can be trained to understand the evolution over time of systems like a swinging pendulum, a river flowing downhill, or a person loading a dishwasher. Learning these dynamic behaviors helps in two ways: it enables to predict natural phenomena and teaches machines (like robots) to execute physical tasks. The data from these real-world events reflects underlying physical laws, which imprint recognizable patterns. If a model misses these patterns, it struggles to learn the behavior, especially with limited data.

We address this challenge with a model that lets users incorporate partial knowledge about the dynamics, for example by indicating whether the behavior should be periodic or eventually settle down. The model uses this guidance to fit the available data in a way that remains physically consistent, thanks to built-in constraints that preserve the specified behavior during prediction. These constraints are based on contact Hamiltonian theory, a mathematical framework that describes physical systems as motion through a geometric space. In this context, adapting the suggested dynamics while respecting physical laws is known as a contactomorphism.

Our novel model, the first to use contactomorphisms for reconstructing and controlling physical dynamics, surpasses previous approaches in performance, advancing the state of the art.

Link To Code: https://sites.google.com/view/geometric-contact-flows

Primary Area: General Machine Learning->Sequential, Network, and Time Series Modeling

Keywords: Contact geometry, Riemannian geometry, contact Hamiltonian dynamics, Dynamical systems, Diffeomorphisms

Submission Number: 12913