TY - JOUR
T1 - On learning Hamiltonian systems from data
AU - Bertalan, Tom
AU - Dietrich, Felix
AU - Mezić, Igor
AU - Kevrekidis, Ioannis G.
N1 - Publisher Copyright:
© 2019 Author(s).
PY - 2019/12/1
Y1 - 2019/12/1
N2 - Concise, accurate descriptions of physical systems through their conserved quantities abound in the natural sciences. In data science, however, current research often focuses on regression problems, without routinely incorporating additional assumptions about the system that generated the data. Here, we propose to explore a particular type of underlying structure in the data: Hamiltonian systems, where an "energy" is conserved. Given a collection of observations of such a Hamiltonian system over time, we extract phase space coordinates and a Hamiltonian function of them that acts as the generator of the system dynamics. The approach employs an autoencoder neural network component to estimate the transformation from observations to the phase space of a Hamiltonian system. An additional neural network component is used to approximate the Hamiltonian function on this constructed space, and the two components are trained jointly. As an alternative approach, we also demonstrate the use of Gaussian processes for the estimation of such a Hamiltonian. After two illustrative examples, we extract an underlying phase space as well as the generating Hamiltonian from a collection of movies of a pendulum. The approach is fully data-driven and does not assume a particular form of the Hamiltonian function.
AB - Concise, accurate descriptions of physical systems through their conserved quantities abound in the natural sciences. In data science, however, current research often focuses on regression problems, without routinely incorporating additional assumptions about the system that generated the data. Here, we propose to explore a particular type of underlying structure in the data: Hamiltonian systems, where an "energy" is conserved. Given a collection of observations of such a Hamiltonian system over time, we extract phase space coordinates and a Hamiltonian function of them that acts as the generator of the system dynamics. The approach employs an autoencoder neural network component to estimate the transformation from observations to the phase space of a Hamiltonian system. An additional neural network component is used to approximate the Hamiltonian function on this constructed space, and the two components are trained jointly. As an alternative approach, we also demonstrate the use of Gaussian processes for the estimation of such a Hamiltonian. After two illustrative examples, we extract an underlying phase space as well as the generating Hamiltonian from a collection of movies of a pendulum. The approach is fully data-driven and does not assume a particular form of the Hamiltonian function.
UR - http://www.scopus.com/inward/record.url?scp=85077543837&partnerID=8YFLogxK
U2 - 10.1063/1.5128231
DO - 10.1063/1.5128231
M3 - Article
AN - SCOPUS:85077543837
SN - 1054-1500
VL - 29
JO - Chaos
JF - Chaos
IS - 12
M1 - 121107
ER -