On learning Hamiltonian systems from data

Area of Science:

• Data Science
• Physics
• Machine Learning

Background:

• Scientific descriptions often leverage conserved quantities, but data science typically lacks this.
• Current data science methods often overlook underlying physical system assumptions.

Purpose of the Study:

• To develop a data-driven approach for identifying Hamiltonian systems and their properties.
• To extract phase space coordinates and the generating Hamiltonian function from observational data.

Main Methods:

• Utilizing an autoencoder neural network to map observations to phase space coordinates.
• Employing a second neural network to approximate the Hamiltonian function, trained jointly with the autoencoder.
• Exploring Gaussian processes as an alternative for Hamiltonian estimation.

Main Results:

• Successfully extracted phase space and generating Hamiltonians from observational data.
• Demonstrated the approach with two illustrative examples and a pendulum system.
• The method is fully data-driven, requiring no prior assumptions on the Hamiltonian's form.

Conclusions:

• This data-driven framework effectively identifies and characterizes Hamiltonian systems.
• The approach offers a novel way to integrate physical principles into data analysis.
• Enables the discovery of underlying dynamics and conserved quantities in complex systems.